\(\int \frac {\sin ^2(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [233]

Optimal result

Integrand size = 24, antiderivative size = 347 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\left (12 a-14 \sqrt {a} \sqrt {b}+5 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {\left (12 a+14 \sqrt {a} \sqrt {b}+5 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac {5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[Out]

1/64*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(12*a+5*b-14*a^(1/2)*b^(1/2))/a^(9/4)/d/(a^(1/2)-b^(1/
2))^(5/2)/b^(1/2)-1/64*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(12*a+5*b+14*a^(1/2)*b^(1/2))/a^(9/4
)/d/b^(1/2)/(a^(1/2)+b^(1/2))^(5/2)-1/8*b*tan(d*x+c)*(a*(a+3*b)+(a^2+6*a*b+b^2)*tan(d*x+c)^2)/a/(a-b)^3/d/(a+2
*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)^2-1/32*tan(d*x+c)*(2*a*(5*a^2-9*a*b-4*b^2)/(a-b)^3+5*(2*a^2+3*a*b-b^2)*tan
(d*x+c)^2/(a-b)^2)/a^2/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1347, 1692, 1180, 211} \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\left (-14 \sqrt {a} \sqrt {b}+12 a+5 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \sqrt {b} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\left (14 \sqrt {a} \sqrt {b}+12 a+5 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \sqrt {b} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\tan (c+d x) \left (\frac {5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}\right )}{32 a^2 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {b \tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 a d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

[In]

Int[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a
] - Sqrt[b])^(5/2)*Sqrt[b]*d) - ((12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x
])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b +
b^2)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((
2*a*(5*a^2 - 9*a*b - 4*b^2))/(a - b)^3 + (5*(2*a^2 + 3*a*b - b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2
*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1347

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d +
e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*
f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)
^(p + 1)*Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x
] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]

Rule 1692

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 +
c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 3296

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 \left (1+x^2\right )^4}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-\frac {2 a^2 b^2 (a+3 b)}{(a-b)^3}-\frac {2 a b \left (8 a^3-29 a^2 b+18 a b^2-5 b^3\right ) x^2}{(a-b)^3}-\frac {32 a^2 (a-2 b) b x^4}{(a-b)^2}-\frac {16 a^2 b x^6}{a-b}}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{16 a^2 b d} \\ & = -\frac {b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac {5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {\frac {8 a^3 (5 a-2 b) b^2}{(a-b)^2}+\frac {4 a^2 b^2 \left (22 a^2-15 a b+5 b^2\right ) x^2}{(a-b)^2}}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d} \\ & = -\frac {b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac {5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \left (12 a-14 \sqrt {a} \sqrt {b}+5 b\right )\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a^2 \left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {b} d}-\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \left (12 a+14 \sqrt {a} \sqrt {b}+5 b\right )\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a^2 \left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} d} \\ & = \frac {\left (12 a-14 \sqrt {a} \sqrt {b}+5 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {\left (12 a+14 \sqrt {a} \sqrt {b}+5 b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac {5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.17 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {\left (-12 a^{5/2} \sqrt {b}+10 a^2 b+11 a^{3/2} b^{3/2}-4 a b^2-5 \sqrt {a} b^{5/2}\right ) \arctan \left (\frac {\left (\sqrt {a} \sqrt {b}+b\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}{64 a^{5/2} \sqrt {a+\sqrt {a} \sqrt {b}} (a-b)^2 b d}-\frac {\left (12 a^{5/2} \sqrt {b}+10 a^2 b-11 a^{3/2} b^{3/2}-4 a b^2+5 \sqrt {a} b^{5/2}\right ) \text {arctanh}\left (\frac {\left (\sqrt {a} \sqrt {b}-b\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}{64 a^{5/2} \sqrt {-a+\sqrt {a} \sqrt {b}} (a-b)^2 b d}+\frac {-4 a \sin (2 (c+d x))-2 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{a (a-b) d (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}+\frac {24 a^2 \sin (2 (c+d x))+22 a b \sin (2 (c+d x))-10 b^2 \sin (2 (c+d x))-11 a b \sin (4 (c+d x))+5 b^2 \sin (4 (c+d x))}{32 a^2 (a-b)^2 d (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))} \]

[In]

Integrate[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((-12*a^(5/2)*Sqrt[b] + 10*a^2*b + 11*a^(3/2)*b^(3/2) - 4*a*b^2 - 5*Sqrt[a]*b^(5/2))*ArcTan[((Sqrt[a]*Sqrt[b]
+ b)*Tan[c + d*x])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(64*a^(5/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*(a - b)^2*b*d)
- ((12*a^(5/2)*Sqrt[b] + 10*a^2*b - 11*a^(3/2)*b^(3/2) - 4*a*b^2 + 5*Sqrt[a]*b^(5/2))*ArcTanh[((Sqrt[a]*Sqrt[b
] - b)*Tan[c + d*x])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(64*a^(5/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*(a - b)^2*b
*d) + (-4*a*Sin[2*(c + d*x)] - 2*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(a*(a - b)*d*(-8*a + 3*b - 4*b*Cos[2
*(c + d*x)] + b*Cos[4*(c + d*x)])^2) + (24*a^2*Sin[2*(c + d*x)] + 22*a*b*Sin[2*(c + d*x)] - 10*b^2*Sin[2*(c +
d*x)] - 11*a*b*Sin[4*(c + d*x)] + 5*b^2*Sin[4*(c + d*x)])/(32*a^2*(a - b)^2*d*(-8*a + 3*b - 4*b*Cos[2*(c + d*x
)] + b*Cos[4*(c + d*x)]))

Maple [A] (verified)

Time = 5.49 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {\frac {-\frac {5 \left (2 a^{2}+3 a b -b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 a^{2} \left (a -b \right )}-\frac {3 \left (5 a^{2}+2 a b -3 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{16 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (10 a^{2}+a b -3 b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (5 a -2 b \right ) \tan \left (d x +c \right )}{16 \left (a^{2}-2 a b +b^{2}\right )}}{{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )}^{2}}+\frac {\left (a -b \right ) \left (\frac {\left (22 a^{2} \sqrt {a b}-15 a b \sqrt {a b}+5 b^{2} \sqrt {a b}-12 a^{3}+a^{2} b -a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (22 a^{2} \sqrt {a b}-15 a b \sqrt {a b}+5 b^{2} \sqrt {a b}+12 a^{3}-a^{2} b +a \,b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 a^{2} \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(422\)
default	\(\frac {\frac {-\frac {5 \left (2 a^{2}+3 a b -b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 a^{2} \left (a -b \right )}-\frac {3 \left (5 a^{2}+2 a b -3 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{16 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (10 a^{2}+a b -3 b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (5 a -2 b \right ) \tan \left (d x +c \right )}{16 \left (a^{2}-2 a b +b^{2}\right )}}{{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )}^{2}}+\frac {\left (a -b \right ) \left (\frac {\left (22 a^{2} \sqrt {a b}-15 a b \sqrt {a b}+5 b^{2} \sqrt {a b}-12 a^{3}+a^{2} b -a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (22 a^{2} \sqrt {a b}-15 a b \sqrt {a b}+5 b^{2} \sqrt {a b}+12 a^{3}-a^{2} b +a \,b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 a^{2} \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(422\)
risch	\(\text {Expression too large to display}\)	\(2554\)

[In]

int(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((-5/32*(2*a^2+3*a*b-b^2)/a^2/(a-b)*tan(d*x+c)^7-3/16*(5*a^2+2*a*b-3*b^2)/a/(a^2-2*a*b+b^2)*tan(d*x+c)^5-3
/32*(10*a^2+a*b-3*b^2)/a/(a^2-2*a*b+b^2)*tan(d*x+c)^3-1/16*(5*a-2*b)/(a^2-2*a*b+b^2)*tan(d*x+c))/(tan(d*x+c)^4
*a-b*tan(d*x+c)^4+2*a*tan(d*x+c)^2+a)^2+1/32/a^2/(a^2-2*a*b+b^2)*(a-b)*(1/2*(22*a^2*(a*b)^(1/2)-15*a*b*(a*b)^(
1/2)+5*b^2*(a*b)^(1/2)-12*a^3+a^2*b-a*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(
d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+1/2*(22*a^2*(a*b)^(1/2)-15*a*b*(a*b)^(1/2)+5*b^2*(a*b)^(1/2)+12*a^3-a^2*
b+a*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2)
)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6215 vs. \(2 (295) = 590\).

Time = 2.37 (sec) , antiderivative size = 6215, normalized size of antiderivative = 17.91 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**2/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\int { -\frac {\sin \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \]

[In]

integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/16*(4*(96*a^3*b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((12*a^2*b^3 - 11*a*
b^4 + 5*b^5)*sin(14*d*x + 14*c) - (104*a^2*b^3 - 85*a*b^4 + 35*b^5)*sin(12*d*x + 12*c) - (320*a^3*b^2 - 652*a^
2*b^3 + 407*a*b^4 - 105*b^5)*sin(10*d*x + 10*c) + (1408*a^3*b^2 - 1696*a^2*b^3 + 865*a*b^4 - 175*b^5)*sin(8*d*
x + 8*c) + (320*a^3*b^2 + 756*a^2*b^3 - 849*a*b^4 + 175*b^5)*sin(6*d*x + 6*c) - (248*a^2*b^3 - 383*a*b^4 + 105
*b^5)*sin(4*d*x + 4*c) - (12*a^2*b^3 + 77*a*b^4 - 35*b^5)*sin(2*d*x + 2*c))*cos(16*d*x + 16*c) + 2*(2*(96*a^3*
b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*sin(12*d*x + 12*c) + 8*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4 - 35*b^5)*s
in(10*d*x + 10*c) - 3*(512*a^4*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*sin(8*d*x + 8*c) - 16*(1
28*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*sin(6*d*x + 6*c) + 2*(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4 + 17
5*b^5)*sin(4*d*x + 4*c) + 24*(4*a^2*b^3 + 11*a*b^4 - 5*b^5)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 2*(2*(2560*
a^4*b - 4128*a^3*b^2 + 3644*a^2*b^3 - 1379*a*b^4 + 245*b^5)*sin(10*d*x + 10*c) - (9216*a^4*b - 25984*a^3*b^2 +
21304*a^2*b^3 - 8575*a*b^4 + 1225*b^5)*sin(8*d*x + 8*c) - 2*(2560*a^4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*a
*b^4 - 735*b^5)*sin(6*d*x + 6*c) + 4*(576*a^3*b^2 - 1696*a^2*b^3 + 1323*a*b^4 - 245*b^5)*sin(4*d*x + 4*c) + 2*
(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4 + 175*b^5)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 2*((40960*a^5 - 24064*
a^4*b - 22080*a^3*b^2 + 27516*a^2*b^3 - 11095*a*b^4 + 1225*b^5)*sin(8*d*x + 8*c) + 8*(5120*a^4*b - 1408*a^3*b^
2 - 3900*a^2*b^3 + 2107*a*b^4 - 245*b^5)*sin(6*d*x + 6*c) - 2*(2560*a^4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*
a*b^4 - 735*b^5)*sin(4*d*x + 4*c) - 16*(128*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*sin(2*d*x + 2*c))*cos(
10*d*x + 10*c) + 2*((40960*a^5 - 24064*a^4*b - 22080*a^3*b^2 + 27516*a^2*b^3 - 11095*a*b^4 + 1225*b^5)*sin(6*d
*x + 6*c) - (9216*a^4*b - 25984*a^3*b^2 + 21304*a^2*b^3 - 8575*a*b^4 + 1225*b^5)*sin(4*d*x + 4*c) - 3*(512*a^4
*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 4*((2560*a^4*b -
4128*a^3*b^2 + 3644*a^2*b^3 - 1379*a*b^4 + 245*b^5)*sin(4*d*x + 4*c) + 4*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4
- 35*b^5)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 16*((a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(16*d*x + 16*c)^2 + 6
4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^
3*b^5 + 49*a^2*b^6)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^
2*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^8 - 57344*a^7*b + 83712*a^6*b^2 - 67648*a^5*b^3 + 32841*a^4*b^4 - 9
170*a^3*b^5 + 1225*a^2*b^6)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 +
49*a^2*b^6)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*c
os(4*d*x + 4*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c)^2 + (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*
d*sin(16*d*x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^6*b^2 - 240*a^5*
b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4
*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^8 - 57344*a^7*b + 83712*a^6*b^2 - 67648*a
^5*b^3 + 32841*a^4*b^4 - 9170*a^3*b^5 + 1225*a^2*b^6)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 7
53*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210
*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x
+ 4*c)*sin(2*d*x + 2*c) + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x + 2*c)^2 - 16*(a^4*b^4 - 2*a^3*b^5 +
a^2*b^6)*d*cos(2*d*x + 2*c) + (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d - 2*(8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(
14*d*x + 14*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^5*b^3 - 39
*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3
*b^5 + 35*a^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c
) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)
*d*cos(2*d*x + 2*c) - (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^5*b^3 - 23*a^4*b^4 +
22*a^3*b^5 - 7*a^2*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d
*x + 10*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16
*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 -
7*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c) - (a^4*b^4 - 2*a^3*b^5 +
a^2*b^6)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*c
os(10*d*x + 10*c) + 2*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d
*cos(8*d*x + 8*c) + 8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6*c)
- 4*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(8*a^5*b^3 - 23
*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*c
os(12*d*x + 12*c) + 16*(2*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^
6)*d*cos(8*d*x + 8*c) + 8*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6
*c) - 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(16*a^5*b^
3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^
6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*
a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*
a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*cos(2*d
*x + 2*c) + (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(
128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(16*a^5*b^3 - 39*a^
4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) - (16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d)*cos
(6*d*x + 6*c) + 8*(8*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) - (8*a^5*b^3 - 23*a^
4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*cos(4*d*x + 4*c) - 4*(4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(14*d*x + 14*c
) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30
*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b
^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^5*b^
3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c) + 4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x +
2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(12*d*x + 12*c) - 4*(1
6*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b
^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin
(6*d*x + 6*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c) + 4*(a^4*b^4 - 2*a^3*b^
5 + a^2*b^6)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3
*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c) + (1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*
b^5 - 245*a^2*b^6)*d*sin(8*d*x + 8*c) + 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)
*d*sin(6*d*x + 6*c) - 2*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c)
- 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^7*
b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*sin(8*d*x + 8*c) + 4*(256*a^6*b
^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c) - 2*(128*a^6*b^2 - 424*a^5*b^3 +
513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*
b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 +
1722*a^3*b^5 - 245*a^2*b^6)*d*sin(6*d*x + 6*c) - (1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1
442*a^3*b^5 - 245*a^2*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*
a^2*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 4
9*a^2*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(6
*d*x + 6*c))*integrate(-1/8*(4*(12*a^2*b - 11*a*b^2 + 5*b^3)*cos(6*d*x + 6*c)^2 - 4*(256*a^3 - 248*a^2*b + 97*
a*b^2 - 15*b^3)*cos(4*d*x + 4*c)^2 + 4*(12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c)^2 + 4*(12*a^2*b - 11*a*b
^2 + 5*b^3)*sin(6*d*x + 6*c)^2 - 4*(256*a^3 - 248*a^2*b + 97*a*b^2 - 15*b^3)*sin(4*d*x + 4*c)^2 + 2*(96*a^3 -
252*a^2*b + 149*a*b^2 - 35*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(12*a^2*b - 11*a*b^2 + 5*b^3)*sin(2*d*x
+ 2*c)^2 - ((12*a^2*b - 11*a*b^2 + 5*b^3)*cos(6*d*x + 6*c) - 2*(32*a^2*b - 19*a*b^2 + 5*b^3)*cos(4*d*x + 4*c)
+ (12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (12*a^2*b - 11*a*b^2 + 5*b^3 - 2*(96*a^3
- 252*a^2*b + 149*a*b^2 - 35*b^3)*cos(4*d*x + 4*c) - 8*(12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*cos(6*d
*x + 6*c) + 2*(32*a^2*b - 19*a*b^2 + 5*b^3 + (96*a^3 - 252*a^2*b + 149*a*b^2 - 35*b^3)*cos(2*d*x + 2*c))*cos(4
*d*x + 4*c) - (12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c) - ((12*a^2*b - 11*a*b^2 + 5*b^3)*sin(6*d*x + 6*c)
- 2*(32*a^2*b - 19*a*b^2 + 5*b^3)*sin(4*d*x + 4*c) + (12*a^2*b - 11*a*b^2 + 5*b^3)*sin(2*d*x + 2*c))*sin(8*d*
x + 8*c) + 2*((96*a^3 - 252*a^2*b + 149*a*b^2 - 35*b^3)*sin(4*d*x + 4*c) + 4*(12*a^2*b - 11*a*b^2 + 5*b^3)*sin
(2*d*x + 2*c))*sin(6*d*x + 6*c))/(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(8*d*x +
8*c)^2 + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a^6 - 176*a^5*b + 169*a^4*b^2 - 66*a^3*
b^3 + 9*a^2*b^4)*cos(4*d*x + 4*c)^2 + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(2*d*x + 2*c)^2 + (a^4*b^2 - 2*a^3
*b^3 + a^2*b^4)*sin(8*d*x + 8*c)^2 + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^6 - 176*a
^5*b + 169*a^4*b^2 - 66*a^3*b^3 + 9*a^2*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^
2*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(2*d*x + 2*c)^2 + 2*(a^4*b^2
- 2*a^3*b^3 + a^2*b^4 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(6*d*x + 6*c) - 2*(8*a^5*b - 19*a^4*b^2 + 14*a^3*
b^3 - 3*a^2*b^4)*cos(4*d*x + 4*c) - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(
a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 2*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4)*cos(4*d*x + 4*c) - 4*(a^4*b^
2 - 2*a^3*b^3 + a^2*b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4
- 4*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^4*b^2 - 2*a^3*b
^3 + a^2*b^4)*cos(2*d*x + 2*c) - 4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(6*d*x + 6*c) + (8*a^5*b - 19*a^4*b^2
+ 14*a^3*b^3 - 3*a^2*b^4)*sin(4*d*x + 4*c) + 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(2*d*x + 2*c))*sin(8*d*x +
8*c) + 16*((8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4)*sin(4*d*x + 4*c) + 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4
)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - (11*a*b^4 - 5*b^5 + (12*a^2*b^3 - 11*a*b^4 + 5*b^5)*cos(14*d*x + 1
4*c) - (104*a^2*b^3 - 85*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) - (320*a^3*b^2 - 652*a^2*b^3 + 407*a*b^4 - 105*b^5
)*cos(10*d*x + 10*c) + (1408*a^3*b^2 - 1696*a^2*b^3 + 865*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) + (320*a^3*b^2 + 7
56*a^2*b^3 - 849*a*b^4 + 175*b^5)*cos(6*d*x + 6*c) - (248*a^2*b^3 - 383*a*b^4 + 105*b^5)*cos(4*d*x + 4*c) - (1
2*a^2*b^3 + 77*a*b^4 - 35*b^5)*cos(2*d*x + 2*c))*sin(16*d*x + 16*c) + (12*a^2*b^3 + 77*a*b^4 - 35*b^5 - 4*(96*
a^3*b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) - 16*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4 - 35*b
^5)*cos(10*d*x + 10*c) + 6*(512*a^4*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) +
32*(128*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*cos(6*d*x + 6*c) - 4*(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4
+ 175*b^5)*cos(4*d*x + 4*c) - 48*(4*a^2*b^3 + 11*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*sin(14*d*x + 14*c) + (248*a
^2*b^3 - 383*a*b^4 + 105*b^5 - 4*(2560*a^4*b - 4128*a^3*b^2 + 3644*a^2*b^3 - 1379*a*b^4 + 245*b^5)*cos(10*d*x
+ 10*c) + 2*(9216*a^4*b - 25984*a^3*b^2 + 21304*a^2*b^3 - 8575*a*b^4 + 1225*b^5)*cos(8*d*x + 8*c) + 4*(2560*a^
4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*a*b^4 - 735*b^5)*cos(6*d*x + 6*c) - 8*(576*a^3*b^2 - 1696*a^2*b^3 + 13
23*a*b^4 - 245*b^5)*cos(4*d*x + 4*c) - 4*(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4 + 175*b^5)*cos(2*d*x + 2*c))*si
n(12*d*x + 12*c) - (320*a^3*b^2 + 756*a^2*b^3 - 849*a*b^4 + 175*b^5 + 2*(40960*a^5 - 24064*a^4*b - 22080*a^3*b
^2 + 27516*a^2*b^3 - 11095*a*b^4 + 1225*b^5)*cos(8*d*x + 8*c) + 16*(5120*a^4*b - 1408*a^3*b^2 - 3900*a^2*b^3 +
2107*a*b^4 - 245*b^5)*cos(6*d*x + 6*c) - 4*(2560*a^4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*a*b^4 - 735*b^5)*c
os(4*d*x + 4*c) - 32*(128*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - (
1408*a^3*b^2 - 1696*a^2*b^3 + 865*a*b^4 - 175*b^5 + 2*(40960*a^5 - 24064*a^4*b - 22080*a^3*b^2 + 27516*a^2*b^3
- 11095*a*b^4 + 1225*b^5)*cos(6*d*x + 6*c) - 2*(9216*a^4*b - 25984*a^3*b^2 + 21304*a^2*b^3 - 8575*a*b^4 + 122
5*b^5)*cos(4*d*x + 4*c) - 6*(512*a^4*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*cos(2*d*x + 2*c))*
sin(8*d*x + 8*c) + (320*a^3*b^2 - 652*a^2*b^3 + 407*a*b^4 - 105*b^5 - 4*(2560*a^4*b - 4128*a^3*b^2 + 3644*a^2*
b^3 - 1379*a*b^4 + 245*b^5)*cos(4*d*x + 4*c) - 16*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4 - 35*b^5)*cos(2*d*x +
2*c))*sin(6*d*x + 6*c) + (104*a^2*b^3 - 85*a*b^4 + 35*b^5 - 4*(96*a^3*b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*co
s(2*d*x + 2*c))*sin(4*d*x + 4*c) - (12*a^2*b^3 - 11*a*b^4 + 5*b^5)*sin(2*d*x + 2*c))/((a^4*b^4 - 2*a^3*b^5 + a
^2*b^6)*d*cos(16*d*x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^6*b^2 -
240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 +
753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^8 - 57344*a^7*b + 83712*a^6*b^2 -
67648*a^5*b^3 + 32841*a^4*b^4 - 9170*a^3*b^5 + 1225*a^2*b^6)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^6*b^2 - 736*a^5
*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b
^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c)^2
+ (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(16*d*x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(14*d*x + 1
4*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(2
56*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^8 - 573
44*a^7*b + 83712*a^6*b^2 - 67648*a^5*b^3 + 32841*a^4*b^4 - 9170*a^3*b^5 + 1225*a^2*b^6)*d*sin(8*d*x + 8*c)^2 +
64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^6*b^2
- 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^5*b^3 - 23*a^4*b^4 + 2
2*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x +
2*c)^2 - 16*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c) + (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d - 2*(8*(a^4
*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(14*d*x + 14*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(1
2*d*x + 12*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^6*b^2 - 3
52*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*
b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(4*d*x + 4*c) +
8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c) - (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d)*cos(16*d*x + 16*c)
+ 16*(4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 +
30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a
^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^
5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*
x + 2*c) - (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b
^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(10*d*x + 10*c) + 2*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b
^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*cos(8*d*x + 8*c) + 8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5
+ 49*a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*co
s(4*d*x + 4*c) - 8*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (8*a^5*b^3 - 23*a^4*
b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a
^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*cos(8*d*x + 8*c) + 8*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3
*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)
*d*cos(4*d*x + 4*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (16*a^5*b^3 -
39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 -
5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 -
3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 1
66*a^3*b^5 + 35*a^2*b^6)*d*cos(2*d*x + 2*c) + (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*
b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(4*
d*x + 4*c) + 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) - (16*a^5*b^3 - 39*a^4*b^
4 + 30*a^3*b^5 - 7*a^2*b^6)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos
(2*d*x + 2*c) - (8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*cos(4*d*x + 4*c) - 4*(4*(a^4*b^4 - 2*a^3*
b^5 + a^2*b^6)*d*sin(14*d*x + 14*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(12*d*x + 12*c)
- 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128*a^6*b^2 - 352*a^5*b^3 + 35
5*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6
)*d*sin(6*d*x + 6*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c) + 4*(a^4*b^4 - 2
*a^3*b^5 + a^2*b^6)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^
2*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128
*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b
^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*
d*x + 4*c) + 4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^6*b^2 - 4
24*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c) + (1024*a^7*b - 3712*a^6*b^2 + 5304*
a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*sin(8*d*x + 8*c) + 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a
^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c) - 2*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^
5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) - 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*s
in(12*d*x + 12*c) + 32*((2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)
*d*sin(8*d*x + 8*c) + 4*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c
) - 2*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) - 4*(16*a^5*b^3
- 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^7*b - 6528*a^6*b
^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*sin(6*d*x + 6*c) - (1024*a^7*b - 3712*a^6*b^2
+ 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^6*b^2 - 352*a^5*b^3
+ 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^6*b^2 - 424*a^5*b
^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*
a^2*b^6)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1184 vs. \(2 (295) = 590\).

Time = 1.98 (sec) , antiderivative size = 1184, normalized size of antiderivative = 3.41 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/64*((30*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 72*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt
(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 + 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 + 36*sqrt(a^2 - a*b + sq
rt(a*b)*(a - b))*sqrt(a*b)*a^4 - 105*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 69*sqrt(a^2 - a*b +
sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 5*sqrt(a^2 -
a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^5 - 2*a^4*
b + a^3*b^2 + sqrt((a^5 - 2*a^4*b + a^3*b^2)^2 - (a^5 - 2*a^4*b + a^3*b^2)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^
3)))/(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3))))*abs(-a + b)/(3*a^9*b - 18*a^8*b^2 + 41*a^7*b^3 - 44*a^6*b^4 + 21
*a^5*b^5 - 2*a^4*b^6 - a^3*b^7) + (30*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 72*sqrt(a^2 - a*b - sqrt(a*b
)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 + 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*
b^4 - 36*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 + 105*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)
*a^3*b - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqr
t(a*b)*a*b^3 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan
(d*x + c)/sqrt((a^5 - 2*a^4*b + a^3*b^2 - sqrt((a^5 - 2*a^4*b + a^3*b^2)^2 - (a^5 - 2*a^4*b + a^3*b^2)*(a^5 -
3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3))))*abs(-a + b)/(3*a^9*b - 18*a^8*b^2 +
41*a^7*b^3 - 44*a^6*b^4 + 21*a^5*b^5 - 2*a^4*b^6 - a^3*b^7) - 2*(10*a^3*tan(d*x + c)^7 + 5*a^2*b*tan(d*x + c)^
7 - 20*a*b^2*tan(d*x + c)^7 + 5*b^3*tan(d*x + c)^7 + 30*a^3*tan(d*x + c)^5 + 12*a^2*b*tan(d*x + c)^5 - 18*a*b^
2*tan(d*x + c)^5 + 30*a^3*tan(d*x + c)^3 + 3*a^2*b*tan(d*x + c)^3 - 9*a*b^2*tan(d*x + c)^3 + 10*a^3*tan(d*x +
c) - 4*a^2*b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^4 - 2*a^3*b +
a^2*b^2)))/d

Mupad [B] (verification not implemented)

Time = 19.03 (sec) , antiderivative size = 6646, normalized size of antiderivative = 19.15 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

int(sin(c + d*x)^2/(a - b*sin(c + d*x)^4)^3,x)

[Out]

- ((tan(c + d*x)*(5*a - 2*b))/(16*(a^2 - 2*a*b + b^2)) + (3*tan(c + d*x)^3*(a*b + 10*a^2 - 3*b^2))/(32*a*(a^2
- 2*a*b + b^2)) + (5*tan(c + d*x)^7*(3*a*b + 2*a^2 - b^2))/(32*a^2*(a - b)) + (3*tan(c + d*x)^5*(2*a*b + 5*a^2
- 3*b^2))/(16*a*(a - b)^2))/(d*(tan(c + d*x)^8*(a^2 - 2*a*b + b^2) + a^2 - tan(c + d*x)^4*(2*a*b - 6*a^2) - t
an(c + d*x)^6*(4*a*b - 4*a^2) + 4*a^2*tan(c + d*x)^2)) - (atan(((((163840*a^9*b + 65536*a^5*b^5 - 360448*a^6*b
^4 + 688128*a^7*b^3 - 557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) - (tan(c + d*x)*(-(384*a^4
*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 3
49*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b
^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a^5*b^6 + 81920*a^6*b^5
- 163840*a^7*b^4 + 163840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a
^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*
a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6
+ 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) - (tan(c + d*x)*(460*a^4*b - 149*a*b^4 + 144*a^5
+ 25*b^5 + 443*a^2*b^3 - 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/2)
+ 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*
b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5
- 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*1i - (((163840*a^9*b + 65536*a^5*b^5 - 360448*a^6*b^4 + 688128
*a^7*b^3 - 557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) + (tan(c + d*x)*(-(384*a^4*(a^9*b^3)^
(1/2) + 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*
(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^1
1*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a^5*b^6 + 81920*a^6*b^5 - 163840*a^
7*b^4 + 163840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/
2) + 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^
9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b
^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) + (tan(c + d*x)*(460*a^4*b - 149*a*b^4 + 144*a^5 + 25*b^5 +
443*a^2*b^3 - 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(
a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2)
- 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*
b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*1i)/((((163840*a^9*b + 65536*a^5*b^5 - 360448*a^6*b^4 + 688128*a^7*b^3 -
557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) - (tan(c + d*x)*(-(384*a^4*(a^9*b^3)^(1/2) + 25*
b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(
1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*
a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a^5*b^6 + 81920*a^6*b^5 - 163840*a^7*b^4 + 163
840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4
*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2
) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^1
2*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) - (tan(c + d*x)*(460*a^4*b - 149*a*b^4 + 144*a^5 + 25*b^5 + 443*a^2*b^3
- 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1
/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3
*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^1
3*b^3 - a^14*b^2)))^(1/2) - (3168*a^4 - 3832*a^3*b - 755*a*b^3 + 125*b^4 + 2410*a^2*b^2)/(16384*(3*a^7*b - a^8
+ a^5*b^3 - 3*a^6*b^2)) + (((163840*a^9*b + 65536*a^5*b^5 - 360448*a^6*b^4 + 688128*a^7*b^3 - 557056*a^8*b^2)
/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) + (tan(c + d*x)*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(
1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^
3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^
13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a^5*b^6 + 81920*a^6*b^5 - 163840*a^7*b^4 + 163840*a^8*b^3 - 8
1920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2
) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(
a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*
b^3 - a^14*b^2)))^(1/2) + (tan(c + d*x)*(460*a^4*b - 149*a*b^4 + 144*a^5 + 25*b^5 + 443*a^2*b^3 - 635*a^3*b^2)
)/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b
+ 15*a^5*b^5 - 94*a^6*b^4 + 155*a^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2
) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^
2)))^(1/2)))*(-(384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) - 144*a^9*b + 15*a^5*b^5 - 94*a^6*b^4 + 155*a
^7*b^3 - 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16
384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*2i)/d - (atan(((((16384
0*a^9*b + 65536*a^5*b^5 - 360448*a^6*b^4 + 688128*a^7*b^3 - 557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 -
3*a^6*b^2)) - (tan(c + d*x)*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a
^6*b^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b
^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^
10*b - 16384*a^5*b^6 + 81920*a^6*b^5 - 163840*a^7*b^4 + 163840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 +
a^3*b^3 - 3*a^4*b^2)))*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b
^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^
(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) - (tan(c + d*
x)*(460*a^4*b - 149*a*b^4 + 144*a^5 + 25*b^5 + 443*a^2*b^3 - 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a
^4*b^2)))*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b
^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*
(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*1i - (((163840*a^9*b + 6553
6*a^5*b^5 - 360448*a^6*b^4 + 688128*a^7*b^3 - 557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) +
(tan(c + d*x)*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a
^7*b^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16
384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a
^5*b^6 + 81920*a^6*b^5 - 163840*a^7*b^4 + 163840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a
^4*b^2)))*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b
^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*
(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) + (tan(c + d*x)*(460*a^4*b
- 149*a*b^4 + 144*a^5 + 25*b^5 + 443*a^2*b^3 - 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*((38
4*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^
2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a
^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*1i)/((((163840*a^9*b + 65536*a^5*b^5 - 36
0448*a^6*b^4 + 688128*a^7*b^3 - 557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) - (tan(c + d*x)*
((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^
8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 -
5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a^5*b^6 + 81920
*a^6*b^5 - 163840*a^7*b^4 + 163840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*((38
4*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^
2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a
^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) - (tan(c + d*x)*(460*a^4*b - 149*a*b^4 +
144*a^5 + 25*b^5 + 443*a^2*b^3 - 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*((384*a^4*(a^9*b^3
)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^
2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a
^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) - (3168*a^4 - 3832*a^3*b - 755*a*b^3 + 125*b^4 + 2410*a
^2*b^2)/(16384*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) + (((163840*a^9*b + 65536*a^5*b^5 - 360448*a^6*b^4 + 688
128*a^7*b^3 - 557056*a^8*b^2)/(32768*(3*a^7*b - a^8 + a^5*b^3 - 3*a^6*b^2)) + (tan(c + d*x)*((384*a^4*(a^9*b^3
)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^
2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a
^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)*(16384*a^10*b - 16384*a^5*b^6 + 81920*a^6*b^5 - 163840*
a^7*b^4 + 163840*a^8*b^3 - 81920*a^9*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*((384*a^4*(a^9*b^3)^(1
/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^2*(a
^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*
b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2) + (tan(c + d*x)*(460*a^4*b - 149*a*b^4 + 144*a^5 + 25*b^5 +
443*a^2*b^3 - 635*a^3*b^2))/(256*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(
a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2)
- 134*a*b^3*(a^9*b^3)^(1/2) - 480*a^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*
b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2)))*((384*a^4*(a^9*b^3)^(1/2) + 25*b^4*(a^9*b^3)^(1/2) + 144*a^9*b - 15*a^5
*b^5 + 94*a^6*b^4 - 155*a^7*b^3 + 76*a^8*b^2 + 349*a^2*b^2*(a^9*b^3)^(1/2) - 134*a*b^3*(a^9*b^3)^(1/2) - 480*a
^3*b*(a^9*b^3)^(1/2))/(16384*(a^9*b^7 - 5*a^10*b^6 + 10*a^11*b^5 - 10*a^12*b^4 + 5*a^13*b^3 - a^14*b^2)))^(1/2
)*2i)/d