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

Optimal. Leaf size=347 \[ \frac {\left (12 a-14 \sqrt {a} \sqrt {b}+5 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {\left (12 a+14 \sqrt {a} \sqrt {b}+5 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac {5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[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)

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Rubi [A]
time = 0.46, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1347, 1692, 1180, 211} \begin {gather*} \frac {\left (-14 \sqrt {a} \sqrt {b}+12 a+5 b\right ) \text {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 ) \text {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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\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 ) \tan ^{-1}\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 ) \tan ^{-1}\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*}

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Mathematica [A]
time = 5.37, size = 343, normalized size = 0.99 \begin {gather*} -\frac {\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \left (12 a+14 \sqrt {a} \sqrt {b}+5 b\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \left (12 a-14 \sqrt {a} \sqrt {b}+5 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {4 \left (12 a^2+11 a b-5 b^2+b (-11 a+5 b) \cos (2 (c+d x))\right ) \sin (2 (c+d x))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}+\frac {128 a (a-b) (2 a+b-b \cos (2 (c+d x))) \sin (2 (c+d x))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}}{64 a^2 (a-b)^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/64*(((Sqrt[a] - Sqrt[b])^2*(12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt
[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + ((Sqrt[a] + Sqrt[b])^2*(12*a - 14*Sqrt[a]*Sqrt[b
] + 5*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*S
qrt[b]) + (4*(12*a^2 + 11*a*b - 5*b^2 + b*(-11*a + 5*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*C
os[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + (128*a*(a - b)*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(-8*a
+ 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2)/(a^2*(a - b)^2*d)

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Maple [A]
time = 1.65, size = 422, normalized size = 1.22

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

Verification of antiderivative is not currently implemented for this CAS.

[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-tan(d*x+c)^4*b+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)
)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6215 vs. \(2 (295) = 590\).
time = 2.43, size = 6215, normalized size = 17.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256*(((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6
- 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4
)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*
b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456
*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700
*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13
*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*
b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(13824*a^6 - 24576*a^5*b + 24084*a^4*b^2 - 14455*a^3*b^3 + 22509/4*a^2*b^4
- 2625/2*a*b^5 + 625/4*b^6 - 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5
250*a*b^5 + 625*b^6)*cos(d*x + c)^2 + 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*
b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*
b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17
*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 +
a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^
4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3
- 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a
^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^
8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12
*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5
- a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5
+ 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*
a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 4379
52*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 +
45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10
*b^10 + a^9*b^11)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^
4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 +
3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 +
76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^
6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 3
5406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*
a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 1
0*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(13824*a^6 - 24576*a^5*b + 24084*a^4*b^2 - 14455*a^3*b^
3 + 22509/4*a^2*b^4 - 2625/2*a*b^5 + 625/4*b^6 - 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3
+ 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 - 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a
^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^
6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10
*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b
^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a
^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*
a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((14
7456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 -
6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*
a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*
a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8
*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2
325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1184 vs. \(2 (295) = 590\).
time = 1.41, size = 1184, normalized size = 3.41 \begin {gather*} \frac {\frac {{\left (30 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b - 72 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 14 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + 4 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{4} + 36 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} - 105 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b + 69 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 19 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{5} - 2 \, a^{4} b + a^{3} b^{2} + \sqrt {{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}^{2} - {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )}}}{a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}}}}\right )\right )} {\left | -a + b \right |}}{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}} + \frac {{\left (30 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b - 72 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 14 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + 4 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{4} - 36 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} + 105 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b - 69 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} + 19 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} + 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{5} - 2 \, a^{4} b + a^{3} b^{2} - \sqrt {{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}^{2} - {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )}}}{a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}}}}\right )\right )} {\left | -a + b \right |}}{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}} - \frac {2 \, {\left (10 \, a^{3} \tan \left (d x + c\right )^{7} + 5 \, a^{2} b \tan \left (d x + c\right )^{7} - 20 \, a b^{2} \tan \left (d x + c\right )^{7} + 5 \, b^{3} \tan \left (d x + c\right )^{7} + 30 \, a^{3} \tan \left (d x + c\right )^{5} + 12 \, a^{2} b \tan \left (d x + c\right )^{5} - 18 \, a b^{2} \tan \left (d x + c\right )^{5} + 30 \, a^{3} \tan \left (d x + c\right )^{3} + 3 \, a^{2} b \tan \left (d x + c\right )^{3} - 9 \, a b^{2} \tan \left (d x + c\right )^{3} + 10 \, a^{3} \tan \left (d x + c\right ) - 4 \, a^{2} b \tan \left (d x + c\right )\right )}}{{\left (a \tan \left (d x + c\right )^{4} - b \tan \left (d x + c\right )^{4} + 2 \, a \tan \left (d x + c\right )^{2} + a\right )}^{2} {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [B]
time = 19.90, size = 2500, normalized size = 7.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 - ...

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