\(\int \frac {\sin ^4(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [232]

Optimal result

Integrand size = 24, antiderivative size = 313 \[ \int \frac {\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {3 \left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {3 \left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {b \tan (c+d x) \left (3 a+b+4 (a+b) \tan ^2(c+d x)\right )}{8 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\tan (c+d x) \left (\frac {9 a^2-24 a b-b^2}{(a-b)^3}+\frac {(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[Out]

3/64*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)-b^(1/2))/a^(7/4)/d/(a^(1/2)-b^(1/2))^(5/2)/
b^(1/2)-3/64*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)+b^(1/2))/a^(7/4)/d/b^(1/2)/(a^(1/2)
+b^(1/2))^(5/2)-1/8*b*tan(d*x+c)*(3*a+b+4*(a+b)*tan(d*x+c)^2)/(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)*((9*a^2-24*a*b-b^2)/(a-b)^3+(17*a+3*b)*tan(d*x+c)^2/(a-b)^2)/a/d/(a+2*a*tan(d*x+c)^2+(a-b)
*tan(d*x+c)^4)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 313, 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} \[ \int \frac {\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {3 \left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \sqrt {b} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \sqrt {b} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\tan (c+d x) \left (\frac {9 a^2-24 a b-b^2}{(a-b)^3}+\frac {(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {b \tan (c+d x) \left (4 (a+b) \tan ^2(c+d x)+3 a+b\right )}{8 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

[In]

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

[Out]

(3*(2*Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] - Sqrt[b
])^(5/2)*Sqrt[b]*d) - (3*(2*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(
7/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(3*a + b + 4*(a + b)*Tan[c + d*x]^2))/(8*(a - b)^3
*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((9*a^2 - 24*a*b - b^2)/(a - b)^3 + ((
17*a + 3*b)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*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^4 \left (1+x^2\right )^3}{\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 (3 a+b+4 (a+b) \tan ^2(c+d x)\right )}{8 (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 (3 a+b)}{(a-b)^3}+\frac {8 a^2 (3 a-b) b^2 x^2}{(a-b)^3}-\frac {16 a^2 (a-3 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 (3 a+b+4 (a+b) \tan ^2(c+d x)\right )}{8 (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 {9 a^2-24 a b-b^2}{(a-b)^3}+\frac {(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a 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 {12 a^3 (3 a-b) b^2}{(a-b)^2}+\frac {12 a^3 (5 a-b) b^2 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 (3 a+b+4 (a+b) \tan ^2(c+d x)\right )}{8 (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 {9 a^2-24 a b-b^2}{(a-b)^3}+\frac {(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\left (3 \left (2 a-\sqrt {a} \sqrt {b}-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^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} d}+\frac {\left (3 \left (2 a+\sqrt {a} \sqrt {b}-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^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {b} d} \\ & = \frac {3 \left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {3 \left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} \sqrt {b} d}-\frac {b \tan (c+d x) \left (3 a+b+4 (a+b) \tan ^2(c+d x)\right )}{8 (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 {9 a^2-24 a b-b^2}{(a-b)^3}+\frac {(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.44 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\frac {-\frac {3 \left (2 a^{3/2}-3 a \sqrt {b}+b^{3/2}\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{a^{3/2} \sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {3 \left (2 a^{3/2}+3 a \sqrt {b}-b^{3/2}\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{a^{3/2} \sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}+\frac {8 (-7 a-2 b+(2 a+b) \cos (2 (c+d x))) \sin (2 (c+d x))}{a (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}+\frac {64 (a-b) (-6 \sin (2 (c+d x))+\sin (4 (c+d x)))}{(-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}}{64 (a-b)^2 d} \]

[In]

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

[Out]

((-3*(2*a^(3/2) - 3*a*Sqrt[b] + b^(3/2))*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])
/(a^(3/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (3*(2*a^(3/2) + 3*a*Sqrt[b] - b^(3/2))*ArcTanh[((Sqrt[a] - Sqrt
[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a^(3/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) + (8*(-7*a - 2*b
+ (2*a + b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) +
(64*(a - b)*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])
^2)/(64*(a - b)^2*d)

Maple [A] (verified)

Time = 4.44 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (17 a +3 b \right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 a \left (a -b \right )}-\frac {\left (43 a^{2}-18 a b -b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (35 a -11 b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a -b \right ) \tan \left (d x +c \right )}{32 \left (a^{2}-2 a b +b^{2}\right )}}{{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )}^{2}}+\frac {3 \left (a -b \right ) \left (\frac {\left (5 a \sqrt {a b}-\sqrt {a b}\, b -2 a^{2}-3 a b +b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (5 a \sqrt {a b}-\sqrt {a b}\, b +2 a^{2}+3 a b -b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(372\)
default	\(\frac {\frac {-\frac {\left (17 a +3 b \right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 a \left (a -b \right )}-\frac {\left (43 a^{2}-18 a b -b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (35 a -11 b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (3 a -b \right ) \tan \left (d x +c \right )}{32 \left (a^{2}-2 a b +b^{2}\right )}}{{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )}^{2}}+\frac {3 \left (a -b \right ) \left (\frac {\left (5 a \sqrt {a b}-\sqrt {a b}\, b -2 a^{2}-3 a b +b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (5 a \sqrt {a b}-\sqrt {a b}\, b +2 a^{2}+3 a b -b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(372\)
risch	\(\text {Expression too large to display}\)	\(1716\)

[In]

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

[Out]

1/d*((-1/32*(17*a+3*b)/a/(a-b)*tan(d*x+c)^7-1/32*(43*a^2-18*a*b-b^2)/a/(a^2-2*a*b+b^2)*tan(d*x+c)^5-1/32*(35*a
-11*b)/(a^2-2*a*b+b^2)*tan(d*x+c)^3-3/32*(3*a-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+3/32/a/(a^2-2*a*b+b^2)*(a-b)*(1/2*(5*a*(a*b)^(1/2)-(a*b)^(1/2)*b-2*a^2-3*a*b+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*(5*a*(a*b
)^(1/2)-(a*b)^(1/2)*b+2*a^2+3*a*b-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. 5510 vs. \(2 (261) = 522\).

Time = 1.44 (sec) , antiderivative size = 5510, normalized size of antiderivative = 17.60 \[ \int \frac {\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/8*(3*a*b^3*sin(2*d*x + 2*c) - 12*(8*a^2*b^2 + 13*a*b^3 - 2*b^4)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - (3*a*b^
3*sin(14*d*x + 14*c) - 3*(10*a*b^3 - b^4)*sin(12*d*x + 12*c) - (80*a^2*b^2 - 111*a*b^3 + 16*b^4)*sin(10*d*x +
10*c) + (256*a^3*b - 64*a^2*b^2 - 26*a*b^3 + 35*b^4)*sin(8*d*x + 8*c) + (336*a^2*b^2 - 95*a*b^3 - 40*b^4)*sin(
6*d*x + 6*c) - (64*a^2*b^2 - 54*a*b^3 - 25*b^4)*sin(4*d*x + 4*c) - (19*a*b^3 + 8*b^4)*sin(2*d*x + 2*c))*cos(16
*d*x + 16*c) - 2*(6*(8*a^2*b^2 + 13*a*b^3 - 2*b^4)*sin(12*d*x + 12*c) + 8*(16*a^2*b^2 - 45*a*b^3 + 8*b^4)*sin(
10*d*x + 10*c) - (1408*a^3*b - 544*a^2*b^2 + a*b^3 + 140*b^4)*sin(8*d*x + 8*c) - 16*(96*a^2*b^2 - 29*a*b^3 - 1
0*b^4)*sin(6*d*x + 6*c) + 2*(152*a^2*b^2 - 129*a*b^3 - 50*b^4)*sin(4*d*x + 4*c) + 8*(11*a*b^3 + 4*b^4)*sin(2*d
*x + 2*c))*cos(14*d*x + 14*c) - 2*(2*(640*a^3*b - 488*a^2*b^2 + 389*a*b^3 - 70*b^4)*sin(10*d*x + 10*c) - (4096
*a^4 - 8448*a^3*b + 3744*a^2*b^2 - 414*a*b^3 - 385*b^4)*sin(8*d*x + 8*c) - 2*(2688*a^3*b - 4072*a^2*b^2 + 861*
a*b^3 + 238*b^4)*sin(6*d*x + 6*c) + 4*(256*a^3*b - 560*a^2*b^2 + 206*a*b^3 + 77*b^4)*sin(4*d*x + 4*c) + 2*(152
*a^2*b^2 - 129*a*b^3 - 50*b^4)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) - 2*((26624*a^4 - 33152*a^3*b + 15632*a^2*
b^2 - 2453*a*b^3 - 420*b^4)*sin(8*d*x + 8*c) + 8*(3328*a^3*b - 3104*a^2*b^2 + 529*a*b^3 + 84*b^4)*sin(6*d*x +
6*c) - 2*(2688*a^3*b - 4072*a^2*b^2 + 861*a*b^3 + 238*b^4)*sin(4*d*x + 4*c) - 16*(96*a^2*b^2 - 29*a*b^3 - 10*b
^4)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) - 2*((26624*a^4 - 33152*a^3*b + 15632*a^2*b^2 - 2453*a*b^3 - 420*b^4)
*sin(6*d*x + 6*c) - (4096*a^4 - 8448*a^3*b + 3744*a^2*b^2 - 414*a*b^3 - 385*b^4)*sin(4*d*x + 4*c) - (1408*a^3*
b - 544*a^2*b^2 + a*b^3 + 140*b^4)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) - 4*((640*a^3*b - 488*a^2*b^2 + 389*a*b^
3 - 70*b^4)*sin(4*d*x + 4*c) + 4*(16*a^2*b^2 - 45*a*b^3 + 8*b^4)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) - 8*((a^3*
b^4 - 2*a^2*b^5 + a*b^6)*d*cos(16*d*x + 16*c)^2 + 64*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(14*d*x + 14*c)^2 + 16
*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^5*b^2 -
736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^7 - 57344*a^6*b + 8371
2*a^5*b^2 - 67648*a^4*b^3 + 32841*a^3*b^4 - 9170*a^2*b^5 + 1225*a*b^6)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^5*b^2
- 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^5*b^2 - 240*a^4*b^3 + 33
7*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c)^2 + 64*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*c)
^2 + (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(16*d*x + 16*c)^2 + 64*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(14*d*x + 14
*c)^2 + 16*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(256*
a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^7 - 57344*a^
6*b + 83712*a^5*b^2 - 67648*a^4*b^3 + 32841*a^3*b^4 - 9170*a^2*b^5 + 1225*a*b^6)*d*sin(8*d*x + 8*c)^2 + 64*(25
6*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^5*b^2 - 240*a^
4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 -
7*a*b^6)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(2*d*x + 2*c)^2 - 16*(a^
3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*c) + (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d - 2*(8*(a^3*b^4 - 2*a^2*b^5 +
a*b^6)*d*cos(14*d*x + 14*c) + 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a
^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4
- 166*a^2*b^5 + 35*a*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(6*d*x
+ 6*c) + 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(4*d*x + 4*c) + 8*(a^3*b^4 - 2*a^2*b^5 + a*b^6
)*d*cos(2*d*x + 2*c) - (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^4*b^3 - 23*a^3*b^4 + 2
2*a^2*b^5 - 7*a*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(10*d*x +
10*c) - 2*(128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^
3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*
d*cos(4*d*x + 4*c) + 8*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*c) - (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d)*cos
(14*d*x + 14*c) - 8*(8*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*cos(10*d*x + 10*c)
+ 2*(1024*a^6*b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^3*b^4 + 1442*a^2*b^5 - 245*a*b^6)*d*cos(8*d*x + 8*c) +
8*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*cos(6*d*x + 6*c) - 4*(64*a^5*b^2 - 240
*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c) - 8*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 -
7*a*b^6)*d*cos(2*d*x + 2*c) + (8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d)*cos(12*d*x + 12*c) + 16*(2*(
2048*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1722*a^2*b^5 - 245*a*b^6)*d*cos(8*d*x + 8*c) + 8*(25
6*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*cos(6*d*x + 6*c) - 4*(128*a^5*b^2 - 424*a^4*
b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c) - 8*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a
*b^6)*d*cos(2*d*x + 2*c) + (16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048
*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1722*a^2*b^5 - 245*a*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a
^6*b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^3*b^4 + 1442*a^2*b^5 - 245*a*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^5*
b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*cos(2*d*x + 2*c) + (128*a^5*b^2 - 352*a^4*b^3 + 35
5*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266
*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c) + 8*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(2*d*x + 2*c
) - (16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^4*b^3 - 23*a^3*b^4 + 22*a
^2*b^5 - 7*a*b^6)*d*cos(2*d*x + 2*c) - (8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d)*cos(4*d*x + 4*c) - 4
*(4*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(14*d*x + 14*c) + 2*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*s
in(12*d*x + 12*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(10*d*x + 10*c) - (128*a^5*b^2 - 3
52*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^
5 - 7*a*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(4*d*x + 4*c) + 4*(a^
3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5
- 7*a*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(10*d*x + 10*c) - (
128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^3 - 39*a^3*
b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(4*d*x
+ 4*c) + 4*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^5*b^2 - 424*a^
4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*sin(10*d*x + 10*c) + (1024*a^6*b - 3712*a^5*b^2 + 5304*a^4*b^3
- 3813*a^3*b^4 + 1442*a^2*b^5 - 245*a*b^6)*d*sin(8*d*x + 8*c) + 4*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 -
266*a^2*b^5 + 49*a*b^6)*d*sin(6*d*x + 6*c) - 2*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^
6)*d*sin(4*d*x + 4*c) - 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*
c) + 32*((2048*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1722*a^2*b^5 - 245*a*b^6)*d*sin(8*d*x + 8*
c) + 4*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*sin(6*d*x + 6*c) - 2*(128*a^5*b^2
- 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*sin(4*d*x + 4*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2
*b^5 - 7*a*b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 514
1*a^3*b^4 + 1722*a^2*b^5 - 245*a*b^6)*d*sin(6*d*x + 6*c) - (1024*a^6*b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^
3*b^4 + 1442*a^2*b^5 - 245*a*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^
5 + 35*a*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^
5 + 49*a*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(2*d*x + 2*c))*sin(
6*d*x + 6*c))*integrate(-3/4*(4*a*b*cos(6*d*x + 6*c)^2 + 4*a*b*cos(2*d*x + 2*c)^2 + 4*a*b*sin(6*d*x + 6*c)^2 +
4*a*b*sin(2*d*x + 2*c)^2 - 4*(32*a^2 - 20*a*b + 3*b^2)*cos(4*d*x + 4*c)^2 - a*b*cos(2*d*x + 2*c) - 4*(32*a^2
- 20*a*b + 3*b^2)*sin(4*d*x + 4*c)^2 + 2*(8*a^2 - 19*a*b + 4*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (a*b*cos
(6*d*x + 6*c) + a*b*cos(2*d*x + 2*c) - 2*(4*a*b - b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + (8*a*b*cos(2*d*x +
2*c) - a*b + 2*(8*a^2 - 19*a*b + 4*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) + 2*(4*a*b - b^2 + (8*a^2 - 19*a*b
+ 4*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b*sin(6*d*x + 6*c) + a*b*sin(2*d*x + 2*c) - 2*(4*a*b - b^2)*
sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 2*(4*a*b*sin(2*d*x + 2*c) + (8*a^2 - 19*a*b + 4*b^2)*sin(4*d*x + 4*c))*si
n(6*d*x + 6*c))/(a^3*b^2 - 2*a^2*b^3 + a*b^4 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*cos(8*d*x + 8*c)^2 + 16*(a^3*b^2
- 2*a^2*b^3 + a*b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 176*a^4*b + 169*a^3*b^2 - 66*a^2*b^3 + 9*a*b^4)*cos(4*d*
x + 4*c)^2 + 16*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*cos(2*d*x + 2*c)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*sin(8*d*x + 8
*c)^2 + 16*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 176*a^4*b + 169*a^3*b^2 - 66*a^2*b^3
+ 9*a*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a^4*b - 19*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4)*sin(4*d*x + 4*c)*sin(2*d*x +
2*c) + 16*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*sin(2*d*x + 2*c)^2 + 2*(a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*(a^3*b^2 - 2*
a^2*b^3 + a*b^4)*cos(6*d*x + 6*c) - 2*(8*a^4*b - 19*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4)*cos(4*d*x + 4*c) - 4*(a^3*
b^2 - 2*a^2*b^3 + a*b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^3*b^2 - 2*a^2*b^3 + a*b^4 - 2*(8*a^4*b - 19
*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4)*cos(4*d*x + 4*c) - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*cos(2*d*x + 2*c))*cos(6*d*
x + 6*c) - 4*(8*a^4*b - 19*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4 - 4*(8*a^4*b - 19*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4)*co
s(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*cos(2*d*x + 2*c) - 4*(2*(a^3*b^2 - 2*a^2*b^
3 + a*b^4)*sin(6*d*x + 6*c) + (8*a^4*b - 19*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4)*sin(4*d*x + 4*c) + 2*(a^3*b^2 - 2*
a^2*b^3 + a*b^4)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^4*b - 19*a^3*b^2 + 14*a^2*b^3 - 3*a*b^4)*sin(4*
d*x + 4*c) + 2*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) + (3*a*b^3*cos(14*d*x + 1
4*c) + 2*a*b^3 + b^4 - 3*(10*a*b^3 - b^4)*cos(12*d*x + 12*c) - (80*a^2*b^2 - 111*a*b^3 + 16*b^4)*cos(10*d*x +
10*c) + (256*a^3*b - 64*a^2*b^2 - 26*a*b^3 + 35*b^4)*cos(8*d*x + 8*c) + (336*a^2*b^2 - 95*a*b^3 - 40*b^4)*cos(
6*d*x + 6*c) - (64*a^2*b^2 - 54*a*b^3 - 25*b^4)*cos(4*d*x + 4*c) - (19*a*b^3 + 8*b^4)*cos(2*d*x + 2*c))*sin(16
*d*x + 16*c) - (19*a*b^3 + 8*b^4 - 12*(8*a^2*b^2 + 13*a*b^3 - 2*b^4)*cos(12*d*x + 12*c) - 16*(16*a^2*b^2 - 45*
a*b^3 + 8*b^4)*cos(10*d*x + 10*c) + 2*(1408*a^3*b - 544*a^2*b^2 + a*b^3 + 140*b^4)*cos(8*d*x + 8*c) + 32*(96*a
^2*b^2 - 29*a*b^3 - 10*b^4)*cos(6*d*x + 6*c) - 4*(152*a^2*b^2 - 129*a*b^3 - 50*b^4)*cos(4*d*x + 4*c) - 16*(11*
a*b^3 + 4*b^4)*cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - (64*a^2*b^2 - 54*a*b^3 - 25*b^4 - 4*(640*a^3*b - 488*a^2
*b^2 + 389*a*b^3 - 70*b^4)*cos(10*d*x + 10*c) + 2*(4096*a^4 - 8448*a^3*b + 3744*a^2*b^2 - 414*a*b^3 - 385*b^4)
*cos(8*d*x + 8*c) + 4*(2688*a^3*b - 4072*a^2*b^2 + 861*a*b^3 + 238*b^4)*cos(6*d*x + 6*c) - 8*(256*a^3*b - 560*
a^2*b^2 + 206*a*b^3 + 77*b^4)*cos(4*d*x + 4*c) - 4*(152*a^2*b^2 - 129*a*b^3 - 50*b^4)*cos(2*d*x + 2*c))*sin(12
*d*x + 12*c) + (336*a^2*b^2 - 95*a*b^3 - 40*b^4 + 2*(26624*a^4 - 33152*a^3*b + 15632*a^2*b^2 - 2453*a*b^3 - 42
0*b^4)*cos(8*d*x + 8*c) + 16*(3328*a^3*b - 3104*a^2*b^2 + 529*a*b^3 + 84*b^4)*cos(6*d*x + 6*c) - 4*(2688*a^3*b
- 4072*a^2*b^2 + 861*a*b^3 + 238*b^4)*cos(4*d*x + 4*c) - 32*(96*a^2*b^2 - 29*a*b^3 - 10*b^4)*cos(2*d*x + 2*c)
)*sin(10*d*x + 10*c) + (256*a^3*b - 64*a^2*b^2 - 26*a*b^3 + 35*b^4 + 2*(26624*a^4 - 33152*a^3*b + 15632*a^2*b^
2 - 2453*a*b^3 - 420*b^4)*cos(6*d*x + 6*c) - 2*(4096*a^4 - 8448*a^3*b + 3744*a^2*b^2 - 414*a*b^3 - 385*b^4)*co
s(4*d*x + 4*c) - 2*(1408*a^3*b - 544*a^2*b^2 + a*b^3 + 140*b^4)*cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - (80*a^2*b
^2 - 111*a*b^3 + 16*b^4 - 4*(640*a^3*b - 488*a^2*b^2 + 389*a*b^3 - 70*b^4)*cos(4*d*x + 4*c) - 16*(16*a^2*b^2 -
45*a*b^3 + 8*b^4)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) - 3*(10*a*b^3 - b^4 - 4*(8*a^2*b^2 + 13*a*b^3 - 2*b^4)*c
os(2*d*x + 2*c))*sin(4*d*x + 4*c))/((a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(16*d*x + 16*c)^2 + 64*(a^3*b^4 - 2*a^2
*b^5 + a*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*
cos(12*d*x + 12*c)^2 + 64*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*cos(10*d*x + 10
*c)^2 + 4*(16384*a^7 - 57344*a^6*b + 83712*a^5*b^2 - 67648*a^4*b^3 + 32841*a^3*b^4 - 9170*a^2*b^5 + 1225*a*b^6
)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*cos(6*d*x + 6
*c)^2 + 16*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c)^2 + 64*(a^3*b^
4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*c)^2 + (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(16*d*x + 16*c)^2 + 64*(a^3*b
^4 - 2*a^2*b^5 + a*b^6)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49
*a*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*sin(1
0*d*x + 10*c)^2 + 4*(16384*a^7 - 57344*a^6*b + 83712*a^5*b^2 - 67648*a^4*b^3 + 32841*a^3*b^4 - 9170*a^2*b^5 +
1225*a*b^6)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*sin
(6*d*x + 6*c)^2 + 16*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*sin(4*d*x + 4*c)^2 +
64*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^3*b^4 - 2*a^2*b
^5 + a*b^6)*d*sin(2*d*x + 2*c)^2 - 16*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*c) + (a^3*b^4 - 2*a^2*b^5
+ a*b^6)*d - 2*(8*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(14*d*x + 14*c) + 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5
- 7*a*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(10*d*x + 10*c) - 2*
(128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^3 - 39*a^3
*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(4*d*
x + 4*c) + 8*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*c) - (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d)*cos(16*d*x +
16*c) + 16*(4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^3 - 39*a^3*b^
4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35
*a*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^4*
b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(4*d*x + 4*c) + 8*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*cos(2*d*x + 2*
c) - (a^3*b^4 - 2*a^2*b^5 + a*b^6)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266
*a^2*b^5 + 49*a*b^6)*d*cos(10*d*x + 10*c) + 2*(1024*a^6*b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^3*b^4 + 1442*
a^2*b^5 - 245*a*b^6)*d*cos(8*d*x + 8*c) + 8*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)
*d*cos(6*d*x + 6*c) - 4*(64*a^5*b^2 - 240*a^4*b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c) -
8*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(2*d*x + 2*c) + (8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 -
7*a*b^6)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1722*a^2*b^
5 - 245*a*b^6)*d*cos(8*d*x + 8*c) + 8*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*b^6)*d*cos
(6*d*x + 6*c) - 4*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c) - 8*(1
6*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*cos(2*d*x + 2*c) + (16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*
a*b^6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1722*a^2*b^5 -
245*a*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a^6*b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^3*b^4 + 1442*a^2*b^5 - 24
5*a*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*cos(2*d*x
+ 2*c) + (128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(128*
a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*cos(4*d*x + 4*c) + 8*(16*a^4*b^3 - 39*a^3*b^4
+ 30*a^2*b^5 - 7*a*b^6)*d*cos(2*d*x + 2*c) - (16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d)*cos(6*d*x + 6
*c) + 8*(8*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*cos(2*d*x + 2*c) - (8*a^4*b^3 - 23*a^3*b^4 + 22*a
^2*b^5 - 7*a*b^6)*d)*cos(4*d*x + 4*c) - 4*(4*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(14*d*x + 14*c) + 2*(8*a^4*b^3
- 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6
)*d*sin(10*d*x + 10*c) - (128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*sin(8*d*x + 8*c)
- 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*
b^5 - 7*a*b^6)*d*sin(4*d*x + 4*c) + 4*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 3
2*(2*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^
2*b^5 - 7*a*b^6)*d*sin(10*d*x + 10*c) - (128*a^5*b^2 - 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*s
in(8*d*x + 8*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^3 - 23*a^
3*b^4 + 22*a^2*b^5 - 7*a*b^6)*d*sin(4*d*x + 4*c) + 4*(a^3*b^4 - 2*a^2*b^5 + a*b^6)*d*sin(2*d*x + 2*c))*sin(14*
d*x + 14*c) - 16*(4*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*sin(10*d*x + 10*c) +
(1024*a^6*b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^3*b^4 + 1442*a^2*b^5 - 245*a*b^6)*d*sin(8*d*x + 8*c) + 4*(1
28*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*sin(6*d*x + 6*c) - 2*(64*a^5*b^2 - 240*a^4*
b^3 + 337*a^3*b^4 - 210*a^2*b^5 + 49*a*b^6)*d*sin(4*d*x + 4*c) - 4*(8*a^4*b^3 - 23*a^3*b^4 + 22*a^2*b^5 - 7*a*
b^6)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1
722*a^2*b^5 - 245*a*b^6)*d*sin(8*d*x + 8*c) + 4*(256*a^5*b^2 - 736*a^4*b^3 + 753*a^3*b^4 - 322*a^2*b^5 + 49*a*
b^6)*d*sin(6*d*x + 6*c) - 2*(128*a^5*b^2 - 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*sin(4*d*x + 4
*c) - 4*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^2*b^5 - 7*a*b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*
a^6*b - 6528*a^5*b^2 + 8144*a^4*b^3 - 5141*a^3*b^4 + 1722*a^2*b^5 - 245*a*b^6)*d*sin(6*d*x + 6*c) - (1024*a^6*
b - 3712*a^5*b^2 + 5304*a^4*b^3 - 3813*a^3*b^4 + 1442*a^2*b^5 - 245*a*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^5*b^2
- 352*a^4*b^3 + 355*a^3*b^4 - 166*a^2*b^5 + 35*a*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^5*b^2
- 424*a^4*b^3 + 513*a^3*b^4 - 266*a^2*b^5 + 49*a*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^4*b^3 - 39*a^3*b^4 + 30*a^
2*b^5 - 7*a*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. 1986 vs. \(2 (261) = 522\).

Time = 1.41 (sec) , antiderivative size = 1986, normalized size of antiderivative = 6.35 \[ \int \frac {\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/64*(3*((15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 33*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*a^2*b^2 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*b^4)*(a^3 - 2*a^2*b + a*b^2)^2*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b - 48*sqrt(a^2
- a*b + sqrt(a*b)*(a - b))*a^6*b^2 + 93*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^3 - 80*sqrt(a^2 - a*b + sqr
t(a*b)*(a - b))*a^4*b^4 + 27*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^5 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))
*a*b^7)*abs(a^3 - 2*a^2*b + a*b^2)*abs(-a + b) - (6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10 - 27*sq
rt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b + 25*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 +
53*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 131*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*
a^6*b^4 + 103*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 29*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*s
qrt(a*b)*a^4*b^6 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))
*sqrt(a*b)*a^2*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 2*a^3*b + a^2
*b^2 + sqrt((a^4 - 2*a^3*b + a^2*b^2)^2 - (a^4 - 2*a^3*b + a^2*b^2)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)))/(a^4
- 3*a^3*b + 3*a^2*b^2 - a*b^3))))/((3*a^12*b - 27*a^11*b^2 + 104*a^10*b^3 - 224*a^9*b^4 + 294*a^8*b^5 - 238*a
^7*b^6 + 112*a^6*b^7 - 24*a^5*b^8 - a^4*b^9 + a^3*b^10)*abs(a^3 - 2*a^2*b + a*b^2)) - 3*((15*sqrt(a^2 - a*b -
sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 33*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + sqrt(a^2 - a*b
- sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 2*a^2*b + a*
b^2)^2*abs(-a + b) + (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b - 48*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6
*b^2 + 93*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^3 - 80*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^4 + 27*sq
rt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^5 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^7)*abs(a^3 - 2*a^2*b + a*b
^2)*abs(-a + b) - (6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10 - 27*sqrt(a^2 - a*b - sqrt(a*b)*(a - b
))*sqrt(a*b)*a^9*b + 25*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 53*sqrt(a^2 - a*b - sqrt(a*b)*
(a - b))*sqrt(a*b)*a^7*b^3 - 131*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 103*sqrt(a^2 - a*b -
sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 29*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^6 - sqrt(a^2 - a
*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^8)*abs(-a + b)
)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 2*a^3*b + a^2*b^2 - sqrt((a^4 - 2*a^3*b + a^
2*b^2)^2 - (a^4 - 2*a^3*b + a^2*b^2)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)))/(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)
)))/((3*a^12*b - 27*a^11*b^2 + 104*a^10*b^3 - 224*a^9*b^4 + 294*a^8*b^5 - 238*a^7*b^6 + 112*a^6*b^7 - 24*a^5*b
^8 - a^4*b^9 + a^3*b^10)*abs(a^3 - 2*a^2*b + a*b^2)) + 2*(17*a^2*tan(d*x + c)^7 - 14*a*b*tan(d*x + c)^7 - 3*b^
2*tan(d*x + c)^7 + 43*a^2*tan(d*x + c)^5 - 18*a*b*tan(d*x + c)^5 - b^2*tan(d*x + c)^5 + 35*a^2*tan(d*x + c)^3
- 11*a*b*tan(d*x + c)^3 + 9*a^2*tan(d*x + c) - 3*a*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^3 - 2*a^2*b + a*b^2)))/d

Mupad [B] (verification not implemented)

Time = 18.73 (sec) , antiderivative size = 5892, normalized size of antiderivative = 18.82 \[ \int \frac {\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- (atan(((((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*b^3 - 163840*a^6*b^2))/(32768*(3*a^5*b
- a^6 + a^3*b^3 - 3*a^4*b^2)) - (tan(c + d*x)*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a
^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8
*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*a^9*b - 16384*a^4*b^6 + 81920*a^5*b^5
- 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*((9*(16*a^3*(
a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) +
5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(
1/2) - (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a
^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*
a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a
^11*b^3 - a^12*b^2)))^(1/2)*1i - (((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*b^3 - 163840*a
^6*b^2))/(32768*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (tan(c + d*x)*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*
b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))
/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*a^9*b - 16384*
a^4*b^6 + 81920*a^5*b^5 - 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*
a^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6
*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*
a^11*b^3 - a^12*b^2)))^(1/2) + (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2))/(256*(3*a^4
*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a
^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9
*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*1i)/((((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 1
96608*a^5*b^3 - 163840*a^6*b^2))/(32768*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) - (tan(c + d*x)*((9*(16*a^3*(a^
7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5
*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/
2)*(16384*a^9*b - 16384*a^4*b^6 + 81920*a^5*b^5 - 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^
4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*
a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^
9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) - (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 -
45*a^2*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) +
4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*
b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) - (3*(180*a^2 - 81*a*b + 9*b^2))/(
16384*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*
b^3 - 163840*a^6*b^2))/(32768*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (tan(c + d*x)*((9*(16*a^3*(a^7*b^3)^(1/
2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^
7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*
a^9*b - 16384*a^4*b^6 + 81920*a^5*b^5 - 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5
+ a^2*b^3 - 3*a^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 +
21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10
*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) + (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2
))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) + 4*a^7*b +
a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^
8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)))*((9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3
)^(1/2) + 4*a^7*b + a^4*b^4 - 10*a^5*b^3 + 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(1
6384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*2i)/d - (atan(((((3*(491
52*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*b^3 - 163840*a^6*b^2))/(32768*(3*a^5*b - a^6 + a^3*b^3 -
3*a^4*b^2)) - (tan(c + d*x)*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b
^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5
- 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*a^9*b - 16384*a^4*b^6 + 81920*a^5*b^5 - 163840*a^6*b^4
+ 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) +
b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^
3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) - (tan(c +
d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*(-(9*(
16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)
^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*
b^2)))^(1/2)*1i - (((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*b^3 - 163840*a^6*b^2))/(32768
*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (tan(c + d*x)*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4
*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b
^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*a^9*b - 16384*a^4*b^6 + 8192
0*a^5*b^5 - 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*(-(
9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b
^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^
12*b^2)))^(1/2) + (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2))/(256*(3*a^4*b - a^5 + a^
2*b^3 - 3*a^3*b^2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*
a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^
10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*1i)/((((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*b^
3 - 163840*a^6*b^2))/(32768*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) - (tan(c + d*x)*(-(9*(16*a^3*(a^7*b^3)^(1/2
) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7
*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*a
^9*b - 16384*a^4*b^6 + 81920*a^5*b^5 - 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5 +
a^2*b^3 - 3*a^3*b^2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 -
21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10
*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) - (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2
))/(256*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b -
a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a
^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) - (3*(180*a^2 - 81*a*b + 9*b^2))/(16384*(3*
a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (((3*(49152*a^7*b + 16384*a^3*b^5 - 98304*a^4*b^4 + 196608*a^5*b^3 - 163
840*a^6*b^2))/(32768*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (tan(c + d*x)*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3
*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(
1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*(16384*a^9*b -
16384*a^4*b^6 + 81920*a^5*b^5 - 163840*a^6*b^4 + 163840*a^7*b^3 - 81920*a^8*b^2))/(256*(3*a^4*b - a^5 + a^2*b^
3 - 3*a^3*b^2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*
b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b
^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2) + (tan(c + d*x)*(333*a^3*b - 45*a*b^3 + 36*a^4 + 9*b^4 - 45*a^2*b^2))/(256
*(3*a^4*b - a^5 + a^2*b^3 - 3*a^3*b^2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/2) - 4*a^7*b - a^4*b^
4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*(a^7*b^7 - 5*a^8*b^6
+ 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)))*(-(9*(16*a^3*(a^7*b^3)^(1/2) + b^3*(a^7*b^3)^(1/
2) - 4*a^7*b - a^4*b^4 + 10*a^5*b^3 - 21*a^6*b^2 - 6*a*b^2*(a^7*b^3)^(1/2) + 5*a^2*b*(a^7*b^3)^(1/2)))/(16384*
(a^7*b^7 - 5*a^8*b^6 + 10*a^9*b^5 - 10*a^10*b^4 + 5*a^11*b^3 - a^12*b^2)))^(1/2)*2i)/d - ((3*tan(c + d*x)*(3*a
- b))/(32*(a^2 - 2*a*b + b^2)) + (tan(c + d*x)^3*(35*a - 11*b))/(32*(a^2 - 2*a*b + b^2)) - (tan(c + d*x)^5*(1
8*a*b - 43*a^2 + b^2))/(32*a*(a - b)^2) + (tan(c + d*x)^7*(17*a + 3*b))/(32*a*(a - b)))/(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) - tan(c + d*x)^6*(4*a*b - 4*a^2) + 4*a^2*tan(c + d*x)^
2))