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

Optimal. Leaf size=313 \[ -\frac{\tan (c+d x) \left (\frac{9 a^2-24 a b-b^2}{(a-b)^3}+\frac{(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}+\frac{3 \left (2 \sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \sqrt{b} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}-\frac{3 \left (2 \sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \sqrt{b} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{b \tan (c+d x) \left (4 (a+b) \tan ^2(c+d x)+3 a+b\right )}{8 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

[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))

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Rubi [A]  time = 0.695044, antiderivative size = 313, normalized size of antiderivative = 1., 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 = {3217, 1333, 1678, 1166, 205} \[ -\frac{\tan (c+d x) \left (\frac{9 a^2-24 a b-b^2}{(a-b)^3}+\frac{(17 a+3 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}+\frac{3 \left (2 \sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \sqrt{b} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}-\frac{3 \left (2 \sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{7/4} \sqrt{b} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{b \tan (c+d x) \left (4 (a+b) \tan ^2(c+d x)+3 a+b\right )}{8 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

Antiderivative was successfully verified.

[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 3217

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]

Rule 1333

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 1678

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 1166

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 205

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

Rubi steps

\begin{align*} \int \frac{\sin ^4(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \tan ^{-1}\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 ) \tan ^{-1}\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]  time = 5.04678, size = 316, normalized size = 1.01 \[ \frac{-\frac{3 \left (2 a^{3/2}-3 a \sqrt{b}+b^{3/2}\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{a^{3/2} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{3 \left (2 a^{3/2}+3 a \sqrt{b}-b^{3/2}\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{a^{3/2} \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}+\frac{8 \sin (2 (c+d x)) ((2 a+b) \cos (2 (c+d x))-7 a-2 b)}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{64 (a-b) (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}}{64 d (a-b)^2} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.131, size = 1624, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-17/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)*tan(d*x+c)^7-3/32/d/(tan(d*x+c)^4*a-tan(d*
x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)/a*b*tan(d*x+c)^7-43/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+
a)^2/(a^2-2*a*b+b^2)*tan(d*x+c)^5*a+9/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^2-2*a*b+b
^2)*tan(d*x+c)^5+1/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/a/(a^2-2*a*b+b^2)*tan(d*x+c)^5*b^
2-35/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a/(a^2-2*a*b+b^2)*tan(d*x+c)^3+11/32/d/(tan(d*x
+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^2-2*a*b+b^2)*tan(d*x+c)^3-9/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^
4*b+2*a*tan(d*x+c)^2+a)^2*a/(a^2-2*a*b+b^2)*tan(d*x+c)+3/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+
a)^2/(a^2-2*a*b+b^2)*tan(d*x+c)*b+15/64/d/(a^2-2*a*b+b^2)*a/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*t
an(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-9/32/d*b/(a^2-2*a*b+b^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))+3/32/d/(a^2-2*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))*a^2+3/64/d*b/(a^2-2*a*b+b^2)*a/(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))-3/16/d*b^2/(a^2-2*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))+15/
64/d/(a^2-2*a*b+b^2)*a/(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))-9/32/d*b/(a^2-2*a*b+b^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))-3/32/d/(a^2-2*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))*a^2-3/64/d*b/(a^2-2*a*b+b^2)*a/(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))+3/16/d*b^2/(a^2-2*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))+3/64/d/a*b^2/(a^2-2*a*b+b^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))+3/64/d/a/(a^2-2*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))*b^
3+3/64/d/a*b^2/(a^2-2*a*b+b^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))-3/64/d/a/(a^2-2*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))*b^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [B]  time = 13.6166, size = 12400, normalized size = 39.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(3*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 -
2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos
(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a
^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5
+ b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120
*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^
2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(432*a^4 + 27*a^3*b - 783/4*a^2*b^2 + 135/2*a*b^3
- 27/4*b^4 - 27/4*(64*a^4 + 4*a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 + 27/2*((5*a^12*b - 26*a^11*
b^2 + 55*a^10*b^3 - 60*a^9*b^4 + 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^
2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b
^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(
d*x + c) - (32*a^7 + 58*a^6*b - 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sq
rt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a
^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a
^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21
*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*
(4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10
- 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*
a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*
a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) - 3*((a^3*
b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^
3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (
a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*
b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*
b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*
a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3
- 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(432*a^4 + 27*a^3*b - 783/4*a^2*b^2 + 135/2*a*b^3 - 27/4*b^4 - 27
/4*(64*a^4 + 4*a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 - 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b
^3 - 60*a^9*b^4 + 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3
+ 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b
^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (32*
a^7 + 58*a^6*b - 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^8*b - 5
*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3
*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a
^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b
^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^
9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45
*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^
3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*
a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) + 3*((a^3*b^2 - 2*a^2*b^3
+ a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b
^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b +
6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d
^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2
+ 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8
*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5
*a^4*b^5 - a^3*b^6)*d^2))*log(-432*a^4 - 27*a^3*b + 783/4*a^2*b^2 - 135/2*a*b^3 + 27/4*b^4 + 27/4*(64*a^4 + 4*
a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 + 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b^3 - 60*a^9*b^4
+ 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 -
12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b
^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (32*a^7 + 58*a^6*b
- 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^
6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^
4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^
11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*
b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^9*b + 45*a^8*b^2
- 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^
7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b
^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a
^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) - 3*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(
d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*
cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a
^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6
+ 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 -
120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11
)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b
^6)*d^2))*log(-432*a^4 - 27*a^3*b + 783/4*a^2*b^2 - 135/2*a*b^3 + 27/4*b^4 + 27/4*(64*a^4 + 4*a^3*b - 29*a^2*b
^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 - 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b^3 - 60*a^9*b^4 + 35*a^8*b^5 -
10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/
((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^
8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (32*a^7 + 58*a^6*b - 13*a^5*b^2 - 2
1*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b
^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b
^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^1
0*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 +
10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 +
30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b
^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 +
b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^
10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) - 8*(2*(2*a*b + b^2)*cos(d*x + c)^7 - (17*a*b + 7*b^2)*co
s(d*x + c)^5 - 8*(a^2 - 3*a*b - b^2)*cos(d*x + c)^3 + (17*a^2 - 14*a*b - 3*b^2)*cos(d*x + c))*sin(d*x + c))/((
a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b -
5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2
+ (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError