3.231 \(\int \frac{\sin ^6(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=343 \[ -\frac{\left (-10 \sqrt{a} \sqrt{b}+4 a+3 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (10 \sqrt{a} \sqrt{b}+4 a+3 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{\tan (c+d x) \left (\frac{\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}\right )}{32 a b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac{\tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

[Out]

-((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a
] - Sqrt[b])^(5/2)*b^(3/2)*d) + ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x]
)/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - (Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2
)*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]*((2*a*(
a^2 - a*b - 8*b^2))/(a - b)^3 + ((2*a^2 + 15*a*b + 3*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*b*d*(a + 2*a*Tan[c
+ d*x]^2 + (a - b)*Tan[c + d*x]^4))

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Rubi [A]  time = 0.765268, antiderivative size = 343, 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} \[ -\frac{\left (-10 \sqrt{a} \sqrt{b}+4 a+3 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (10 \sqrt{a} \sqrt{b}+4 a+3 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{\tan (c+d x) \left (\frac{\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}\right )}{32 a b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac{\tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

-((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a
] - Sqrt[b])^(5/2)*b^(3/2)*d) + ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x]
)/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - (Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2
)*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]*((2*a*(
a^2 - a*b - 8*b^2))/(a - b)^3 + ((2*a^2 + 15*a*b + 3*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*b*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 ^6(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (1+x^2\right )^2}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\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-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^3 b (a+3 b)}{(a-b)^3}+\frac{2 a^2 b \left (5 a^2+6 a b-3 b^2\right ) x^2}{(a-b)^3}+\frac{32 a^2 b^2 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{\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-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 (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac{\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b 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{8 a^4 b (a+2 b)}{(a-b)^2}-\frac{4 a^3 b \left (2 a^2-17 a b+3 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{\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-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 (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac{\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b 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 (4 a-10 \sqrt{a} \sqrt{b}+3 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 \left (\sqrt{a}-\sqrt{b}\right )^2 b^{3/2} d}+\frac{\left (-\frac{2 a^3 b \left (2 a^2-17 a b+3 b^2\right )}{(a-b)^2}+\frac{\frac{16 a^4 b (a+2 b)}{a-b}+\frac{8 a^4 b \left (2 a^2-17 a b+3 b^2\right )}{(a-b)^2}}{4 \sqrt{a} \sqrt{b}}\right ) \operatorname{Subst}\left (\int \frac{1}{a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d}\\ &=-\frac{\left (4 a-10 \sqrt{a} \sqrt{b}+3 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} b^{3/2} d}+\frac{\left (4 a+10 \sqrt{a} \sqrt{b}+3 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} b^{3/2} d}-\frac{\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-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 (a^2-a b-8 b^2\right )}{(a-b)^3}+\frac{\left (2 a^2+15 a b+3 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 3.8556, size = 350, normalized size = 1.02 \[ \frac{\frac{4 b \sin (2 (c+d x)) \left (4 a^2+3 b (a+b) \cos (2 (c+d x))-19 a b-3 b^2\right )}{a (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{\sqrt{b} \left (10 \sqrt{a} \sqrt{b}+4 a+3 b\right ) \left (\sqrt{a}-\sqrt{b}\right )^2 \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{a \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{128 b (a-b) \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{\sqrt{b} \left (\sqrt{a}+\sqrt{b}\right )^2 \left (-10 \sqrt{a} \sqrt{b}+4 a+3 b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{a \sqrt{\sqrt{a} \sqrt{b}-a}}}{64 b^2 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

(((Sqrt[a] - Sqrt[b])^2*Sqrt[b]*(4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqr
t[a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((Sqrt[a] + Sqrt[b])^2*Sqrt[b]*(4*a - 10*Sqrt[a]*Sqrt
[b] + 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[-a + Sqrt[a]*Sqrt[b
]]) + (4*b*(4*a^2 - 19*a*b - 3*b^2 + 3*b*(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)])) - (128*(a - b)*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 - b)^2*b^2*d)

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Maple [B]  time = 0.138, size = 1909, normalized size = 5.6 \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)^6/(a-b*sin(d*x+c)^4)^3,x)

[Out]

-1/16/d/b/(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^3-7/32/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))+1/16/d/b/(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^3+7/32/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))+1/32/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+5/16/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)^5-19/32/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)^3-1/8/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)-15/32/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)^7-1/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(
d*x+c)^2+a)^2/(a-b)/b*tan(d*x+c)^7*a-3/16/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*a^2-3/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a^2/b/(a^2-2*a*b+b^2)*tan(d*
x+c)^3-1/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a^2/b/(a^2-2*a*b+b^2)*tan(d*x+c)+19/64/d/(a
^2-2*a*b+b^2)*a/(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))-7/8
/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+19/64/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))+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*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))-5/16/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))-5/16/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))-1/32/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))*a^2+19/64/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^2-1/32/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))*a^2-19/64/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^2-1/64/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))+1/
64/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))

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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)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/16*(4*(32*a^3*b^2 - 84*a^2*b^3 - 83*a*b^4 + 21*b^5)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((4*a^2*b^3 - 13*a*
b^4 + 3*b^5)*sin(14*d*x + 14*c) - 3*(8*a^2*b^3 - 33*a*b^4 + 7*b^5)*sin(12*d*x + 12*c) + (64*a^3*b^2 + 68*a^2*b
^3 - 225*a*b^4 + 63*b^5)*sin(10*d*x + 10*c) - 3*(128*a^3*b^2 + 32*a^2*b^3 - 61*a*b^4 + 35*b^5)*sin(8*d*x + 8*c
) - (64*a^3*b^2 + 452*a^2*b^3 - 9*a*b^4 - 105*b^5)*sin(6*d*x + 6*c) + 3*(40*a^2*b^3 - 29*a*b^4 - 21*b^5)*sin(4
*d*x + 4*c) - (4*a^2*b^3 - 37*a*b^4 - 21*b^5)*sin(2*d*x + 2*c))*cos(16*d*x + 16*c) + 2*(2*(32*a^3*b^2 - 84*a^2
*b^3 - 83*a*b^4 + 21*b^5)*sin(12*d*x + 12*c) - 8*(64*a^3*b^2 - 84*a^2*b^3 - 43*a*b^4 + 21*b^5)*sin(10*d*x + 10
*c) - (512*a^4*b - 3584*a^3*b^2 + 1388*a^2*b^3 - 11*a*b^4 - 315*b^5)*sin(8*d*x + 8*c) + 16*(172*a^2*b^3 - 37*a
*b^4 - 21*b^5)*sin(6*d*x + 6*c) + 2*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 105*b^5)*sin(4*d*x + 4*c) + 8*(4*a
^2*b^3 - 25*a*b^4 - 9*b^5)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) - 2*(2*(512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3
+ 21*a*b^4 - 147*b^5)*sin(10*d*x + 10*c) - 3*(3072*a^4*b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a*b^4 - 245*b^5)
*sin(8*d*x + 8*c) - 2*(512*a^4*b + 3936*a^3*b^2 - 6740*a^2*b^3 + 1281*a*b^4 + 441*b^5)*sin(6*d*x + 6*c) + 12*(
192*a^3*b^2 - 416*a^2*b^3 + 161*a*b^4 + 49*b^5)*sin(4*d*x + 4*c) - 2*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 1
05*b^5)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) - 2*((8192*a^5 + 27136*a^4*b - 37696*a^3*b^2 + 17644*a^2*b^3 - 20
79*a*b^4 - 735*b^5)*sin(8*d*x + 8*c) + 8*(1024*a^4*b + 3712*a^3*b^2 - 3692*a^2*b^3 + 483*a*b^4 + 147*b^5)*sin(
6*d*x + 6*c) - 2*(512*a^4*b + 3936*a^3*b^2 - 6740*a^2*b^3 + 1281*a*b^4 + 441*b^5)*sin(4*d*x + 4*c) - 16*(172*a
^2*b^3 - 37*a*b^4 - 21*b^5)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) - 2*((8192*a^5 + 27136*a^4*b - 37696*a^3*b^2
+ 17644*a^2*b^3 - 2079*a*b^4 - 735*b^5)*sin(6*d*x + 6*c) - 3*(3072*a^4*b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a
*b^4 - 245*b^5)*sin(4*d*x + 4*c) + (512*a^4*b - 3584*a^3*b^2 + 1388*a^2*b^3 - 11*a*b^4 - 315*b^5)*sin(2*d*x +
2*c))*cos(8*d*x + 8*c) - 4*((512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3 + 21*a*b^4 - 147*b^5)*sin(4*d*x + 4*c) + 4
*(64*a^3*b^2 - 84*a^2*b^3 - 43*a*b^4 + 21*b^5)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 16*((a^3*b^5 - 2*a^2*b^6 +
a*b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^5*b^3 - 24
0*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753
*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^7*b - 57344*a^6*b^2 + 83712*a^5*b^3 - 6
7648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1225*a*b^7)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4
+ 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 2
10*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c)^2 + (a^3*b^5
- 2*a^2*b^6 + a*b^7)*d*sin(16*d*x + 16*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x + 14*c)^2 + 16*(6
4*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^5*b^3 - 736
*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^7*b - 57344*a^6*b^2 + 837
12*a^5*b^3 - 67648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1225*a*b^7)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^5*b^3
- 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 3
37*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)
*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c)^2 - 16*(a^3*b^5 - 2
*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) + (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d - 2*(8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*
cos(14*d*x + 14*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^4 -
39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2
*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c) +
4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4*d*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2
*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6
- 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x + 10*c) - 2
*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*a^
3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4*d
*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*cos(14*d*x +
14*c) - 8*(8*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x + 10*c) + 2*(102
4*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*cos(8*d*x + 8*c) + 8*(128
*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c) - 4*(64*a^5*b^3 - 240*a^4*b^
4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) - 8*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^
7)*d*cos(2*d*x + 2*c) + (8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^
6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*cos(8*d*x + 8*c) + 8*(256*a^5
*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c) - 4*(128*a^5*b^3 - 424*a^4*b^4 +
513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)
*d*cos(2*d*x + 2*c) + (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^6*
b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*cos(6*d*x + 6*c) - 4*(1024*a^6*
b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*cos(4*d*x + 4*c) - 8*(128*a^5*b
^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(2*d*x + 2*c) + (128*a^5*b^3 - 352*a^4*b^4 + 355
*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*
a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) + 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c)
- (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^
2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) - (8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d)*cos(4*d*x + 4*c) - 4*
(4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x + 14*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*si
n(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(10*d*x + 10*c) - (128*a^5*b^3 - 35
2*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6
- 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c) + 4*(a^3
*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6
- 7*a*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(10*d*x + 10*c) - (1
28*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(8*d*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3*b
^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x
+ 4*c) + 4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^5*b^3 - 424*a^4
*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(10*d*x + 10*c) + (1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^
4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x + 8*c) + 4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 -
266*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b
^7)*d*sin(4*d*x + 4*c) - 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12
*c) + 32*((2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x +
8*c) + 4*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2*(128*a^5*b
^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*
a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4
- 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*sin(6*d*x + 6*c) - (1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 -
3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(4*d*x + 4*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166
*a^2*b^6 + 35*a*b^7)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266
*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) + 2*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c
))*sin(6*d*x + 6*c))*integrate(-1/8*(4*(4*a^2*b - 13*a*b^2 + 3*b^3)*cos(6*d*x + 6*c)^2 + 12*(56*a^2*b - 29*a*b
^2 + 3*b^3)*cos(4*d*x + 4*c)^2 + 4*(4*a^2*b - 13*a*b^2 + 3*b^3)*cos(2*d*x + 2*c)^2 + 4*(4*a^2*b - 13*a*b^2 + 3
*b^3)*sin(6*d*x + 6*c)^2 + 12*(56*a^2*b - 29*a*b^2 + 3*b^3)*sin(4*d*x + 4*c)^2 + 2*(32*a^3 - 116*a^2*b + 147*a
*b^2 - 21*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(4*a^2*b - 13*a*b^2 + 3*b^3)*sin(2*d*x + 2*c)^2 - ((4*a^2
*b - 13*a*b^2 + 3*b^3)*cos(6*d*x + 6*c) + 6*(7*a*b^2 - b^3)*cos(4*d*x + 4*c) + (4*a^2*b - 13*a*b^2 + 3*b^3)*co
s(2*d*x + 2*c))*cos(8*d*x + 8*c) - (4*a^2*b - 13*a*b^2 + 3*b^3 - 2*(32*a^3 - 116*a^2*b + 147*a*b^2 - 21*b^3)*c
os(4*d*x + 4*c) - 8*(4*a^2*b - 13*a*b^2 + 3*b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 2*(21*a*b^2 - 3*b^3 - (3
2*a^3 - 116*a^2*b + 147*a*b^2 - 21*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (4*a^2*b - 13*a*b^2 + 3*b^3)*cos(
2*d*x + 2*c) - ((4*a^2*b - 13*a*b^2 + 3*b^3)*sin(6*d*x + 6*c) + 6*(7*a*b^2 - b^3)*sin(4*d*x + 4*c) + (4*a^2*b
- 13*a*b^2 + 3*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((32*a^3 - 116*a^2*b + 147*a*b^2 - 21*b^3)*sin(4*d*
x + 4*c) + 4*(4*a^2*b - 13*a*b^2 + 3*b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))/(a^3*b^3 - 2*a^2*b^4 + a*b^5 + (
a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(8*d*x + 8*c)^2 + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(6*d*x + 6*c)^2 + 4*(64*
a^5*b - 176*a^4*b^2 + 169*a^3*b^3 - 66*a^2*b^4 + 9*a*b^5)*cos(4*d*x + 4*c)^2 + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^5
)*cos(2*d*x + 2*c)^2 + (a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(8*d*x + 8*c)^2 + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin
(6*d*x + 6*c)^2 + 4*(64*a^5*b - 176*a^4*b^2 + 169*a^3*b^3 - 66*a^2*b^4 + 9*a*b^5)*sin(4*d*x + 4*c)^2 + 16*(8*a
^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^3*b^3 - 2*a^2*b^4 + a*b^
5)*sin(2*d*x + 2*c)^2 + 2*(a^3*b^3 - 2*a^2*b^4 + a*b^5 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(6*d*x + 6*c) - 2*
(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*cos(4*d*x + 4*c) - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(2*d*x +
2*c))*cos(8*d*x + 8*c) - 8*(a^3*b^3 - 2*a^2*b^4 + a*b^5 - 2*(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*c
os(4*d*x + 4*c) - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^4*b^2 - 19*a^3*b
^3 + 14*a^2*b^4 - 3*a*b^5 - 4*(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*cos(2*d*x + 2*c))*cos(4*d*x + 4*
c) - 8*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*cos(2*d*x + 2*c) - 4*(2*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(6*d*x + 6*c) +
(8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*sin(4*d*x + 4*c) + 2*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 16*((8*a^4*b^2 - 19*a^3*b^3 + 14*a^2*b^4 - 3*a*b^5)*sin(4*d*x + 4*c) + 2*(a^3*b^3 -
2*a^2*b^4 + a*b^5)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) + (3*a*b^4 + 3*b^5 - (4*a^2*b^3 - 13*a*b^4 + 3*b^5)
*cos(14*d*x + 14*c) + 3*(8*a^2*b^3 - 33*a*b^4 + 7*b^5)*cos(12*d*x + 12*c) - (64*a^3*b^2 + 68*a^2*b^3 - 225*a*b
^4 + 63*b^5)*cos(10*d*x + 10*c) + 3*(128*a^3*b^2 + 32*a^2*b^3 - 61*a*b^4 + 35*b^5)*cos(8*d*x + 8*c) + (64*a^3*
b^2 + 452*a^2*b^3 - 9*a*b^4 - 105*b^5)*cos(6*d*x + 6*c) - 3*(40*a^2*b^3 - 29*a*b^4 - 21*b^5)*cos(4*d*x + 4*c)
+ (4*a^2*b^3 - 37*a*b^4 - 21*b^5)*cos(2*d*x + 2*c))*sin(16*d*x + 16*c) + (4*a^2*b^3 - 37*a*b^4 - 21*b^5 - 4*(3
2*a^3*b^2 - 84*a^2*b^3 - 83*a*b^4 + 21*b^5)*cos(12*d*x + 12*c) + 16*(64*a^3*b^2 - 84*a^2*b^3 - 43*a*b^4 + 21*b
^5)*cos(10*d*x + 10*c) + 2*(512*a^4*b - 3584*a^3*b^2 + 1388*a^2*b^3 - 11*a*b^4 - 315*b^5)*cos(8*d*x + 8*c) - 3
2*(172*a^2*b^3 - 37*a*b^4 - 21*b^5)*cos(6*d*x + 6*c) - 4*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 105*b^5)*cos(
4*d*x + 4*c) - 16*(4*a^2*b^3 - 25*a*b^4 - 9*b^5)*cos(2*d*x + 2*c))*sin(14*d*x + 14*c) - (120*a^2*b^3 - 87*a*b^
4 - 63*b^5 - 4*(512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3 + 21*a*b^4 - 147*b^5)*cos(10*d*x + 10*c) + 6*(3072*a^4*
b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a*b^4 - 245*b^5)*cos(8*d*x + 8*c) + 4*(512*a^4*b + 3936*a^3*b^2 - 6740*a
^2*b^3 + 1281*a*b^4 + 441*b^5)*cos(6*d*x + 6*c) - 24*(192*a^3*b^2 - 416*a^2*b^3 + 161*a*b^4 + 49*b^5)*cos(4*d*
x + 4*c) + 4*(32*a^3*b^2 - 372*a^2*b^3 + 289*a*b^4 + 105*b^5)*cos(2*d*x + 2*c))*sin(12*d*x + 12*c) + (64*a^3*b
^2 + 452*a^2*b^3 - 9*a*b^4 - 105*b^5 + 2*(8192*a^5 + 27136*a^4*b - 37696*a^3*b^2 + 17644*a^2*b^3 - 2079*a*b^4
- 735*b^5)*cos(8*d*x + 8*c) + 16*(1024*a^4*b + 3712*a^3*b^2 - 3692*a^2*b^3 + 483*a*b^4 + 147*b^5)*cos(6*d*x +
6*c) - 4*(512*a^4*b + 3936*a^3*b^2 - 6740*a^2*b^3 + 1281*a*b^4 + 441*b^5)*cos(4*d*x + 4*c) - 32*(172*a^2*b^3 -
37*a*b^4 - 21*b^5)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) + (384*a^3*b^2 + 96*a^2*b^3 - 183*a*b^4 + 105*b^5 + 2
*(8192*a^5 + 27136*a^4*b - 37696*a^3*b^2 + 17644*a^2*b^3 - 2079*a*b^4 - 735*b^5)*cos(6*d*x + 6*c) - 6*(3072*a^
4*b - 6272*a^3*b^2 + 2920*a^2*b^3 - 413*a*b^4 - 245*b^5)*cos(4*d*x + 4*c) + 2*(512*a^4*b - 3584*a^3*b^2 + 1388
*a^2*b^3 - 11*a*b^4 - 315*b^5)*cos(2*d*x + 2*c))*sin(8*d*x + 8*c) - (64*a^3*b^2 + 68*a^2*b^3 - 225*a*b^4 + 63*
b^5 - 4*(512*a^4*b - 672*a^3*b^2 + 1228*a^2*b^3 + 21*a*b^4 - 147*b^5)*cos(4*d*x + 4*c) - 16*(64*a^3*b^2 - 84*a
^2*b^3 - 43*a*b^4 + 21*b^5)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) + (24*a^2*b^3 - 99*a*b^4 + 21*b^5 - 4*(32*a^3*b
^2 - 84*a^2*b^3 - 83*a*b^4 + 21*b^5)*cos(2*d*x + 2*c))*sin(4*d*x + 4*c) - (4*a^2*b^3 - 13*a*b^4 + 3*b^5)*sin(2
*d*x + 2*c))/((a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(14
*d*x + 14*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(12*d*x + 12*c)^2 +
64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^7*b
- 57344*a^6*b^2 + 83712*a^5*b^3 - 67648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1225*a*b^7)*d*cos(8*d*x + 8*
c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^5
*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a*
b^7)*d*cos(2*d*x + 2*c)^2 + (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(16*d*x + 16*c)^2 + 64*(a^3*b^5 - 2*a^2*b^6 + a
*b^7)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(12*d
*x + 12*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(10*d*x + 10*c)^2 +
4*(16384*a^7*b - 57344*a^6*b^2 + 83712*a^5*b^3 - 67648*a^4*b^4 + 32841*a^3*b^5 - 9170*a^2*b^6 + 1225*a*b^7)*d*
sin(8*d*x + 8*c)^2 + 64*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c)^
2 + 16*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^4*b^4
- 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*
sin(2*d*x + 2*c)^2 - 16*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) + (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d - 2
*(8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(14*d*x + 14*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*c
os(12*d*x + 12*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^3 -
352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*
b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4*d*x + 4*c) + 8*(
a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) - (a^3*b^5 - 2*a^2*b^6 + a*b^7)*d)*cos(16*d*x + 16*c) + 16*(4*
(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6
- 7*a*b^7)*d*cos(10*d*x + 10*c) - 2*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(
8*d*x + 8*c) - 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^4*b^4 - 23*a^3*b
^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(4*d*x + 4*c) + 8*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*cos(2*d*x + 2*c) - (a^3*b^5
- 2*a^2*b^6 + a*b^7)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*
a*b^7)*d*cos(10*d*x + 10*c) + 2*(1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 24
5*a*b^7)*d*cos(8*d*x + 8*c) + 8*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x
+ 6*c) - 4*(64*a^5*b^3 - 240*a^4*b^4 + 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) - 8*(8*a^4*b^
4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) + (8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d)
*cos(12*d*x + 12*c) + 16*(2*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*
b^7)*d*cos(8*d*x + 8*c) + 8*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^7)*d*cos(6*d*x + 6
*c) - 4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) - 8*(16*a^4*b^4
- 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) + (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d)*
cos(10*d*x + 10*c) + 4*(8*(2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^
7)*d*cos(6*d*x + 6*c) - 4*(1024*a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^
7)*d*cos(4*d*x + 4*c) - 8*(128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*cos(2*d*x + 2*c
) + (128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^5*b^
3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*cos(4*d*x + 4*c) + 8*(16*a^4*b^4 - 39*a^3*b^5 + 30*a
^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) - (16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d)*cos(6*d*x + 6*c) +
8*(8*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*cos(2*d*x + 2*c) - (8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6
- 7*a*b^7)*d)*cos(4*d*x + 4*c) - 4*(4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(14*d*x + 14*c) + 2*(8*a^4*b^4 - 23*
a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*si
n(10*d*x + 10*c) - (128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(8*d*x + 8*c) - 4*(
16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 -
7*a*b^7)*d*sin(4*d*x + 4*c) + 4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(
8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)*d*sin(12*d*x + 12*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6
- 7*a*b^7)*d*sin(10*d*x + 10*c) - (128*a^5*b^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(8*d
*x + 8*c) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(6*d*x + 6*c) + 2*(8*a^4*b^4 - 23*a^3*b^5
+ 22*a^2*b^6 - 7*a*b^7)*d*sin(4*d*x + 4*c) + 4*(a^3*b^5 - 2*a^2*b^6 + a*b^7)*d*sin(2*d*x + 2*c))*sin(14*d*x +
14*c) - 16*(4*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(10*d*x + 10*c) + (1024*
a^6*b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x + 8*c) + 4*(128*a
^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(6*d*x + 6*c) - 2*(64*a^5*b^3 - 240*a^4*b^4
+ 337*a^3*b^5 - 210*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) - 4*(8*a^4*b^4 - 23*a^3*b^5 + 22*a^2*b^6 - 7*a*b^7)
*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 172
2*a^2*b^6 - 245*a*b^7)*d*sin(8*d*x + 8*c) + 4*(256*a^5*b^3 - 736*a^4*b^4 + 753*a^3*b^5 - 322*a^2*b^6 + 49*a*b^
7)*d*sin(6*d*x + 6*c) - 2*(128*a^5*b^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c
) - 4*(16*a^4*b^4 - 39*a^3*b^5 + 30*a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^
6*b^2 - 6528*a^5*b^3 + 8144*a^4*b^4 - 5141*a^3*b^5 + 1722*a^2*b^6 - 245*a*b^7)*d*sin(6*d*x + 6*c) - (1024*a^6*
b^2 - 3712*a^5*b^3 + 5304*a^4*b^4 - 3813*a^3*b^5 + 1442*a^2*b^6 - 245*a*b^7)*d*sin(4*d*x + 4*c) - 2*(128*a^5*b
^3 - 352*a^4*b^4 + 355*a^3*b^5 - 166*a^2*b^6 + 35*a*b^7)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^5*b
^3 - 424*a^4*b^4 + 513*a^3*b^5 - 266*a^2*b^6 + 49*a*b^7)*d*sin(4*d*x + 4*c) + 2*(16*a^4*b^4 - 39*a^3*b^5 + 30*
a^2*b^6 - 7*a*b^7)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B]  time = 21.1073, size = 13869, normalized size = 40.43 \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)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/256*(((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*
(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*c
os(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2
+ 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^
6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b
^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*
a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(3
20*a^5 - 2724*a^4*b + 6243*a^3*b^2 - 9389/4*a^2*b^3 + 729/2*a*b^4 - 81/4*b^5 - 1/4*(1280*a^5 - 10896*a^4*b + 2
4972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 + 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b
^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104
361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 12
0*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*
d^4))*cos(d*x + c)*sin(d*x + c) + (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2
*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*
a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 5
3212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 21
0*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^
3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a
^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b -
156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400
*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^1
4*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 -
10*a^6*b^12 + a^5*b^13)*d^4))) - ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*
b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^
3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4
- 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 -
a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b
^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^
8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*
b^7 - a^2*b^8)*d^2))*log(320*a^5 - 2724*a^4*b + 6243*a^3*b^2 - 9389/4*a^2*b^3 + 729/2*a*b^4 - 81/4*b^5 - 1/4*(
1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 - 1/2*((2*a^11*b^3
- 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((64
00*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a
^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11
- 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^
4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^
3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*
a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^
13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 +
a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^
10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(
d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 -
9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 8
1*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120
*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))) + ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4
*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)
^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4
+ a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 -
10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a
^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*
b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5
*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(-320*a^5 + 2724*a^4*b - 6243*a^3*b^2 + 9389/4*a^2*b^3 - 729
/2*a*b^4 + 81/4*b^5 + 1/4*(1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*
x + c)^2 + 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9
- 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5
+ 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 -
120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (320*a^7*b - 2404*a^6*b
^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^
3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*
d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^1
5*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 +
45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2
*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^
4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 4
86*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 128
14*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a
^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))) - ((a^3*b^3 - 2*a^2*b^4 +
a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^
4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*
b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3
- 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^
2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6
+ 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a
^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(-320*a^5 + 2724*a^4*b - 6243*a^3
*b^2 + 9389/4*a^2*b^3 - 729/2*a*b^4 + 81/4*b^5 + 1/4*(1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 +
1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 - 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7
- 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 +
12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 25
2*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c
) - (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x +
c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*
b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 -
1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 21
0*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 1
0*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6
*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 96
5*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^
4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12
*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))
- 8*(3*(a*b^2 + b^3)*cos(d*x + c)^7 + (2*a^2*b - 17*a*b^2 - 9*b^3)*cos(d*x + c)^5 - (11*a^2*b - 26*a*b^2 - 9*
b^3)*cos(d*x + c)^3 + (2*a^3 + 13*a^2*b - 12*a*b^2 - 3*b^3)*cos(d*x + c))*sin(d*x + c))/((a^3*b^3 - 2*a^2*b^4
+ a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*
b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^
4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*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)**6/(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)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError