3.393 \(\int \frac{\cos ^7(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\)

Optimal. Leaf size=288 \[ -\frac{\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac{\left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac{2 \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{2 \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d} \]

[Out]

(-2*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]
*a^(5/3)*b^(7/3)*d) + (2*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(
7/3)*d) - ((2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]
^2])/(9*a^(5/3)*b^(7/3)*d) - Sin[c + d*x]/(b^2*d) - (Sin[c + d*x]*(a^2 - b^2 + 3*a*b*Sin[c + d*x] + 3*b^2*Sin[
c + d*x]^2))/(3*a*b^2*d*(a + b*Sin[c + d*x]^3))

________________________________________________________________________________________

Rubi [A]  time = 0.335041, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3223, 1858, 1887, 1860, 31, 634, 617, 204, 628} \[ -\frac{\sin (c+d x) \left (a^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)-b^2\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac{\left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac{2 \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{2 \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

(-2*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]
*a^(5/3)*b^(7/3)*d) + (2*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(
7/3)*d) - ((2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]
^2])/(9*a^(5/3)*b^(7/3)*d) - Sin[c + d*x]/(b^2*d) - (Sin[c + d*x]*(a^2 - b^2 + 3*a*b*Sin[c + d*x] + 3*b^2*Sin[
c + d*x]^2))/(3*a*b^2*d*(a + b*Sin[c + d*x]^3))

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-a^2-2 b^2-6 a b x+3 a b x^3}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d}\\ &=-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \left (3 a-\frac{2 \left (2 a^2+b^2+3 a b x\right )}{a+b x^3}\right ) \, dx,x,\sin (c+d x)\right )}{3 a b^2 d}\\ &=-\frac{\sin (c+d x)}{b^2 d}-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{2 a^2+b^2+3 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a b^2 d}\\ &=-\frac{\sin (c+d x)}{b^2 d}-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (3 a^{4/3} b+2 \sqrt [3]{b} \left (2 a^2+b^2\right )\right )+\sqrt [3]{b} \left (3 a^{4/3} b-\sqrt [3]{b} \left (2 a^2+b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac{\left (2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^2 d}\\ &=\frac{2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d}-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}-\frac{\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}+\frac{\left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{4/3} b^2 d}\\ &=\frac{2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d}-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac{\left (2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{7/3} d}\\ &=-\frac{2 \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{5/3} b^{7/3} d}+\frac{2 \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{\left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{9 a^{5/3} b^{7/3} d}-\frac{\sin (c+d x)}{b^2 d}-\frac{\sin (c+d x) \left (a^2-b^2+3 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{3 a b^2 d \left (a+b \sin ^3(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 3.59769, size = 402, normalized size = 1.4 \[ \frac{\frac{6 \left (1-\frac{a^2}{b^2}\right ) \sin (c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}+\frac{6 \sqrt [3]{-1} \left (2 \sqrt [3]{-1} a^{2/3}+3 b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac{6 \left (2 a^{2/3}-3 b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}-\frac{6 \sqrt [3]{-1} \left (2 a^{2/3}+3 \sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}{\sqrt [3]{a} b^{7/3}}+\frac{2 \left (a^2-b^2\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{a^{5/3} b^{7/3}}-\frac{27 \sin ^2(c+d x) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right )}{a b}+\frac{18}{b \left (a+b \sin ^3(c+d x)\right )}-\frac{18 \sin (c+d x)}{b^2}}{18 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((6*(-1)^(1/3)*(2*(-1)^(1/3)*a^(2/3) + 3*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c + d*x]])/(a^(1/3)*
b^(7/3)) + (6*(2*a^(2/3) - 3*b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) - (6*(-1)^(1/3)*(
2*a^(2/3) + 3*(-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]])/(a^(1/3)*b^(7/3)) + (2*(a^2
- b^2)*(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Sin[c
+ d*x]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(a^(5/3)*b^(7/3)) - (18*Sin[
c + d*x])/b^2 - (27*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*b) + 18/(b*(a +
b*Sin[c + d*x]^3)) + (6*(1 - a^2/b^2)*Sin[c + d*x])/(a*(a + b*Sin[c + d*x]^3)))/(18*d)

________________________________________________________________________________________

Maple [B]  time = 0.145, size = 490, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-sin(d*x+c)/b^2/d-1/d/b/(a+b*sin(d*x+c)^3)*sin(d*x+c)^2-1/3/d/b^2/(a+b*sin(d*x+c)^3)*sin(d*x+c)*a+1/3*sin(d*x+
c)/a/d/(a+b*sin(d*x+c)^3)+1/d/b/(a+b*sin(d*x+c)^3)+4/9/d/b^3*a/(a/b)^(2/3)*ln(sin(d*x+c)+(a/b)^(1/3))+2/9/d/b/
a/(a/b)^(2/3)*ln(sin(d*x+c)+(a/b)^(1/3))-2/9/d/b^3*a/(a/b)^(2/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^
(2/3))-1/9/d/b/a/(a/b)^(2/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))+4/9/d/b^3*a/(a/b)^(2/3)*3^(1/
2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1))+2/9/d/b/a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*sin(d*x+c)-1))-2/3/d/b^2/(a/b)^(1/3)*ln(sin(d*x+c)+(a/b)^(1/3))+1/3/d/b^2/(a/b)^(1/3)*ln(sin(d*x+c)^2-(a
/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))+2/3/d/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1
))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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(cos(d*x+c)**7/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.2355, size = 374, normalized size = 1.3 \begin{align*} -\frac{\frac{9 \, \sin \left (d x + c\right )}{b^{2}} + \frac{2 \,{\left (3 \, a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{2} + b^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | -\left (-\frac{a}{b}\right )^{\frac{1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{2} b^{2}} + \frac{2 \, \sqrt{3}{\left (3 \, \left (-a b^{2}\right )^{\frac{2}{3}} a - \left (-a b^{2}\right )^{\frac{1}{3}}{\left (2 \, a^{2} + b^{2}\right )}\right )} \arctan \left (\frac{\sqrt{3}{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a^{2} b^{3}} + \frac{3 \,{\left (3 \, a b \sin \left (d x + c\right )^{2} + a^{2} \sin \left (d x + c\right ) - b^{2} \sin \left (d x + c\right ) - 3 \, a b\right )}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )} a b^{2}} - \frac{{\left (3 \, \left (-a b^{2}\right )^{\frac{2}{3}} a + \left (-a b^{2}\right )^{\frac{1}{3}}{\left (2 \, a^{2} + b^{2}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}} \sin \left (d x + c\right ) + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a^{2} b^{3}}}{9 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")

[Out]

-1/9*(9*sin(d*x + c)/b^2 + 2*(3*a*b*(-a/b)^(1/3) + 2*a^2 + b^2)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x +
c)))/(a^2*b^2) + 2*sqrt(3)*(3*(-a*b^2)^(2/3)*a - (-a*b^2)^(1/3)*(2*a^2 + b^2))*arctan(1/3*sqrt(3)*((-a/b)^(1/
3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b^3) + 3*(3*a*b*sin(d*x + c)^2 + a^2*sin(d*x + c) - b^2*sin(d*x + c) -
3*a*b)/((b*sin(d*x + c)^3 + a)*a*b^2) - (3*(-a*b^2)^(2/3)*a + (-a*b^2)^(1/3)*(2*a^2 + b^2))*log(sin(d*x + c)^
2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b^3))/d