3.324 \(\int \frac{(e+f x)^3 \cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=753 \[ -\frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \left (a^2-b^2\right )^{3/2}}+\frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \left (a^2-b^2\right )^{3/2}}+\frac{3 i f^3 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^4 \left (a^2-b^2\right )}+\frac{3 i f^3 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^4 \left (a^2-b^2\right )}-\frac{3 i a f^3 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^4 \left (a^2-b^2\right )^{3/2}}+\frac{3 i a f^3 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^4 \left (a^2-b^2\right )^{3/2}}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \left (a^2-b^2\right )}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \left (a^2-b^2\right )}-\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}+\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 d^2 \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{3 i f (e+f x)^2}{2 b d^2 \left (a^2-b^2\right )}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2} \]

[Out]

(((3*I)/2)*f*(e + f*x)^2)/(b*(a^2 - b^2)*d^2) - (3*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 -
b^2])])/(b*(a^2 - b^2)*d^3) - (((3*I)/2)*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])]
)/(b*(a^2 - b^2)^(3/2)*d^2) - (3*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 -
b^2)*d^3) + (((3*I)/2)*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(
3/2)*d^2) + ((3*I)*f^3*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) - (3*a*f^2
*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + ((3*I)*f^3*Pol
yLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, (I*b
*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((3*I)*a*f^3*PolyLog[3, (I*b*E^(I*(c + d
*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) + ((3*I)*a*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + S
qrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sin[c + d*x])^2) + (3*f*(e + f*x)^2*Co
s[c + d*x])/(2*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.2742, antiderivative size = 753, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {4422, 3324, 3323, 2264, 2190, 2531, 2282, 6589, 4519, 2279, 2391} \[ -\frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \left (a^2-b^2\right )^{3/2}}+\frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \left (a^2-b^2\right )^{3/2}}+\frac{3 i f^3 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^4 \left (a^2-b^2\right )}+\frac{3 i f^3 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^4 \left (a^2-b^2\right )}-\frac{3 i a f^3 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^4 \left (a^2-b^2\right )^{3/2}}+\frac{3 i a f^3 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^4 \left (a^2-b^2\right )^{3/2}}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \left (a^2-b^2\right )}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \left (a^2-b^2\right )}-\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}+\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 d^2 \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{3 i f (e+f x)^2}{2 b d^2 \left (a^2-b^2\right )}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)/2)*f*(e + f*x)^2)/(b*(a^2 - b^2)*d^2) - (3*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 -
b^2])])/(b*(a^2 - b^2)*d^3) - (((3*I)/2)*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])]
)/(b*(a^2 - b^2)^(3/2)*d^2) - (3*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 -
b^2)*d^3) + (((3*I)/2)*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(
3/2)*d^2) + ((3*I)*f^3*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) - (3*a*f^2
*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + ((3*I)*f^3*Pol
yLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, (I*b
*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((3*I)*a*f^3*PolyLog[3, (I*b*E^(I*(c + d
*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) + ((3*I)*a*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + S
qrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sin[c + d*x])^2) + (3*f*(e + f*x)^2*Co
s[c + d*x])/(2*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x]))

Rule 4422

Int[Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol]
:> Simp[((e + f*x)^m*(a + b*Sin[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e + f*x
)^(m - 1)*(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{(3 f) \int \frac{(e+f x)^2}{(a+b \sin (c+d x))^2} \, dx}{2 b d}\\ &=-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}+\frac{(3 a f) \int \frac{(e+f x)^2}{a+b \sin (c+d x)} \, dx}{2 b \left (a^2-b^2\right ) d}-\frac{\left (3 f^2\right ) \int \frac{(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right ) d^2}\\ &=\frac{3 i f (e+f x)^2}{2 b \left (a^2-b^2\right ) d^2}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}+\frac{(3 a f) \int \frac{e^{i (c+d x)} (e+f x)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b \left (a^2-b^2\right ) d}-\frac{\left (3 f^2\right ) \int \frac{e^{i (c+d x)} (e+f x)}{a-\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right ) d^2}-\frac{\left (3 f^2\right ) \int \frac{e^{i (c+d x)} (e+f x)}{a+\sqrt{a^2-b^2}-i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right ) d^2}\\ &=\frac{3 i f (e+f x)^2}{2 b \left (a^2-b^2\right ) d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{(3 i a f) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2} d}+\frac{(3 i a f) \int \frac{e^{i (c+d x)} (e+f x)^2}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2} d}+\frac{\left (3 f^3\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right ) d^3}+\frac{\left (3 f^3\right ) \int \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right ) d^3}\\ &=\frac{3 i f (e+f x)^2}{2 b \left (a^2-b^2\right ) d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}+\frac{\left (3 i a f^2\right ) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{\left (3 i a f^2\right ) \int (e+f x) \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^2}-\frac{\left (3 i f^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right ) d^4}-\frac{\left (3 i f^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right ) d^4}\\ &=\frac{3 i f (e+f x)^2}{2 b \left (a^2-b^2\right ) d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}+\frac{3 i f^3 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^4}-\frac{3 a f^2 (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 i f^3 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^4}+\frac{3 a f^2 (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}+\frac{\left (3 a f^3\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{\left (3 a f^3\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \left (a^2-b^2\right )^{3/2} d^3}\\ &=\frac{3 i f (e+f x)^2}{2 b \left (a^2-b^2\right ) d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}+\frac{3 i f^3 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^4}-\frac{3 a f^2 (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 i f^3 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^4}+\frac{3 a f^2 (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{\left (3 i a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{3/2} d^4}+\frac{\left (3 i a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{3/2} d^4}\\ &=\frac{3 i f (e+f x)^2}{2 b \left (a^2-b^2\right ) d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}-\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{3 f^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^3}+\frac{3 i a f (e+f x)^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}+\frac{3 i f^3 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^4}-\frac{3 a f^2 (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}+\frac{3 i f^3 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right ) d^4}+\frac{3 a f^2 (e+f x) \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^3}-\frac{3 i a f^3 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^4}+\frac{3 i a f^3 \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2} d^4}-\frac{(e+f x)^3}{2 b d (a+b \sin (c+d x))^2}+\frac{3 f (e+f x)^2 \cos (c+d x)}{2 \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 19.5174, size = 2311, normalized size = 3.07 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-3*I)*E^(I*c)*f*(2*e*E^(I*c)*f*x + E^(I*c)*f^2*x^2 + (I*a*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^
2]])/(Sqrt[a^2 - b^2]*E^(I*c)) - (I*a*e^2*E^(I*c)*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/Sqrt[a^2
- b^2] + (2*a*e*f*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d*E^(I*c)) + (e*f*ArcTan
[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))])/(d*E^(I*c)) - (e*E^(I*c)*f*ArcTan[(2*a*E^(I*(c + d*x))
)/(b*(-1 + E^((2*I)*(c + d*x))))])/d + ((2*I)*a*e*f*ArcTanh[(-a + I*b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]])/(Sqrt
[a^2 - b^2]*d*E^(I*c)) - ((I/2)*e*f*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2])/(d*E^(I
*c)) + ((I/2)*e*E^(I*c)*f*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x)))^2])/d + (I*a*e*f*x*Lo
g[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] -
(I*a*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a
^2 + b^2)*E^((2*I)*c)] - (I*f^2*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])
])/(d*E^(I*c)) + (I*E^(I*c)*f^2*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])
])/d + ((I/2)*a*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-
a^2 + b^2)*E^((2*I)*c)] - ((I/2)*a*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2
+ b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (I*a*e*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c)
+ Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + (I*a*e*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(
2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - (I*f^2*x*Log[1
+ (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d*E^(I*c)) + (I*E^(I*c)*f^2*x*Log[1
+ (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/d - ((I/2)*a*f^2*x^2*Log[1 + (b*E^(I*
(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + ((I/2)*a*E^((2
*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/Sqrt[(-a^2 + b^2
)*E^((2*I)*c)] - ((-1 + E^((2*I)*c))*f*(-(Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*f) + a*d*E^(I*c)*(e + f*x))*PolyLog[2
, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d^2*E^(I*c)*Sqrt[(-a^2 + b^2)*E^((
2*I)*c)]) + ((-1 + E^((2*I)*c))*f*(Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*f + a*d*E^(I*c)*(e + f*x))*PolyLog[2, -((b*E
^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d^2*E^(I*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)
]) + (I*a*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])])/(d^2*Sqrt[(-
a^2 + b^2)*E^((2*I)*c)]) - (I*a*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 +
b^2)*E^((2*I)*c)])])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) - (I*a*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*
E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]) + (I*a*E^((2*I)*c)*f^2*PolyL
og[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(d^2*Sqrt[(-a^2 + b^2)*E^((2*I
)*c)])))/(b*(-a^2 + b^2)*d^2*(-1 + E^((2*I)*c))) - (e + f*x)^3/(2*b*d*(a + b*Sin[c + d*x])^2) - (3*Csc[c/2]*Se
c[c/2]*(a*e^2*f*Cos[c] + 2*a*e*f^2*x*Cos[c] + a*f^3*x^2*Cos[c] + b*e^2*f*Sin[d*x] + 2*b*e*f^2*x*Sin[d*x] + b*f
^3*x^2*Sin[d*x]))/(4*(a - b)*b*(a + b)*d^2*(a + b*Sin[c + d*x]))

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Maple [F]  time = 1.497, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\cos \left ( dx+c \right ) }{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((f*x+e)^3*cos(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 6.01823, size = 10630, normalized size = 14.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/8*(4*(a^4 - 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 12*(a^4 - 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 12*(a^4 - 2*a^2*b^2 +
b^4)*d^3*e^2*f*x + 4*(a^4 - 2*a^2*b^2 + b^4)*d^3*e^3 - 12*((a^2*b^2 - b^4)*d^2*f^3*x^2 + 2*(a^2*b^2 - b^4)*d^2
*e*f^2*x + (a^2*b^2 - b^4)*d^2*e^2*f)*cos(d*x + c)*sin(d*x + c) - 12*(a*b^3*f^3*cos(d*x + c)^2 - 2*a^2*b^2*f^3
*sin(d*x + c) - (a^3*b + a*b^3)*f^3)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(2*I*a*cos(d*x + c) - 2*a*sin(d*x +
c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 12*(a*b^3*f^3*cos(d*x + c)^2 - 2*a^2*
b^2*f^3*sin(d*x + c) - (a^3*b + a*b^3)*f^3)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(2*I*a*cos(d*x + c) - 2*a*si
n(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 12*(a*b^3*f^3*cos(d*x + c)^2 -
2*a^2*b^2*f^3*sin(d*x + c) - (a^3*b + a*b^3)*f^3)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(I*a*cos(d*x + c) + a*si
n(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 12*(a*b^3*f^3*cos(d*x + c)^2 - 2
*a^2*b^2*f^3*sin(d*x + c) - (a^3*b + a*b^3)*f^3)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(I*a*cos(d*x + c) + a*sin(
d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 12*((a^3*b - a*b^3)*d^2*f^3*x^2 +
2*(a^3*b - a*b^3)*d^2*e*f^2*x + (a^3*b - a*b^3)*d^2*e^2*f)*cos(d*x + c) - (12*I*(a^2*b^2 - b^4)*f^3*cos(d*x +
c)^2 - 24*I*(a^3*b - a*b^3)*f^3*sin(d*x + c) - 12*I*(a^4 - b^4)*f^3 + 2*(6*I*(a^3*b + a*b^3)*d*f^3*x + 6*I*(a^
3*b + a*b^3)*d*e*f^2 + (-6*I*a*b^3*d*f^3*x - 6*I*a*b^3*d*e*f^2)*cos(d*x + c)^2 + (12*I*a^2*b^2*d*f^3*x + 12*I*
a^2*b^2*d*e*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(
b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - (12*I*(a^2*b^2 - b^4)*f^3*cos(d*x +
c)^2 - 24*I*(a^3*b - a*b^3)*f^3*sin(d*x + c) - 12*I*(a^4 - b^4)*f^3 + 2*(-6*I*(a^3*b + a*b^3)*d*f^3*x - 6*I*(a
^3*b + a*b^3)*d*e*f^2 + (6*I*a*b^3*d*f^3*x + 6*I*a*b^3*d*e*f^2)*cos(d*x + c)^2 + (-12*I*a^2*b^2*d*f^3*x - 12*I
*a^2*b^2*d*e*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*
(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - (-12*I*(a^2*b^2 - b^4)*f^3*cos(d*x
+ c)^2 + 24*I*(a^3*b - a*b^3)*f^3*sin(d*x + c) + 12*I*(a^4 - b^4)*f^3 + 2*(-6*I*(a^3*b + a*b^3)*d*f^3*x - 6*I*
(a^3*b + a*b^3)*d*e*f^2 + (6*I*a*b^3*d*f^3*x + 6*I*a*b^3*d*e*f^2)*cos(d*x + c)^2 + (-12*I*a^2*b^2*d*f^3*x - 12
*I*a^2*b^2*d*e*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) +
2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - (-12*I*(a^2*b^2 - b^4)*f^3*cos(d
*x + c)^2 + 24*I*(a^3*b - a*b^3)*f^3*sin(d*x + c) + 12*I*(a^4 - b^4)*f^3 + 2*(6*I*(a^3*b + a*b^3)*d*f^3*x + 6*
I*(a^3*b + a*b^3)*d*e*f^2 + (-6*I*a*b^3*d*f^3*x - 6*I*a*b^3*d*e*f^2)*cos(d*x + c)^2 + (12*I*a^2*b^2*d*f^3*x +
12*I*a^2*b^2*d*e*f^2)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c)
- 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + 6*(2*(a^4 - b^4)*d*e*f^2 - 2*(
a^4 - b^4)*c*f^3 - 2*((a^2*b^2 - b^4)*d*e*f^2 - (a^2*b^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)*d*e
*f^2 - (a^3*b - a*b^3)*c*f^3)*sin(d*x + c) - ((a^3*b + a*b^3)*d^2*e^2*f - 2*(a^3*b + a*b^3)*c*d*e*f^2 + (a^3*b
+ a*b^3)*c^2*f^3 - (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cos(d*x + c)^2 + 2*(a^2*b^2*d^2*e^2*
f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(d*x + c) + 2*I*b*
sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) + 6*(2*(a^4 - b^4)*d*e*f^2 - 2*(a^4 - b^4)*c*f^3 - 2*((a^2*
b^2 - b^4)*d*e*f^2 - (a^2*b^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)*d*e*f^2 - (a^3*b - a*b^3)*c*f^
3)*sin(d*x + c) - ((a^3*b + a*b^3)*d^2*e^2*f - 2*(a^3*b + a*b^3)*c*d*e*f^2 + (a^3*b + a*b^3)*c^2*f^3 - (a*b^3*
d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cos(d*x + c)^2 + 2*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a
^2*b^2*c^2*f^3)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a
^2 - b^2)/b^2) - 2*I*a) + 6*(2*(a^4 - b^4)*d*e*f^2 - 2*(a^4 - b^4)*c*f^3 - 2*((a^2*b^2 - b^4)*d*e*f^2 - (a^2*b
^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)*d*e*f^2 - (a^3*b - a*b^3)*c*f^3)*sin(d*x + c) + ((a^3*b +
a*b^3)*d^2*e^2*f - 2*(a^3*b + a*b^3)*c*d*e*f^2 + (a^3*b + a*b^3)*c^2*f^3 - (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f
^2 + a*b^3*c^2*f^3)*cos(d*x + c)^2 + 2*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*sin(d*x + c
))*sqrt(-(a^2 - b^2)/b^2))*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) +
6*(2*(a^4 - b^4)*d*e*f^2 - 2*(a^4 - b^4)*c*f^3 - 2*((a^2*b^2 - b^4)*d*e*f^2 - (a^2*b^2 - b^4)*c*f^3)*cos(d*x +
c)^2 + 4*((a^3*b - a*b^3)*d*e*f^2 - (a^3*b - a*b^3)*c*f^3)*sin(d*x + c) + ((a^3*b + a*b^3)*d^2*e^2*f - 2*(a^3
*b + a*b^3)*c*d*e*f^2 + (a^3*b + a*b^3)*c^2*f^3 - (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cos(d*
x + c)^2 + 2*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))
*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) + 6*(2*(a^4 - b^4)*d*f^3*x +
2*(a^4 - b^4)*c*f^3 - 2*((a^2*b^2 - b^4)*d*f^3*x + (a^2*b^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)
*d*f^3*x + (a^3*b - a*b^3)*c*f^3)*sin(d*x + c) - ((a^3*b + a*b^3)*d^2*f^3*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f^2*x
+ 2*(a^3*b + a*b^3)*c*d*e*f^2 - (a^3*b + a*b^3)*c^2*f^3 - (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c
*d*e*f^2 - a*b^3*c^2*f^3)*cos(d*x + c)^2 + 2*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^
2 - a^2*b^2*c^2*f^3)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*
(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 6*(2*(a^4 - b^4)*d*f^3*x + 2*(a^4 - b^4
)*c*f^3 - 2*((a^2*b^2 - b^4)*d*f^3*x + (a^2*b^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)*d*f^3*x + (a
^3*b - a*b^3)*c*f^3)*sin(d*x + c) + ((a^3*b + a*b^3)*d^2*f^3*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f^2*x + 2*(a^3*b +
a*b^3)*c*d*e*f^2 - (a^3*b + a*b^3)*c^2*f^3 - (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*
b^3*c^2*f^3)*cos(d*x + c)^2 + 2*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c
^2*f^3)*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x +
c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 6*(2*(a^4 - b^4)*d*f^3*x + 2*(a^4 - b^4)*c*f^3 - 2*(
(a^2*b^2 - b^4)*d*f^3*x + (a^2*b^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)*d*f^3*x + (a^3*b - a*b^3)
*c*f^3)*sin(d*x + c) - ((a^3*b + a*b^3)*d^2*f^3*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f^2*x + 2*(a^3*b + a*b^3)*c*d*e*
f^2 - (a^3*b + a*b^3)*c^2*f^3 - (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*
cos(d*x + c)^2 + 2*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*sin(d
*x + c))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin
(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 6*(2*(a^4 - b^4)*d*f^3*x + 2*(a^4 - b^4)*c*f^3 - 2*((a^2*b^2 - b
^4)*d*f^3*x + (a^2*b^2 - b^4)*c*f^3)*cos(d*x + c)^2 + 4*((a^3*b - a*b^3)*d*f^3*x + (a^3*b - a*b^3)*c*f^3)*sin(
d*x + c) + ((a^3*b + a*b^3)*d^2*f^3*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f^2*x + 2*(a^3*b + a*b^3)*c*d*e*f^2 - (a^3*b
+ a*b^3)*c^2*f^3 - (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cos(d*x + c)
^2 + 2*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*sin(d*x + c))*sqr
t(-(a^2 - b^2)/b^2))*log(1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*s
qrt(-(a^2 - b^2)/b^2) + 2*b)/b))/((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(d*x + c)^2 - 2*(a^5*b^2 - 2*a^3*b^4 + a*
b^6)*d^4*sin(d*x + c) - (a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^4)

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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((f*x+e)**3*cos(d*x+c)/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \cos \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x + e)^3*cos(d*x + c)/(b*sin(d*x + c) + a)^3, x)