3.168 \(\int \frac{(c+d x)^3}{(a+b \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=925 \[ -\frac{6 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right ) f^4}-\frac{6 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right ) f^4}+\frac{6 a \text{PolyLog}\left (4,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right )^{3/2} f^4}-\frac{6 a \text{PolyLog}\left (4,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right )^{3/2} f^4}+\frac{6 i (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right ) f^3}+\frac{6 i (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right ) f^3}-\frac{6 i a (c+d x) \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{6 i a (c+d x) \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right )^{3/2} f^3}-\frac{3 (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right ) f^2}-\frac{3 (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right ) f^2}-\frac{3 a (c+d x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{3 a (c+d x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]

[Out]

(I*(c + d*x)^3)/((a^2 - b^2)*f) - (3*d*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2
- b^2)*f^2) - (I*a*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) -
(3*d*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (I*a*(c + d*x)^3*Lo
g[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + ((6*I)*d^2*(c + d*x)*PolyLog[2, (I
*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) - (3*a*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f
*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + ((6*I)*d^2*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/
(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) + (3*a*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2
- b^2])])/((a^2 - b^2)^(3/2)*f^2) - (6*d^3*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^
2)*f^4) - ((6*I)*a*d^2*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f
^3) - (6*d^3*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^4) + ((6*I)*a*d^2*(c + d*
x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (6*a*d^3*PolyLog[4, (I*b
*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^4) - (6*a*d^3*PolyLog[4, (I*b*E^(I*(e + f*x)))/
(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^4) + (b*(c + d*x)^3*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f
*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.65487, antiderivative size = 925, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3324, 3323, 2264, 2190, 2531, 6609, 2282, 6589, 4519} \[ -\frac{6 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right ) f^4}-\frac{6 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right ) f^4}+\frac{6 a \text{PolyLog}\left (4,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right )^{3/2} f^4}-\frac{6 a \text{PolyLog}\left (4,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^3}{\left (a^2-b^2\right )^{3/2} f^4}+\frac{6 i (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right ) f^3}+\frac{6 i (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right ) f^3}-\frac{6 i a (c+d x) \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{6 i a (c+d x) \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d^2}{\left (a^2-b^2\right )^{3/2} f^3}-\frac{3 (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right ) f^2}-\frac{3 (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right ) f^2}-\frac{3 a (c+d x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{3 a (c+d x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) d}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*(c + d*x)^3)/((a^2 - b^2)*f) - (3*d*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2
- b^2)*f^2) - (I*a*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) -
(3*d*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (I*a*(c + d*x)^3*Lo
g[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + ((6*I)*d^2*(c + d*x)*PolyLog[2, (I
*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) - (3*a*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f
*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + ((6*I)*d^2*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/
(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) + (3*a*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2
- b^2])])/((a^2 - b^2)^(3/2)*f^2) - (6*d^3*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^
2)*f^4) - ((6*I)*a*d^2*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f
^3) - (6*d^3*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^4) + ((6*I)*a*d^2*(c + d*
x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (6*a*d^3*PolyLog[4, (I*b
*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^4) - (6*a*d^3*PolyLog[4, (I*b*E^(I*(e + f*x)))/
(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^4) + (b*(c + d*x)^3*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f
*x]))

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 6609

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

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{(a+b \sin (e+f x))^2} \, dx &=\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{a \int \frac{(c+d x)^3}{a+b \sin (e+f x)} \, dx}{a^2-b^2}-\frac{(3 b d) \int \frac{(c+d x)^2 \cos (e+f x)}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) f}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{(2 a) \int \frac{e^{i (e+f x)} (c+d x)^3}{i b+2 a e^{i (e+f x)}-i b e^{2 i (e+f x)}} \, dx}{a^2-b^2}-\frac{(3 b d) \int \frac{e^{i (e+f x)} (c+d x)^2}{a-\sqrt{a^2-b^2}-i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right ) f}-\frac{(3 b d) \int \frac{e^{i (e+f x)} (c+d x)^2}{a+\sqrt{a^2-b^2}-i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right ) f}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{(2 i a b) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{(2 i a b) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{\left (6 d^2\right ) \int (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}+\frac{\left (6 d^2\right ) \int (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\left (6 i d^3\right ) \int \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^3}-\frac{\left (6 i d^3\right ) \int \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^3}+\frac{(3 i a d) \int (c+d x)^2 \log \left (1-\frac{2 i b e^{i (e+f x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}-\frac{(3 i a d) \int (c+d x)^2 \log \left (1-\frac{2 i b e^{i (e+f x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\left (6 d^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 (e+f x)}\right )}{\left (a^2-b^2\right ) f^4}-\frac{\left (6 d^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 (e+f x)}\right )}{\left (a^2-b^2\right ) f^4}+\frac{\left (6 a d^2\right ) \int (c+d x) \text{Li}_2\left (\frac{2 i b e^{i (e+f x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}-\frac{\left (6 a d^2\right ) \int (c+d x) \text{Li}_2\left (\frac{2 i b e^{i (e+f x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac{6 d^3 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}-\frac{6 i a d^2 (c+d x) \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}-\frac{6 d^3 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}+\frac{6 i a d^2 (c+d x) \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left (6 i a d^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (e+f x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^3}-\frac{\left (6 i a d^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (e+f x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^3}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac{6 d^3 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}-\frac{6 i a d^2 (c+d x) \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}-\frac{6 d^3 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}+\frac{6 i a d^2 (c+d x) \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left (6 a d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^4}-\frac{\left (6 a d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^4}\\ &=\frac{i (c+d x)^3}{\left (a^2-b^2\right ) f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{3 d (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{6 i d^2 (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{3 a d (c+d x)^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac{6 d^3 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}-\frac{6 i a d^2 (c+d x) \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}-\frac{6 d^3 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^4}+\frac{6 i a d^2 (c+d x) \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{6 a d^3 \text{Li}_4\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^4}-\frac{6 a d^3 \text{Li}_4\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^4}+\frac{b (c+d x)^3 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}\\ \end{align*}

Mathematica [A]  time = 3.22821, size = 742, normalized size = 0.8 \[ \frac{-\frac{i a \left (-3 i d \left (f^2 (c+d x)^2 \text{PolyLog}\left (2,-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )+2 i d f (c+d x) \text{PolyLog}\left (3,-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )-2 d^2 \text{PolyLog}\left (4,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )\right )+3 i d \left (f^2 (c+d x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )+2 i d f (c+d x) \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )-2 d^2 \text{PolyLog}\left (4,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )\right )+f^3 (c+d x)^3 \log \left (1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )-f^3 (c+d x)^3 \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )\right )}{\sqrt{a^2-b^2}}+6 i d^2 \left (f (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )+i d \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )\right )+6 i d^2 \left (f (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )+i d \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )\right )-3 d f^2 (c+d x)^2 \log \left (1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )-3 d f^2 (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )+\frac{b f^3 (c+d x)^3 \cos (e+f x)}{a+b \sin (e+f x)}+i f^3 (c+d x)^3}{f^4 \left (a^2-b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*f^3*(c + d*x)^3 - 3*d*f^2*(c + d*x)^2*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - 3*d*f^2*(c +
d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (6*I)*d^2*(f*(c + d*x)*PolyLog[2, (I*b*E^(I*(e +
f*x)))/(a - Sqrt[a^2 - b^2])] + I*d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])]) + (6*I)*d^2*(f*(
c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + I*d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + S
qrt[a^2 - b^2])]) - (I*a*(f^3*(c + d*x)^3*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - f^3*(c + d*x
)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - (3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, ((-I)*b*E^(I*(e
+ f*x)))/(-a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[3, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b
^2])] - 2*d^2*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])]) + (3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, (
I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a
^2 - b^2])] - 2*d^2*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])))/Sqrt[a^2 - b^2] + (b*f^3*(c + d
*x)^3*Cos[e + f*x])/(a + b*Sin[e + f*x]))/((a^2 - b^2)*f^4)

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Maple [F]  time = 1.56, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{3}}{ \left ( a+b\sin \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*x+c)^3/(a+b*sin(f*x+e))^2,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((d*x+c)^3/(a+b*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 7.28598, size = 11266, normalized size = 12.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{3}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^3/(b*sin(f*x + e) + a)^2, x)