\(\int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [224]

Optimal result

Integrand size = 28, antiderivative size = 643 \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {(e+f x)^3 \cos (c+d x)}{b d}-\frac {i a^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {i a^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}-\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^2}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^2}-\frac {6 i a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^3}+\frac {6 i a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^3}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^4}-\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^4}-\frac {6 f^3 \sin (c+d x)}{b d^4}+\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2} \]

[Out]

-1/4*a*(f*x+e)^4/b^2/f+6*f^2*(f*x+e)*cos(d*x+c)/b/d^3-(f*x+e)^3*cos(d*x+c)/b/d-6*f^3*sin(d*x+c)/b/d^4+3*f*(f*x
+e)^2*sin(d*x+c)/b/d^2-I*a^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^2/d/(a^2-b^2)^(1/2)+I*a^
2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^2/d/(a^2-b^2)^(1/2)-3*a^2*f*(f*x+e)^2*polylog(2,I*b
*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^2/d^2/(a^2-b^2)^(1/2)+3*a^2*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a
+(a^2-b^2)^(1/2)))/b^2/d^2/(a^2-b^2)^(1/2)-6*I*a^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)
))/b^2/d^3/(a^2-b^2)^(1/2)+6*I*a^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^2/d^3/(a^2-
b^2)^(1/2)+6*a^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^2/d^4/(a^2-b^2)^(1/2)-6*a^2*f^3*polyl
og(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^2/d^4/(a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {4611, 3377, 2717, 32, 3404, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4 \sqrt {a^2-b^2}}-\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^4 \sqrt {a^2-b^2}}-\frac {6 i a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3 \sqrt {a^2-b^2}}+\frac {6 i a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3 \sqrt {a^2-b^2}}-\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2 \sqrt {a^2-b^2}}+\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2 \sqrt {a^2-b^2}}-\frac {i a^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d \sqrt {a^2-b^2}}+\frac {i a^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d \sqrt {a^2-b^2}}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^3 \sin (c+d x)}{b d^4}+\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac {(e+f x)^3 \cos (c+d x)}{b d} \]

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2320

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 2611

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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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

Rule 3404

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 4611

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

Rule 6724

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 6744

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]

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

Mathematica [A] (verified)

Time = 5.95 (sec) , antiderivative size = 1020, normalized size of antiderivative = 1.59 \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-a d^4 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 b d (e+f x) \left (-6 f^2+d^2 (e+f x)^2\right ) \cos (c+d x)+\frac {4 a^2 \left (2 \sqrt {-a^2+b^2} d^3 e^3 \arctan \left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+\sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-3 \sqrt {a^2-b^2} d^3 e^2 f x \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-3 \sqrt {a^2-b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-\sqrt {a^2-b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-3 i \sqrt {a^2-b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+3 i \sqrt {a^2-b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+6 \sqrt {a^2-b^2} d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+6 \sqrt {a^2-b^2} d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-6 \sqrt {a^2-b^2} d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-6 \sqrt {a^2-b^2} d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )+6 i \sqrt {a^2-b^2} f^3 \operatorname {PolyLog}\left (4,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-6 i \sqrt {a^2-b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )\right )}{\sqrt {-\left (a^2-b^2\right )^2}}+12 b f \left (-2 f^2+d^2 (e+f x)^2\right ) \sin (c+d x)}{4 b^2 d^4} \]

[In]

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

[Out]

(-(a*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)) - 4*b*d*(e + f*x)*(-6*f^2 + d^2*(e + f*x)^2)*Cos[c + d
*x] + (4*a^2*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 3*Sqrt[a^2 - b^2]
*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1
- (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/(
(-I)*a + Sqrt[-a^2 + b^2])] - 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2
])] - 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - Sqrt[a^2 - b^2]*
d^3*f^3*x^3*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2])] - (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*Po
lyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + (3*I)*Sqrt[a^2 - b^2]*d^2*f*(e + f*x)^2*PolyLog[2,
-((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/(
(-I)*a + Sqrt[-a^2 + b^2])] + 6*Sqrt[a^2 - b^2]*d*f^3*x*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b
^2])] - 6*Sqrt[a^2 - b^2]*d*e*f^2*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - 6*Sqrt[a^2 - b
^2]*d*f^3*x*PolyLog[3, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] + (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4,
(b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - (6*I)*Sqrt[a^2 - b^2]*f^3*PolyLog[4, -((b*E^(I*(c + d*x)))
/(I*a + Sqrt[-a^2 + b^2]))]))/Sqrt[-(a^2 - b^2)^2] + 12*b*f*(-2*f^2 + d^2*(e + f*x)^2)*Sin[c + d*x])/(4*b^2*d^
4)

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2671 vs. \(2 (567) = 1134\).

Time = 0.53 (sec) , antiderivative size = 2671, normalized size of antiderivative = 4.15 \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*((a^3 - a*b^2)*d^4*f^3*x^4 + 4*(a^3 - a*b^2)*d^4*e*f^2*x^3 + 6*(a^3 - a*b^2)*d^4*e^2*f*x^2 + 4*(a^3 - a*b
^2)*d^4*e^3*x + 12*I*a^2*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(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*I*a^2*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(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*I*a^2*b
*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-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*I*a^2*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-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) - 6*(I*a^2*b*d^2*f^3*x^2 + 2*I*a^2*b
*d^2*e*f^2*x + I*a^2*b*d^2*e^2*f)*sqrt(-(a^2 - b^2)/b^2)*dilog((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)/b + 1) - 6*(-I*a^2*b*d^2*f^3*x^2 - 2*I*a^2*b*d^2*e*f^2*x
- I*a^2*b*d^2*e^2*f)*sqrt(-(a^2 - b^2)/b^2)*dilog((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)/b + 1) - 6*(-I*a^2*b*d^2*f^3*x^2 - 2*I*a^2*b*d^2*e*f^2*x - I*a^2*b*d
^2*e^2*f)*sqrt(-(a^2 - b^2)/b^2)*dilog((-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)/b + 1) - 6*(I*a^2*b*d^2*f^3*x^2 + 2*I*a^2*b*d^2*e*f^2*x + I*a^2*b*d^2*e^2*f)*sq
rt(-(a^2 - b^2)/b^2)*dilog((-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)/b + 1) - 2*(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)*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) - 2*(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)*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) + 2*(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)*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) + 2*(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)*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) + 2*(a^2*b
*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 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)*sqrt(-(a^2 - b^2)/b^2)*log(-(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)/b) - 2*(a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 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)*sqrt(-(a^2 - b^2)/b^2)*log(-(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)/b) + 2*(a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3
*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 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)*sqrt(-(a^2 - b^2
)/b^2)*log(-(-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)/b) - 2*(a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 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)*sqrt(-(a^2 - b^2)/b^2)*log(-(-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)/b) - 12*(a^2*b*d*f^3*x + a^2*b*d*e*f^2)*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^2*b*d*f^3*x + a^2*b*d*e*f^2)*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^2*b*d*f^3*x + a^2*b*d*e*f^2)*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^2*b*d*f^3*x + a^2*b*d*e*f^2)*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) + 4*((a^2*b - b^3)*d^3*f^3*x^3
+ 3*(a^2*b - b^3)*d^3*e*f^2*x^2 + (a^2*b - b^3)*d^3*e^3 - 6*(a^2*b - b^3)*d*e*f^2 + 3*((a^2*b - b^3)*d^3*e^2*
f - 2*(a^2*b - b^3)*d*f^3)*x)*cos(d*x + c) - 12*((a^2*b - b^3)*d^2*f^3*x^2 + 2*(a^2*b - b^3)*d^2*e*f^2*x + (a^
2*b - b^3)*d^2*e^2*f - 2*(a^2*b - b^3)*f^3)*sin(d*x + c))/((a^2*b^2 - b^4)*d^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sin \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}