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

Optimal. Leaf size=751 \[ \frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d^2 \left (a^2-b^2\right )^{5/2}}-\frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d^2 \left (a^2-b^2\right )^{5/2}}-\frac{3 a f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}+\frac{3 a f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}-\frac{a f}{2 b d^2 \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b d^2 \left (a^2-b^2\right )^2}-\frac{f \log (a+b \sin (c+d x))}{b d^2 \left (a^2-b^2\right )}+\frac{3 i a^3 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d \left (a^2-b^2\right )^{5/2}}-\frac{3 i a^3 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d \left (a^2-b^2\right )^{5/2}}-\frac{3 i a (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d \left (a^2-b^2\right )^{3/2}}+\frac{3 i a (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d \left (a^2-b^2\right )^{3/2}}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac{a (e+f x) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac{(e+f x) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))} \]

[Out]

(((3*I)/2)*a^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) - (((3*
I)/2)*a*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (((3*I)/2)*a
^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*a*(e +
f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*Log[a + b*Sin[c
+ d*x]])/(2*b*(a^2 - b^2)^2*d^2) - (f*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d^2) + (3*a^3*f*PolyLog[2, (I*b
*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*PolyLog[2, (I*b*E^(I*(c + d*x))
)/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) - (3*a^3*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^
2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) + (3*a*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b
*(a^2 - b^2)^(3/2)*d^2) - (a*(e + f*x)*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a*f)/(2*b*(a^
2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) +
((e + f*x)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.95513, antiderivative size = 751, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {6742, 3325, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31, 32} \[ \frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d^2 \left (a^2-b^2\right )^{5/2}}-\frac{3 a^3 f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d^2 \left (a^2-b^2\right )^{5/2}}-\frac{3 a f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}+\frac{3 a f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d^2 \left (a^2-b^2\right )^{3/2}}-\frac{a f}{2 b d^2 \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b d^2 \left (a^2-b^2\right )^2}-\frac{f \log (a+b \sin (c+d x))}{b d^2 \left (a^2-b^2\right )}+\frac{3 i a^3 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d \left (a^2-b^2\right )^{5/2}}-\frac{3 i a^3 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d \left (a^2-b^2\right )^{5/2}}-\frac{3 i a (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b d \left (a^2-b^2\right )^{3/2}}+\frac{3 i a (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{2 b d \left (a^2-b^2\right )^{3/2}}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}-\frac{a (e+f x) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac{(e+f x) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)/2)*a^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) - (((3*
I)/2)*a*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (((3*I)/2)*a
^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*a*(e +
f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*Log[a + b*Sin[c
+ d*x]])/(2*b*(a^2 - b^2)^2*d^2) - (f*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d^2) + (3*a^3*f*PolyLog[2, (I*b
*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*PolyLog[2, (I*b*E^(I*(c + d*x))
)/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) - (3*a^3*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^
2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) + (3*a*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b
*(a^2 - b^2)^(3/2)*d^2) - (a*(e + f*x)*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a*f)/(2*b*(a^
2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) +
((e + f*x)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3325

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

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 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]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

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

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]

Rubi steps

\begin{align*} \int \frac{(e+f x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \left (-\frac{a (e+f x)}{b (a+b \sin (c+d x))^3}+\frac{e+f x}{b (a+b \sin (c+d x))^2}\right ) \, dx\\ &=\frac{\int \frac{e+f x}{(a+b \sin (c+d x))^2} \, dx}{b}-\frac{a \int \frac{e+f x}{(a+b \sin (c+d x))^3} \, dx}{b}\\ &=-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{a \int \frac{(e+f x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )}+\frac{a \int \frac{e+f x}{a+b \sin (c+d x)} \, dx}{b \left (a^2-b^2\right )}-\frac{a^2 \int \frac{e+f x}{(a+b \sin (c+d x))^2} \, dx}{b \left (a^2-b^2\right )}-\frac{f \int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right ) d}+\frac{(a f) \int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right ) d}\\ &=-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a^2 (e+f x) \cos (c+d x)}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{a^3 \int \frac{e+f x}{a+b \sin (c+d x)} \, dx}{b \left (a^2-b^2\right )^2}+\frac{a \int \left (-\frac{a (e+f x)}{b (a+b \sin (c+d x))^2}+\frac{e+f x}{b (a+b \sin (c+d x))}\right ) \, dx}{2 \left (a^2-b^2\right )}+\frac{(2 a) \int \frac{e^{i (c+d x)} (e+f x)}{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 )}-\frac{f \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sin (c+d x)\right )}{b \left (a^2-b^2\right ) d^2}+\frac{(a f) \operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{2 b \left (a^2-b^2\right ) d^2}+\frac{\left (a^2 f\right ) \int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2 d}\\ &=-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{a^2 (e+f x) \cos (c+d x)}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (2 a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)}{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 )^2}-\frac{(2 i a) \int \frac{e^{i (c+d x)} (e+f x)}{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}}+\frac{(2 i a) \int \frac{e^{i (c+d x)} (e+f x)}{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}}+\frac{a \int \frac{e+f x}{a+b \sin (c+d x)} \, dx}{2 b \left (a^2-b^2\right )}-\frac{a^2 \int \frac{e+f x}{(a+b \sin (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}+\frac{\left (a^2 f\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sin (c+d x)\right )}{b \left (a^2-b^2\right )^2 d^2}\\ &=-\frac{i a (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 )^{3/2} d}+\frac{i a (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 )^{3/2} d}+\frac{a^2 f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right )^2 d^2}-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (2 i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{\left (2 i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{a^3 \int \frac{e+f x}{a+b \sin (c+d x)} \, dx}{2 b \left (a^2-b^2\right )^2}+\frac{a \int \frac{e^{i (c+d x)} (e+f x)}{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 )}+\frac{\left (a^2 f\right ) \int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 d}+\frac{(i a f) \int \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}-\frac{(i a f) \int \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}\\ &=\frac{i a^3 (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 )^{5/2} d}-\frac{i a (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 )^{3/2} d}-\frac{i a^3 (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 )^{5/2} d}+\frac{i a (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 )^{3/2} d}+\frac{a^2 f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right )^2 d^2}-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{a^3 \int \frac{e^{i (c+d x)} (e+f x)}{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 )^2}-\frac{(i a) \int \frac{e^{i (c+d x)} (e+f x)}{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}}+\frac{(i a) \int \frac{e^{i (c+d x)} (e+f x)}{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}}+\frac{\left (a^2 f\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sin (c+d x)\right )}{2 b \left (a^2-b^2\right )^2 d^2}+\frac{(a f) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a-2 \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^2}-\frac{(a f) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a+2 \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^2}-\frac{\left (i a^3 f\right ) \int \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 )^{5/2} d}+\frac{\left (i a^3 f\right ) \int \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 )^{5/2} d}\\ &=\frac{i a^3 (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 )^{5/2} d}-\frac{3 i a (e+f x) \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}-\frac{i a^3 (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 )^{5/2} d}+\frac{3 i a (e+f x) \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}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b \left (a^2-b^2\right )^2 d^2}-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}-\frac{a f \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^2}+\frac{a f \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^2}-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{\left (i a^3\right ) \int \frac{e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{5/2}}-\frac{\left (a^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a-2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{\left (a^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a+2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \left (a^2-b^2\right )^{5/2} d^2}+\frac{(i a f) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{2 b \left (a^2-b^2\right )^{3/2} d}-\frac{(i a f) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{2 b \left (a^2-b^2\right )^{3/2} d}\\ &=\frac{3 i a^3 (e+f x) \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 )^{5/2} d}-\frac{3 i a (e+f x) \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}-\frac{3 i a^3 (e+f x) \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 )^{5/2} d}+\frac{3 i a (e+f x) \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}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b \left (a^2-b^2\right )^2 d^2}-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}+\frac{a^3 f \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 )^{5/2} d^2}-\frac{a f \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^2}-\frac{a^3 f \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 )^{5/2} d^2}+\frac{a f \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^2}-\frac{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{(a f) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a-2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{(a f) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a+2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{2 b \left (a^2-b^2\right )^{3/2} d^2}-\frac{\left (i a^3 f\right ) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{2 b \left (a^2-b^2\right )^{5/2} d}+\frac{\left (i a^3 f\right ) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{2 b \left (a^2-b^2\right )^{5/2} d}\\ &=\frac{3 i a^3 (e+f x) \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 )^{5/2} d}-\frac{3 i a (e+f x) \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}-\frac{3 i a^3 (e+f x) \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 )^{5/2} d}+\frac{3 i a (e+f x) \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}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b \left (a^2-b^2\right )^2 d^2}-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}+\frac{a^3 f \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 )^{5/2} d^2}-\frac{3 a f \text{Li}_2\left (\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{a^3 f \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 )^{5/2} d^2}+\frac{3 a f \text{Li}_2\left (\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{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (a^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a-2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{2 b \left (a^2-b^2\right )^{5/2} d^2}+\frac{\left (a^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a+2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{2 b \left (a^2-b^2\right )^{5/2} d^2}\\ &=\frac{3 i a^3 (e+f x) \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 )^{5/2} d}-\frac{3 i a (e+f x) \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}-\frac{3 i a^3 (e+f x) \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 )^{5/2} d}+\frac{3 i a (e+f x) \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}+\frac{3 a^2 f \log (a+b \sin (c+d x))}{2 b \left (a^2-b^2\right )^2 d^2}-\frac{f \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d^2}+\frac{3 a^3 f \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d^2}-\frac{3 a f \text{Li}_2\left (\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 a^3 f \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{2 b \left (a^2-b^2\right )^{5/2} d^2}+\frac{3 a f \text{Li}_2\left (\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{a (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac{a f}{2 b \left (a^2-b^2\right ) d^2 (a+b \sin (c+d x))}-\frac{3 a^2 (e+f x) \cos (c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{(e+f x) \cos (c+d x)}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 15.4607, size = 2408, normalized size = 3.21 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a*d*e*Cos[c + d*x]) + a*c*f*Cos[c + d*x] - a*f*(c + d*x)*Cos[c + d*x])/(2*(a - b)*(a + b)*d^2*(a + b*Sin[c
+ d*x])^2) + (-(a^3*f) + a*b^2*f - a^2*b*d*e*Cos[c + d*x] - 2*b^3*d*e*Cos[c + d*x] + a^2*b*c*f*Cos[c + d*x] +
2*b^3*c*f*Cos[c + d*x] - a^2*b*f*(c + d*x)*Cos[c + d*x] - 2*b^3*f*(c + d*x)*Cos[c + d*x])/(2*(a - b)^2*b*(a +
b)^2*d^2*(a + b*Sin[c + d*x])) + (((-2*(a^2 + 2*b^2)*f*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[
a^2 - b^2] + (2*(a^2*f + 2*b^2*f + a*b*(-3*d*e + 3*c*f))*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqr
t[a^2 - b^2] - ((a^2 + 2*b^2)*f*Log[Sec[(c + d*x)/2]^2])/b + ((a^2 + 2*b^2)*f*Log[Sec[(c + d*x)/2]^2*(a + b*Si
n[c + d*x])])/b + ((3*I)*a*b*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((
-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]))/Sqr
t[-a^2 + b^2] - ((3*I)*a*b*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a
+ b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^
2 + b^2] - ((3*I)*a*b*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a -
b + Sqrt[-a^2 + b^2]))] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^
2] + ((3*I)*a*b*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt
[-a^2 + b^2])] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2])*((
-3*a*b*e)/(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])) + (3*a*b*c*f)/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (3*a
*b*f*(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (a^2*f*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[
c + d*x])) + (b^2*f*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))))/(d*(-(((a^2 + 2*b^2)*f*Tan[(c + d*x
)/2])/b) + ((a^2 + 2*b^2)*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*
Sin[c + d*x])*Tan[(c + d*x)/2]))/(b*(a + b*Sin[c + d*x])) - (a*(a^2 + 2*b^2)*f*Sec[(c + d*x)/2]^2)/((a^2 - b^2
)*(1 + (b + a*Tan[(c + d*x)/2])^2/(a^2 - b^2))) + (a*(a^2*f + 2*b^2*f + a*b*(-3*d*e + 3*c*f))*Sec[(c + d*x)/2]
^2)/((a^2 - b^2)*(1 + (b + a*Tan[(c + d*x)/2])^2/(a^2 - b^2))) - ((3*I)*a*b*f*(((-I/2)*Log[-((b - Sqrt[-a^2 +
b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (Log[
1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(2*(I + Tan[(c + d*x)/2])) +
(a*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2
+ b^2] + ((3*I)*a*b*f*(((I/2)*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*S
ec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - ((I/2)*a*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a
^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^
2)/(2*(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2] + ((3*I)*a*b*f*(((I/2)*Log[1 - (a*(1 - I
*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - ((I/2)*Log[
(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c
+ d*x)/2]) + (a*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(2*(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]
))))/Sqrt[-a^2 + b^2] - ((3*I)*a*b*f*(((-I/2)*Log[1 - (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2
]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + ((I/2)*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*
a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (a*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(
c + d*x)/2]^2)/(2*(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]))))/Sqrt[-a^2 + b^2]))

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Maple [A]  time = 1.699, size = 1084, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*(4*I*b*a^3*d*f*x*exp(I*(d*x+c))+5*I*b^3*a*d*f*x*exp(I*(d*x+c))+4*I*b*a^3*d*e*exp(I*(d*x+c))-3*I*b^3*a*d*e*ex
p(3*I*(d*x+c))+2*a^4*d*f*x*exp(2*I*(d*x+c))+5*b^2*d*f*x*exp(2*I*(d*x+c))*a^2+2*b^4*d*f*x*exp(2*I*(d*x+c))-2*I*
b^2*f*exp(2*I*(d*x+c))*a^2+5*I*b^3*a*d*e*exp(I*(d*x+c))-3*I*b^3*a*d*f*x*exp(3*I*(d*x+c))+2*I*a^4*f*exp(2*I*(d*
x+c))+2*a^4*d*e*exp(2*I*(d*x+c))+b*a^3*f*exp(3*I*(d*x+c))+5*b^2*d*e*exp(2*I*(d*x+c))*a^2-b^3*a*f*exp(3*I*(d*x+
c))+2*b^4*d*e*exp(2*I*(d*x+c))-a^2*b^2*d*f*x-2*b^4*d*f*x-b*a^3*f*exp(I*(d*x+c))-a^2*b^2*d*e+b^3*a*f*exp(I*(d*x
+c))-2*b^4*d*e)/(I*b+2*a*exp(I*(d*x+c))-I*b*exp(2*I*(d*x+c)))^2/(a^2-b^2)^2/d^2/b-1/(-a^2+b^2)^2/d^2/b*a^2*f*l
n(exp(I*(d*x+c)))+1/2/(-a^2+b^2)^2/d^2/b*a^2*f*ln(I*b*exp(2*I*(d*x+c))-2*a*exp(I*(d*x+c))-I*b)-2/(-a^2+b^2)^2/
d^2*b*f*ln(exp(I*(d*x+c)))+1/(-a^2+b^2)^2/d^2*b*f*ln(I*b*exp(2*I*(d*x+c))-2*a*exp(I*(d*x+c))-I*b)+3/2*I/(-a^2+
b^2)^(5/2)/d^2*b*a*f*dilog((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))-3*I/(-a^2+b^2)^(5/2
)/d*b*a*e*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))+3*I/(-a^2+b^2)^(5/2)/d^2*b*a*f*c*arctan(1/2*
(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))-3/2*I/(-a^2+b^2)^(5/2)/d^2*b*a*f*dilog((I*a+b*exp(I*(d*x+c))+(-a^
2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+3/2/(-a^2+b^2)^(5/2)/d*b*a*f*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/
(I*a+(-a^2+b^2)^(1/2)))*x+3/2/(-a^2+b^2)^(5/2)/d^2*b*a*f*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2
+b^2)^(1/2)))*c-3/2/(-a^2+b^2)^(5/2)/d*b*a*f*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2))
)*x-3/2/(-a^2+b^2)^(5/2)/d^2*b*a*f*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*c

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.517, size = 5449, normalized size = 7.26 \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)*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4*((-3*I*a*b^5*f*cos(d*x + c)^2 + 6*I*a^2*b^4*f*sin(d*x + c) + 3*I*(a^3*b^3 + a*b^5)*f)*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) + (3*I*a*b^5*f*cos(d*x + c)^2 - 6*I*a^2*b^4*f*sin(d*x + c) - 3*I*(a^3*b^3 + a*b^5)*f)*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))*s
qrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (3*I*a*b^5*f*cos(d*x + c)^2 - 6*I*a^2*b^4*f*sin(d*x + c) - 3*I*(a^3*b^3
+ a*b^5)*f)*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) + (-3*I*a*b^5*f*cos(d*x + c)^2 + 6*I*a^2*b^4*f*sin(d*x +
c) + 3*I*(a^3*b^3 + a*b^5)*f)*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) + 3*((a^3*b^3 + a*b^5)*d*f*x + (a^3*b^3
+ a*b^5)*c*f - (a*b^5*d*f*x + a*b^5*c*f)*cos(d*x + c)^2 + 2*(a^2*b^4*d*f*x + a^2*b^4*c*f)*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) - 3*((a^3*b^3 + a*b^5)*d*f*x + (a^3*b^3 + a*b^5)*c*f - (a*b^5*d*f*x + a*b^5*c*f)*c
os(d*x + c)^2 + 2*(a^2*b^4*d*f*x + a^2*b^4*c*f)*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) + 3*((a^3*b^3 +
a*b^5)*d*f*x + (a^3*b^3 + a*b^5)*c*f - (a*b^5*d*f*x + a*b^5*c*f)*cos(d*x + c)^2 + 2*(a^2*b^4*d*f*x + a^2*b^4*
c*f)*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) - 3*((a^3*b^3 + a*b^5)*d*f*x + (a^3*b^3 + a*b^5)*c*f - (a
*b^5*d*f*x + a*b^5*c*f)*cos(d*x + c)^2 + 2*(a^2*b^4*d*f*x + a^2*b^4*c*f)*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) + 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*f + 2*((2*a^5*b - a^3*b^3 - a*b^5)*d*f*x + (2*a^5*b - a^3*b^3 - a*b
^5)*d*e)*cos(d*x + c) + ((a^4*b^2 + a^2*b^4 - 2*b^6)*f*cos(d*x + c)^2 - 2*(a^5*b + a^3*b^3 - 2*a*b^5)*f*sin(d*
x + c) - (a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f + 3*((a^3*b^3 + a*b^5)*d*e - (a^3*b^3 + a*b^5)*c*f - (a*b^5*d*e
- a*b^5*c*f)*cos(d*x + c)^2 + 2*(a^2*b^4*d*e - a^2*b^4*c*f)*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) + ((a^4*b^2 + a^2*b^4 - 2*b^6)*f*cos(d*x
+ c)^2 - 2*(a^5*b + a^3*b^3 - 2*a*b^5)*f*sin(d*x + c) - (a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f + 3*((a^3*b^3 +
a*b^5)*d*e - (a^3*b^3 + a*b^5)*c*f - (a*b^5*d*e - a*b^5*c*f)*cos(d*x + c)^2 + 2*(a^2*b^4*d*e - a^2*b^4*c*f)*si
n(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) + ((a^4*b^2 + a^2*b^4 - 2*b^6)*f*cos(d*x + c)^2 - 2*(a^5*b + a^3*b^3 - 2*a*b^5)*f*sin(d*x + c) - (a^6 +
2*a^4*b^2 - a^2*b^4 - 2*b^6)*f - 3*((a^3*b^3 + a*b^5)*d*e - (a^3*b^3 + a*b^5)*c*f - (a*b^5*d*e - a*b^5*c*f)*co
s(d*x + c)^2 + 2*(a^2*b^4*d*e - a^2*b^4*c*f)*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) + ((a^4*b^2 + a^2*b^4 - 2*b^6)*f*cos(d*x + c)^2 - 2*(a^5
*b + a^3*b^3 - 2*a*b^5)*f*sin(d*x + c) - (a^6 + 2*a^4*b^2 - a^2*b^4 - 2*b^6)*f - 3*((a^3*b^3 + a*b^5)*d*e - (a
^3*b^3 + a*b^5)*c*f - (a*b^5*d*e - a*b^5*c*f)*cos(d*x + c)^2 + 2*(a^2*b^4*d*e - a^2*b^4*c*f)*sin(d*x + c))*sqr
t(-(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^
5*b - 2*a^3*b^3 + a*b^5)*f + ((a^4*b^2 + a^2*b^4 - 2*b^6)*d*f*x + (a^4*b^2 + a^2*b^4 - 2*b^6)*d*e)*cos(d*x + c
))*sin(d*x + c))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^2*cos(d*x + c)^2 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*
b^6 - a*b^8)*d^2*sin(d*x + c) - (a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9)*d^2)

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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)*sin(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 )} \sin \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)*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

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