3.3.30 \(\int \frac {(e+f x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) [230]

Optimal. Leaf size=382 \[ \frac {a^2 e x}{b^3}+\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}+\frac {f x^2}{4 b}+\frac {a (e+f x) \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {a^3 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {a^3 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {a f \sin (c+d x)}{b^2 d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f \sin ^2(c+d x)}{4 b d^2} \]

[Out]

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

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Rubi [A]
time = 0.44, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {4611, 3391, 3377, 2717, 3404, 2296, 2221, 2317, 2438} \begin {gather*} \frac {a^3 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2 \sqrt {a^2-b^2}}-\frac {a^3 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^2 \sqrt {a^2-b^2}}+\frac {a^2 e x}{b^3}+\frac {a^2 f x^2}{2 b^3}+\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^3 d \sqrt {a^2-b^2}}-\frac {i a^3 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d \sqrt {a^2-b^2}}-\frac {a f \sin (c+d x)}{b^2 d^2}+\frac {a (e+f x) \cos (c+d x)}{b^2 d}+\frac {f \sin ^2(c+d x)}{4 b d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 b d}+\frac {e x}{2 b}+\frac {f x^2}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2317

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 2438

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 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 3391

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

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]

Rubi steps

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

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Mathematica [A]
time = 4.92, size = 752, normalized size = 1.97 \begin {gather*} -\frac {2 \left (2 a^2+b^2\right ) (c+d x) (c f-d (2 e+f x))-8 a b d (e+f x) \cos (c+d x)+b^2 f \cos (2 (c+d x))+\frac {8 a^3 d (e+f x) \left (\frac {2 (d e-c f) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {-b+\sqrt {-a^2+b^2}-a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )\right )}{\sqrt {-a^2+b^2}}-\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}\right )}{d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}+8 a b f \sin (c+d x)+2 b^2 d (e+f x) \sin (2 (c+d x))}{8 b^3 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/8*(2*(2*a^2 + b^2)*(c + d*x)*(c*f - d*(2*e + f*x)) - 8*a*b*d*(e + f*x)*Cos[c + d*x] + b^2*f*Cos[2*(c + d*x)
] + (8*a^3*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*
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] + (I*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] + (I*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 + Ta
n[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] - (I*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]))/(d*e - c*f + I*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1
+ I*Tan[(c + d*x)/2]]) + 8*a*b*f*Sin[c + d*x] + 2*b^2*d*(e + f*x)*Sin[2*(c + d*x)])/(b^3*d^2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (342 ) = 684\).
time = 0.69, size = 686, normalized size = 1.80

method	result	size
risch	\(\frac {a^{2} f \,x^{2}}{2 b^{3}}+\frac {f \,x^{2}}{4 b}+\frac {a^{2} e x}{b^{3}}+\frac {e x}{2 b}+\frac {a \left (d x f +d e +i f \right ) {\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d^{2}}+\frac {a \left (d x f +d e -i f \right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d^{2}}-\frac {2 i a^{3} e \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (d x +c \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{b^{3} d \sqrt {-a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) x}{b^{3} d \sqrt {-a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) c}{b^{3} d^{2} \sqrt {-a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) x}{b^{3} d \sqrt {-a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) c}{b^{3} d^{2} \sqrt {-a^{2}+b^{2}}}-\frac {i a^{3} f \dilog \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right )}{b^{3} d^{2} \sqrt {-a^{2}+b^{2}}}+\frac {i a^{3} f \dilog \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right )}{b^{3} d^{2} \sqrt {-a^{2}+b^{2}}}+\frac {2 i a^{3} f c \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (d x +c \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{b^{3} d^{2} \sqrt {-a^{2}+b^{2}}}-\frac {f \cos \left (2 d x +2 c \right )}{8 b \,d^{2}}-\frac {\left (f x +e \right ) \sin \left (2 d x +2 c \right )}{4 d b}\)	\(686\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*f*x^2/b^3+1/4*f*x^2/b+a^2*e*x/b^3+1/2*e*x/b+1/2*a*(d*x*f+I*f+d*e)/b^2/d^2*exp(I*(d*x+c))+1/2*a*(d*x*f-
I*f+d*e)/b^2/d^2*exp(-I*(d*x+c))-2*I*a^3/b^3/d*e/(-a^2+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+
b^2)^(1/2))-a^3/b^3/d*f/(-a^2+b^2)^(1/2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*x-
a^3/b^3/d^2*f/(-a^2+b^2)^(1/2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*c+a^3/b^3/d*
f/(-a^2+b^2)^(1/2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x+a^3/b^3/d^2*f/(-a^2+b^
2)^(1/2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c-I*a^3/b^3/d^2*f/(-a^2+b^2)^(1/2)
*dilog((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+I*a^3/b^3/d^2*f/(-a^2+b^2)^(1/2)*dilog(
(I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))+2*I*a^3/b^3/d^2*f*c/(-a^2+b^2)^(1/2)*arctan(1/
2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))-1/8*f/b/d^2*cos(2*d*x+2*c)-1/4*(f*x+e)/d/b*sin(2*d*x+2*c)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1254 vs. \(2 (340) = 680\).
time = 0.56, size = 1254, normalized size = 3.28 \begin {gather*} -\frac {2 i \, a^{3} b f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 i \, a^{3} b f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 i \, a^{3} b f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 i \, a^{3} b f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{2} f x^{2} - 2 \, {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{2} x e + {\left (a^{2} b^{2} - b^{4}\right )} f \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{3} b c f - a^{3} b d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) - 2 \, {\left (a^{3} b c f - a^{3} b d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) + 2 \, {\left (a^{3} b c f - a^{3} b d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) + 2 \, {\left (a^{3} b c f - a^{3} b d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) - 2 \, {\left (a^{3} b d f x + a^{3} b c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, {\left (a^{3} b d f x + a^{3} b c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left (a^{3} b d f x + a^{3} b c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, {\left (a^{3} b d f x + a^{3} b c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \log \left (-\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) - 4 \, {\left ({\left (a^{3} b - a b^{3}\right )} d f x + {\left (a^{3} b - a b^{3}\right )} d e\right )} \cos \left (d x + c\right ) + 2 \, {\left (2 \, {\left (a^{3} b - a b^{3}\right )} f + {\left ({\left (a^{2} b^{2} - b^{4}\right )} d f x + {\left (a^{2} b^{2} - b^{4}\right )} d e\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{2} b^{3} - b^{5}\right )} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*I*a^3*b*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) - 2*I*a^3*b*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) - 2*I*a^3*b*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) + 2*I*a^3*b*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) - (2*a^4 - a^2*b^2 - b^4)*d^2*f*x^2 - 2*(2*a^4 - a
^2*b^2 - b^4)*d^2*x*e + (a^2*b^2 - b^4)*f*cos(d*x + c)^2 - 2*(a^3*b*c*f - a^3*b*d*e)*sqrt(-(a^2 - b^2)/b^2)*lo
g(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^3*b*c*f - a^3*b*d*e)*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^3*b*
c*f - a^3*b*d*e)*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^3*b*c*f - a^3*b*d*e)*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^3*b*d*f*x + a^3*b*c*f)*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^3*b*d*f*x + a^3*b*
c*f)*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^3*b*d*f*x + a^3*b*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(-(-I*a*cos(d*x + c) - a*si
n(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) + 2*(a^3*b*d*f*x + a^3*b*c*f)*
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) - 4*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*d*e)*cos(d*x + c) + 2*(2*(a^3*b - a*b^3)*f
+ ((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*d*e)*cos(d*x + c))*sin(d*x + c))/((a^2*b^3 - b^5)*d^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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