3.2.13 \(\int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx\) [113]

Optimal. Leaf size=305 \[ \frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \text {PolyLog}\left (3,-i e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f} \]

[Out]

1/3*(d*x+c)^3/a^2/f-2*d*(d*x+c)^2*ln(1+I*exp(f*x+e))/a^2/f^2+4*d^3*ln(cosh(1/2*e+1/4*I*Pi+1/2*f*x))/a^2/f^4-4*
d^2*(d*x+c)*polylog(2,-I*exp(f*x+e))/a^2/f^3+4*d^3*polylog(3,-I*exp(f*x+e))/a^2/f^4+1/2*d*(d*x+c)^2*sech(1/2*e
+1/4*I*Pi+1/2*f*x)^2/a^2/f^2-2*d^2*(d*x+c)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/f^3+1/3*(d*x+c)^3*tanh(1/2*e+1/4*I
*Pi+1/2*f*x)/a^2/f+1/6*(d*x+c)^3*sech(1/2*e+1/4*I*Pi+1/2*f*x)^2*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/f

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Rubi [A]
time = 0.29, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3399, 4271, 4269, 3556, 3797, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {4 d^2 (c+d x) \text {Li}_2\left (-i e^{e+f x}\right )}{a^2 f^3}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a^2 f^3}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{6 a^2 f}+\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \text {Li}_3\left (-i e^{e+f x}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{a^2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^3/(3*a^2*f) - (2*d*(c + d*x)^2*Log[1 + I*E^(e + f*x)])/(a^2*f^2) + (4*d^3*Log[Cosh[e/2 + (I/4)*Pi +
(f*x)/2]])/(a^2*f^4) - (4*d^2*(c + d*x)*PolyLog[2, (-I)*E^(e + f*x)])/(a^2*f^3) + (4*d^3*PolyLog[3, (-I)*E^(e
+ f*x)])/(a^2*f^4) + (d*(c + d*x)^2*Sech[e/2 + (I/4)*Pi + (f*x)/2]^2)/(2*a^2*f^2) - (2*d^2*(c + d*x)*Tanh[e/2
+ (I/4)*Pi + (f*x)/2])/(a^2*f^3) + ((c + d*x)^3*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sech[
e/2 + (I/4)*Pi + (f*x)/2]^2*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(6*a^2*f)

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 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 3399

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

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]

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx &=\frac {\int (c+d x)^3 \csc ^4\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\int (c+d x)^3 \text {csch}^2\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int (c+d x) \text {csch}^2\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f^2}\\ &=\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (2 d^3\right ) \int \coth \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f^3}-\frac {d \int (c+d x)^2 \coth \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {(2 i d) \int \frac {e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^2\right ) \int (c+d x) \log \left (1+i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {Li}_2\left (-i e^{e+f x}\right )}{a^2 f^3}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^3\right ) \int \text {Li}_2\left (-i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {Li}_2\left (-i e^{e+f x}\right )}{a^2 f^3}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4}\\ &=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \text {Li}_2\left (-i e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \text {Li}_3\left (-i e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}

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Mathematica [A]
time = 4.13, size = 508, normalized size = 1.67 \begin {gather*} \frac {\frac {2 d \left (-6 d^2 e^e f x+3 c^2 e^e f^3 x+3 c d e^e f^3 x^2+d^2 e^e f^3 x^3-6 i d^2 \log \left (i-e^{e+f x}\right )+6 d^2 e^e \log \left (i-e^{e+f x}\right )+3 i c^2 f^2 \log \left (i-e^{e+f x}\right )-3 c^2 e^e f^2 \log \left (i-e^{e+f x}\right )+6 i c d f^2 x \log \left (1+i e^{e+f x}\right )-6 c d e^e f^2 x \log \left (1+i e^{e+f x}\right )+3 i d^2 f^2 x^2 \log \left (1+i e^{e+f x}\right )-3 d^2 e^e f^2 x^2 \log \left (1+i e^{e+f x}\right )-6 d \left (-i+e^e\right ) f (c+d x) \text {PolyLog}\left (2,-i e^{e+f x}\right )+6 d^2 \left (-i+e^e\right ) \text {PolyLog}\left (3,-i e^{e+f x}\right )\right )}{-i+e^e}+\frac {f (c+d x) \left (3 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+6 i d^2 \cosh \left (e+\frac {f x}{2}\right )+i \left (c^2 f^2+2 c d f^2 x+d^2 \left (-6+f^2 x^2\right )\right ) \cosh \left (e+\frac {3 f x}{2}\right )+3 \left (c^2 f^2+2 c d f^2 x+d^2 \left (-4+f^2 x^2\right )\right ) \sinh \left (\frac {f x}{2}\right )+3 i d f (c+d x) \sinh \left (e+\frac {f x}{2}\right )\right )}{\left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3}}{3 a^2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*d*(-6*d^2*E^e*f*x + 3*c^2*E^e*f^3*x + 3*c*d*E^e*f^3*x^2 + d^2*E^e*f^3*x^3 - (6*I)*d^2*Log[I - E^(e + f*x)]
+ 6*d^2*E^e*Log[I - E^(e + f*x)] + (3*I)*c^2*f^2*Log[I - E^(e + f*x)] - 3*c^2*E^e*f^2*Log[I - E^(e + f*x)] +
(6*I)*c*d*f^2*x*Log[1 + I*E^(e + f*x)] - 6*c*d*E^e*f^2*x*Log[1 + I*E^(e + f*x)] + (3*I)*d^2*f^2*x^2*Log[1 + I*
E^(e + f*x)] - 3*d^2*E^e*f^2*x^2*Log[1 + I*E^(e + f*x)] - 6*d*(-I + E^e)*f*(c + d*x)*PolyLog[2, (-I)*E^(e + f*
x)] + 6*d^2*(-I + E^e)*PolyLog[3, (-I)*E^(e + f*x)]))/(-I + E^e) + (f*(c + d*x)*(3*d*f*(c + d*x)*Cosh[(f*x)/2]
+ (6*I)*d^2*Cosh[e + (f*x)/2] + I*(c^2*f^2 + 2*c*d*f^2*x + d^2*(-6 + f^2*x^2))*Cosh[e + (3*f*x)/2] + 3*(c^2*f
^2 + 2*c*d*f^2*x + d^2*(-4 + f^2*x^2))*Sinh[(f*x)/2] + (3*I)*d*f*(c + d*x)*Sinh[e + (f*x)/2]))/((Cosh[e/2] + I
*Sinh[e/2])*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3))/(3*a^2*f^4)

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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. 722 vs. \(2 (249 ) = 498\).
time = 3.25, size = 723, normalized size = 2.37

method	result	size
risch	\(\frac {2 f^{2} d^{3} x^{3} {\mathrm e}^{f x +e}-2 f \,d^{3} x^{2} {\mathrm e}^{f x +e}-2 f \,c^{2} d \,{\mathrm e}^{f x +e}-4 i d^{3} x \,{\mathrm e}^{2 f x +2 e}+4 i d^{3} x +4 i c \,d^{2}+6 f^{2} c \,d^{2} x^{2} {\mathrm e}^{f x +e}+6 f^{2} c^{2} d x \,{\mathrm e}^{f x +e}-4 f c \,d^{2} x \,{\mathrm e}^{f x +e}-4 i f c \,d^{2} x \,{\mathrm e}^{2 f x +2 e}-\frac {2 i f^{2} c^{3}}{3}-2 i f^{2} c \,d^{2} x^{2}-2 i f^{2} c^{2} d x -2 i f \,c^{2} d \,{\mathrm e}^{2 f x +2 e}-4 i c \,d^{2} {\mathrm e}^{2 f x +2 e}-8 d^{3} x \,{\mathrm e}^{f x +e}-8 c \,d^{2} {\mathrm e}^{f x +e}-2 i f \,d^{3} x^{2} {\mathrm e}^{2 f x +2 e}+2 f^{2} c^{3} {\mathrm e}^{f x +e}-\frac {2 i f^{2} d^{3} x^{3}}{3}}{\left ({\mathrm e}^{f x +e}-i\right )^{3} f^{3} a^{2}}+\frac {2 d^{2} c \,x^{2}}{a^{2} f}+\frac {2 d^{2} c \,e^{2}}{a^{2} f^{3}}+\frac {2 d^{3} x^{3}}{3 a^{2} f}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}-i\right ) c^{2}}{a^{2} f^{2}}+\frac {4 d^{2} c e x}{a^{2} f^{2}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}\right ) c e}{a^{2} f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}-i\right ) c e}{a^{2} f^{3}}+\frac {2 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) e^{2}}{a^{2} f^{4}}-\frac {2 d^{3} e^{2} x}{a^{2} f^{3}}-\frac {4 d^{2} \ln \left (1+i {\mathrm e}^{f x +e}\right ) c x}{a^{2} f^{2}}-\frac {4 d^{2} \ln \left (1+i {\mathrm e}^{f x +e}\right ) c e}{a^{2} f^{3}}+\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right ) c^{2}}{a^{2} f^{2}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}-i\right ) e^{2}}{a^{2} f^{4}}+\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}\right ) e^{2}}{a^{2} f^{4}}-\frac {4 d^{2} c \polylog \left (2, -i {\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}-\frac {4 d^{3} \polylog \left (2, -i {\mathrm e}^{f x +e}\right ) x}{a^{2} f^{3}}-\frac {2 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) x^{2}}{a^{2} f^{2}}-\frac {4 d^{3} e^{3}}{3 a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}-i\right )}{a^{2} f^{4}}+\frac {4 d^{3} \polylog \left (3, -i {\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}\)	\(723\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+I*a*sinh(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/3*(3*f^2*d^3*x^3*exp(f*x+e)-3*f*d^3*x^2*exp(f*x+e)-3*f*c^2*d*exp(f*x+e)-6*I*d^3*x*exp(2*f*x+2*e)+6*I*d^3*x+6
*I*c*d^2+9*f^2*c*d^2*x^2*exp(f*x+e)+9*f^2*c^2*d*x*exp(f*x+e)-6*f*c*d^2*x*exp(f*x+e)-6*I*f*c*d^2*x*exp(2*f*x+2*
e)-I*f^2*c^3-3*I*f^2*c*d^2*x^2-3*I*f^2*c^2*d*x-3*I*f*c^2*d*exp(2*f*x+2*e)-6*I*c*d^2*exp(2*f*x+2*e)-12*d^3*x*ex
p(f*x+e)-12*c*d^2*exp(f*x+e)-3*I*f*d^3*x^2*exp(2*f*x+2*e)+3*f^2*c^3*exp(f*x+e)-I*f^2*d^3*x^3)/(exp(f*x+e)-I)^3
/f^3/a^2+2/a^2/f*d^2*c*x^2+2/a^2/f^3*d^2*c*e^2+2/3/a^2/f*d^3*x^3-2/a^2/f^2*d*ln(exp(f*x+e)-I)*c^2+4/a^2/f^2*d^
2*c*e*x-4/a^2/f^3*d^2*ln(exp(f*x+e))*c*e+4/a^2/f^3*d^2*ln(exp(f*x+e)-I)*c*e+2/a^2/f^4*d^3*ln(1+I*exp(f*x+e))*e
^2-2/a^2/f^3*d^3*e^2*x-4/a^2/f^2*d^2*ln(1+I*exp(f*x+e))*c*x-4/a^2/f^3*d^2*ln(1+I*exp(f*x+e))*c*e+2/a^2/f^2*d*l
n(exp(f*x+e))*c^2-2/a^2/f^4*d^3*ln(exp(f*x+e)-I)*e^2+2/a^2/f^4*d^3*ln(exp(f*x+e))*e^2-4/a^2/f^3*d^2*c*polylog(
2,-I*exp(f*x+e))-4/a^2/f^3*d^3*polylog(2,-I*exp(f*x+e))*x-2/a^2/f^2*d^3*ln(1+I*exp(f*x+e))*x^2-4/3/a^2/f^4*d^3
*e^3-4/a^2/f^4*d^3*ln(exp(f*x+e))+4/a^2/f^4*d^3*ln(exp(f*x+e)-I)+4*d^3*polylog(3,-I*exp(f*x+e))/a^2/f^4

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (248) = 496\).
time = 0.46, size = 659, normalized size = 2.16 \begin {gather*} 2 \, c^{2} d {\left (\frac {f x e^{\left (3 \, f x + 3 \, e\right )} - {\left (3 i \, f x e^{\left (2 \, e\right )} + i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + i \, a^{2} f^{2}} - \frac {\log \left (-i \, {\left (i \, e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac {2}{3} \, c^{3} {\left (\frac {3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} - 3 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + i \, a^{2}\right )} f} + \frac {i}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} - 3 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + i \, a^{2}\right )} f}\right )} - \frac {2 \, {\left (i \, d^{3} f^{2} x^{3} + 3 i \, c d^{2} f^{2} x^{2} - 6 i \, d^{3} x - 6 i \, c d^{2} - 3 \, {\left (-i \, d^{3} f x^{2} e^{\left (2 \, e\right )} - 2 i \, c d^{2} e^{\left (2 \, e\right )} + 2 \, {\left (-i \, c d^{2} f - i \, d^{3}\right )} x e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 3 \, {\left (d^{3} f^{2} x^{3} e^{e} - 4 \, c d^{2} e^{e} + {\left (3 \, c d^{2} f^{2} - d^{3} f\right )} x^{2} e^{e} - 2 \, {\left (c d^{2} f + 2 \, d^{3}\right )} x e^{e}\right )} e^{\left (f x\right )}\right )}}{3 \, {\left (a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{3} e^{\left (f x + e\right )} + i \, a^{2} f^{3}\right )}} - \frac {4 \, {\left (f x \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a^{2} f^{3}} - \frac {4 \, d^{3} x}{a^{2} f^{3}} - \frac {2 \, {\left (f^{2} x^{2} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (f x + e\right )})\right )} d^{3}}{a^{2} f^{4}} + \frac {4 \, d^{3} \log \left (e^{\left (f x + e\right )} - i\right )}{a^{2} f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{3 \, a^{2} f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c^2*d*((f*x*e^(3*f*x + 3*e) - (3*I*f*x*e^(2*e) + I*e^(2*e))*e^(2*f*x) - e^(f*x + e))/(a^2*f^2*e^(3*f*x + 3*e
) - 3*I*a^2*f^2*e^(2*f*x + 2*e) - 3*a^2*f^2*e^(f*x + e) + I*a^2*f^2) - log(-I*(I*e^(f*x + e) + 1)*e^(-e))/(a^2
*f^2)) + 2/3*c^3*(3*e^(-f*x - e)/((3*a^2*e^(-f*x - e) - 3*I*a^2*e^(-2*f*x - 2*e) - a^2*e^(-3*f*x - 3*e) + I*a^
2)*f) + I/((3*a^2*e^(-f*x - e) - 3*I*a^2*e^(-2*f*x - 2*e) - a^2*e^(-3*f*x - 3*e) + I*a^2)*f)) - 2/3*(I*d^3*f^2
*x^3 + 3*I*c*d^2*f^2*x^2 - 6*I*d^3*x - 6*I*c*d^2 - 3*(-I*d^3*f*x^2*e^(2*e) - 2*I*c*d^2*e^(2*e) + 2*(-I*c*d^2*f
- I*d^3)*x*e^(2*e))*e^(2*f*x) - 3*(d^3*f^2*x^3*e^e - 4*c*d^2*e^e + (3*c*d^2*f^2 - d^3*f)*x^2*e^e - 2*(c*d^2*f
+ 2*d^3)*x*e^e)*e^(f*x))/(a^2*f^3*e^(3*f*x + 3*e) - 3*I*a^2*f^3*e^(2*f*x + 2*e) - 3*a^2*f^3*e^(f*x + e) + I*a
^2*f^3) - 4*(f*x*log(I*e^(f*x + e) + 1) + dilog(-I*e^(f*x + e)))*c*d^2/(a^2*f^3) - 4*d^3*x/(a^2*f^3) - 2*(f^2*
x^2*log(I*e^(f*x + e) + 1) + 2*f*x*dilog(-I*e^(f*x + e)) - 2*polylog(3, -I*e^(f*x + e)))*d^3/(a^2*f^4) + 4*d^3
*log(e^(f*x + e) - I)/(a^2*f^4) + 2/3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2)/(a^2*f^4)

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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. 938 vs. \(2 (248) = 496\).
time = 0.34, size = 938, normalized size = 3.08 \begin {gather*} -\frac {2 \, {\left (i \, c^{3} f^{3} + 3 i \, c d^{2} f e^{2} - 6 i \, c d^{2} f - i \, d^{3} e^{3} + 6 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (d^{3} f x + c d^{2} f\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{\left (f x + e\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) + 3 \, {\left (-i \, c^{2} d f^{2} + 2 i \, d^{3}\right )} e - {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} - 3 \, c d^{2} f e^{2} + d^{3} e^{3} + 3 \, {\left (c^{2} d f^{3} - 2 \, d^{3} f\right )} x + 3 \, {\left (c^{2} d f^{2} - 2 \, d^{3}\right )} e\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (i \, d^{3} f^{3} x^{3} + i \, c^{2} d f^{2} - 3 i \, c d^{2} f e^{2} + 2 i \, c d^{2} f + i \, d^{3} e^{3} + {\left (3 i \, c d^{2} f^{3} + i \, d^{3} f^{2}\right )} x^{2} + {\left (3 i \, c^{2} d f^{3} + 2 i \, c d^{2} f^{2} - 4 i \, d^{3} f\right )} x + 3 \, {\left (i \, c^{2} d f^{2} - 2 i \, d^{3}\right )} e\right )} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, {\left (d^{3} f^{2} x^{2} - c^{3} f^{3} + c^{2} d f^{2} - 3 \, c d^{2} f e^{2} + 4 \, c d^{2} f + d^{3} e^{3} + 2 \, {\left (c d^{2} f^{2} - d^{3} f\right )} x + 3 \, {\left (c^{2} d f^{2} - 2 \, d^{3}\right )} e\right )} e^{\left (f x + e\right )} + 3 \, {\left (i \, c^{2} d f^{2} - 2 i \, c d^{2} f e + i \, d^{3} e^{2} - 2 i \, d^{3} + {\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2} - 2 \, d^{3}\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, c^{2} d f^{2} + 2 i \, c d^{2} f e - i \, d^{3} e^{2} + 2 i \, d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2} - 2 \, d^{3}\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 3 \, {\left (i \, d^{3} f^{2} x^{2} + 2 i \, c d^{2} f^{2} x + 2 i \, c d^{2} f e - i \, d^{3} e^{2} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + 2 \, c d^{2} f e - d^{3} e^{2}\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, d^{3} f^{2} x^{2} - 2 i \, c d^{2} f^{2} x - 2 i \, c d^{2} f e + i \, d^{3} e^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + 2 \, c d^{2} f e - d^{3} e^{2}\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) - 6 \, {\left (d^{3} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{3} e^{\left (f x + e\right )} + i \, d^{3}\right )} {\rm polylog}\left (3, -i \, e^{\left (f x + e\right )}\right )\right )}}{3 \, {\left (a^{2} f^{4} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{4} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{4} e^{\left (f x + e\right )} + i \, a^{2} f^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(I*c^3*f^3 + 3*I*c*d^2*f*e^2 - 6*I*c*d^2*f - I*d^3*e^3 + 6*(I*d^3*f*x + I*c*d^2*f + (d^3*f*x + c*d^2*f)*e
^(3*f*x + 3*e) + 3*(-I*d^3*f*x - I*c*d^2*f)*e^(2*f*x + 2*e) - 3*(d^3*f*x + c*d^2*f)*e^(f*x + e))*dilog(-I*e^(f
*x + e)) + 3*(-I*c^2*d*f^2 + 2*I*d^3)*e - (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 - 3*c*d^2*f*e^2 + d^3*e^3 + 3*(c^2*d*
f^3 - 2*d^3*f)*x + 3*(c^2*d*f^2 - 2*d^3)*e)*e^(3*f*x + 3*e) + 3*(I*d^3*f^3*x^3 + I*c^2*d*f^2 - 3*I*c*d^2*f*e^2
+ 2*I*c*d^2*f + I*d^3*e^3 + (3*I*c*d^2*f^3 + I*d^3*f^2)*x^2 + (3*I*c^2*d*f^3 + 2*I*c*d^2*f^2 - 4*I*d^3*f)*x +
3*(I*c^2*d*f^2 - 2*I*d^3)*e)*e^(2*f*x + 2*e) + 3*(d^3*f^2*x^2 - c^3*f^3 + c^2*d*f^2 - 3*c*d^2*f*e^2 + 4*c*d^2
*f + d^3*e^3 + 2*(c*d^2*f^2 - d^3*f)*x + 3*(c^2*d*f^2 - 2*d^3)*e)*e^(f*x + e) + 3*(I*c^2*d*f^2 - 2*I*c*d^2*f*e
+ I*d^3*e^2 - 2*I*d^3 + (c^2*d*f^2 - 2*c*d^2*f*e + d^3*e^2 - 2*d^3)*e^(3*f*x + 3*e) + 3*(-I*c^2*d*f^2 + 2*I*c
*d^2*f*e - I*d^3*e^2 + 2*I*d^3)*e^(2*f*x + 2*e) - 3*(c^2*d*f^2 - 2*c*d^2*f*e + d^3*e^2 - 2*d^3)*e^(f*x + e))*l
og(e^(f*x + e) - I) + 3*(I*d^3*f^2*x^2 + 2*I*c*d^2*f^2*x + 2*I*c*d^2*f*e - I*d^3*e^2 + (d^3*f^2*x^2 + 2*c*d^2*
f^2*x + 2*c*d^2*f*e - d^3*e^2)*e^(3*f*x + 3*e) + 3*(-I*d^3*f^2*x^2 - 2*I*c*d^2*f^2*x - 2*I*c*d^2*f*e + I*d^3*e
^2)*e^(2*f*x + 2*e) - 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*e - d^3*e^2)*e^(f*x + e))*log(I*e^(f*x + e) +
1) - 6*(d^3*e^(3*f*x + 3*e) - 3*I*d^3*e^(2*f*x + 2*e) - 3*d^3*e^(f*x + e) + I*d^3)*polylog(3, -I*e^(f*x + e))
)/(a^2*f^4*e^(3*f*x + 3*e) - 3*I*a^2*f^4*e^(2*f*x + 2*e) - 3*a^2*f^4*e^(f*x + e) + I*a^2*f^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {- 2 i c^{3} f^{2} - 6 i c^{2} d f^{2} x - 6 i c d^{2} f^{2} x^{2} + 12 i c d^{2} - 2 i d^{3} f^{2} x^{3} + 12 i d^{3} x + \left (- 6 i c^{2} d f e^{2 e} - 12 i c d^{2} f x e^{2 e} - 12 i c d^{2} e^{2 e} - 6 i d^{3} f x^{2} e^{2 e} - 12 i d^{3} x e^{2 e}\right ) e^{2 f x} + \left (6 c^{3} f^{2} e^{e} + 18 c^{2} d f^{2} x e^{e} - 6 c^{2} d f e^{e} + 18 c d^{2} f^{2} x^{2} e^{e} - 12 c d^{2} f x e^{e} - 24 c d^{2} e^{e} + 6 d^{3} f^{2} x^{3} e^{e} - 6 d^{3} f x^{2} e^{e} - 24 d^{3} x e^{e}\right ) e^{f x}}{3 a^{2} f^{3} e^{3 e} e^{3 f x} - 9 i a^{2} f^{3} e^{2 e} e^{2 f x} - 9 a^{2} f^{3} e^{e} e^{f x} + 3 i a^{2} f^{3}} - \frac {2 i d \left (\int \left (- \frac {2 d^{2}}{e^{e} e^{f x} - i}\right )\, dx + \int \frac {c^{2} f^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {d^{2} f^{2} x^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {2 c d f^{2} x}{e^{e} e^{f x} - i}\, dx\right )}{a^{2} f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+I*a*sinh(f*x+e))**2,x)

[Out]

(-2*I*c**3*f**2 - 6*I*c**2*d*f**2*x - 6*I*c*d**2*f**2*x**2 + 12*I*c*d**2 - 2*I*d**3*f**2*x**3 + 12*I*d**3*x +
(-6*I*c**2*d*f*exp(2*e) - 12*I*c*d**2*f*x*exp(2*e) - 12*I*c*d**2*exp(2*e) - 6*I*d**3*f*x**2*exp(2*e) - 12*I*d*
*3*x*exp(2*e))*exp(2*f*x) + (6*c**3*f**2*exp(e) + 18*c**2*d*f**2*x*exp(e) - 6*c**2*d*f*exp(e) + 18*c*d**2*f**2
*x**2*exp(e) - 12*c*d**2*f*x*exp(e) - 24*c*d**2*exp(e) + 6*d**3*f**2*x**3*exp(e) - 6*d**3*f*x**2*exp(e) - 24*d
**3*x*exp(e))*exp(f*x))/(3*a**2*f**3*exp(3*e)*exp(3*f*x) - 9*I*a**2*f**3*exp(2*e)*exp(2*f*x) - 9*a**2*f**3*exp
(e)*exp(f*x) + 3*I*a**2*f**3) - 2*I*d*(Integral(-2*d**2/(exp(e)*exp(f*x) - I), x) + Integral(c**2*f**2/(exp(e)
*exp(f*x) - I), x) + Integral(d**2*f**2*x**2/(exp(e)*exp(f*x) - I), x) + Integral(2*c*d*f**2*x/(exp(e)*exp(f*x
) - I), x))/(a**2*f**3)

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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((d*x+c)^3/(a+I*a*sinh(f*x+e))^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + a*sinh(e + f*x)*1i)^2,x)

[Out]

int((c + d*x)^3/(a + a*sinh(e + f*x)*1i)^2, x)

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