3.2.93 \(\int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [193]

Optimal. Leaf size=241 \[ -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

-(f*x+e)^3/a/d+1/4*(f*x+e)^4/a/f-6*I*f^2*(f*x+e)*cosh(d*x+c)/a/d^3-I*(f*x+e)^3*cosh(d*x+c)/a/d+6*f*(f*x+e)^2*l
n(1+I*exp(d*x+c))/a/d^2+12*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3-12*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+6*
I*f^3*sinh(d*x+c)/a/d^4+3*I*f*(f*x+e)^2*sinh(d*x+c)/a/d^2-(f*x+e)^3*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d

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Rubi [A]
time = 0.38, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {5676, 3377, 2717, 32, 3399, 4269, 3797, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {12 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e + f*x)^3/(a*d)) + (e + f*x)^4/(4*a*f) - ((6*I)*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) - (I*(e + f*x)^3*Cosh
[c + d*x])/(a*d) + (6*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c +
d*x)])/(a*d^3) - (12*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*f^3*Sinh[c + d*x])/(a*d^4) + ((3*I)*f
*(e + f*x)^2*Sinh[c + d*x])/(a*d^2) - ((e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)

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 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 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 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 5676

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[(e + f*x)^m*(Sinh[c + d*x]^(n
- 1)/(a + b*Sinh[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]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac {(e+f x)^3 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^3 \sinh (c+d x) \, dx}{a}\\ &=-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {\int (e+f x)^3 \, dx}{a}+\frac {(3 i f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a d}-\int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {(e+f x)^4}{4 a f}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {\int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}-\frac {\left (6 i f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a d^2}\\ &=\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 f) \int (e+f x)^2 \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}+\frac {\left (6 i f^3\right ) \int \cosh (c+d x) \, dx}{a d^3}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(6 i f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^3\right ) \int \text {Li}_2\left (-i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(872\) vs. \(2(241)=482\).
time = 4.52, size = 872, normalized size = 3.62 \begin {gather*} \frac {-\frac {8 i f \left (d^2 \left (-i d e^c x \left (3 e^2+3 e f x+f^2 x^2\right )+3 \left (1+i e^c\right ) (e+f x)^2 \log \left (1+i e^{c+d x}\right )\right )+6 d \left (1+i e^c\right ) f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )-6 i \left (-i+e^c\right ) f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )\right )}{-i+e^c}+\frac {\left (12 f^3+6 d^2 f (e+f x)^2+d^4 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right ) \cosh \left (\frac {d x}{2}\right )-2 i d (e+f x) \left (6 f^2+d^2 (e+f x)^2\right ) \cosh \left (c+\frac {d x}{2}\right )-2 i d (e+f x) \left (6 f^2+d^2 (e+f x)^2\right ) \cosh \left (c+\frac {3 d x}{2}\right )-6 d^2 e^2 f \cosh \left (2 c+\frac {3 d x}{2}\right )-12 f^3 \cosh \left (2 c+\frac {3 d x}{2}\right )-12 d^2 e f^2 x \cosh \left (2 c+\frac {3 d x}{2}\right )-6 d^2 f^3 x^2 \cosh \left (2 c+\frac {3 d x}{2}\right )-10 d^3 e^3 \sinh \left (\frac {d x}{2}\right )-12 d e f^2 \sinh \left (\frac {d x}{2}\right )-30 d^3 e^2 f x \sinh \left (\frac {d x}{2}\right )-12 d f^3 x \sinh \left (\frac {d x}{2}\right )-30 d^3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )-10 d^3 f^3 x^3 \sinh \left (\frac {d x}{2}\right )+6 i d^2 e^2 f \sinh \left (c+\frac {d x}{2}\right )+12 i f^3 \sinh \left (c+\frac {d x}{2}\right )+4 i d^4 e^3 x \sinh \left (c+\frac {d x}{2}\right )+12 i d^2 e f^2 x \sinh \left (c+\frac {d x}{2}\right )+6 i d^4 e^2 f x^2 \sinh \left (c+\frac {d x}{2}\right )+6 i d^2 f^3 x^2 \sinh \left (c+\frac {d x}{2}\right )+4 i d^4 e f^2 x^3 \sinh \left (c+\frac {d x}{2}\right )+i d^4 f^3 x^4 \sinh \left (c+\frac {d x}{2}\right )+6 i d^2 e^2 f \sinh \left (c+\frac {3 d x}{2}\right )+12 i f^3 \sinh \left (c+\frac {3 d x}{2}\right )+12 i d^2 e f^2 x \sinh \left (c+\frac {3 d x}{2}\right )+6 i d^2 f^3 x^2 \sinh \left (c+\frac {3 d x}{2}\right )+2 d^3 e^3 \sinh \left (2 c+\frac {3 d x}{2}\right )+12 d e f^2 \sinh \left (2 c+\frac {3 d x}{2}\right )+6 d^3 e^2 f x \sinh \left (2 c+\frac {3 d x}{2}\right )+12 d f^3 x \sinh \left (2 c+\frac {3 d x}{2}\right )+6 d^3 e f^2 x^2 \sinh \left (2 c+\frac {3 d x}{2}\right )+2 d^3 f^3 x^3 \sinh \left (2 c+\frac {3 d x}{2}\right )}{\left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}}{4 a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-8*I)*f*(d^2*((-I)*d*E^c*x*(3*e^2 + 3*e*f*x + f^2*x^2) + 3*(1 + I*E^c)*(e + f*x)^2*Log[1 + I*E^(c + d*x)])
+ 6*d*(1 + I*E^c)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] - (6*I)*(-I + E^c)*f^2*PolyLog[3, (-I)*E^(c + d*x)]
))/(-I + E^c) + ((12*f^3 + 6*d^2*f*(e + f*x)^2 + d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))*Cosh[(d*x)
/2] - (2*I)*d*(e + f*x)*(6*f^2 + d^2*(e + f*x)^2)*Cosh[c + (d*x)/2] - (2*I)*d*(e + f*x)*(6*f^2 + d^2*(e + f*x)
^2)*Cosh[c + (3*d*x)/2] - 6*d^2*e^2*f*Cosh[2*c + (3*d*x)/2] - 12*f^3*Cosh[2*c + (3*d*x)/2] - 12*d^2*e*f^2*x*Co
sh[2*c + (3*d*x)/2] - 6*d^2*f^3*x^2*Cosh[2*c + (3*d*x)/2] - 10*d^3*e^3*Sinh[(d*x)/2] - 12*d*e*f^2*Sinh[(d*x)/2
] - 30*d^3*e^2*f*x*Sinh[(d*x)/2] - 12*d*f^3*x*Sinh[(d*x)/2] - 30*d^3*e*f^2*x^2*Sinh[(d*x)/2] - 10*d^3*f^3*x^3*
Sinh[(d*x)/2] + (6*I)*d^2*e^2*f*Sinh[c + (d*x)/2] + (12*I)*f^3*Sinh[c + (d*x)/2] + (4*I)*d^4*e^3*x*Sinh[c + (d
*x)/2] + (12*I)*d^2*e*f^2*x*Sinh[c + (d*x)/2] + (6*I)*d^4*e^2*f*x^2*Sinh[c + (d*x)/2] + (6*I)*d^2*f^3*x^2*Sinh
[c + (d*x)/2] + (4*I)*d^4*e*f^2*x^3*Sinh[c + (d*x)/2] + I*d^4*f^3*x^4*Sinh[c + (d*x)/2] + (6*I)*d^2*e^2*f*Sinh
[c + (3*d*x)/2] + (12*I)*f^3*Sinh[c + (3*d*x)/2] + (12*I)*d^2*e*f^2*x*Sinh[c + (3*d*x)/2] + (6*I)*d^2*f^3*x^2*
Sinh[c + (3*d*x)/2] + 2*d^3*e^3*Sinh[2*c + (3*d*x)/2] + 12*d*e*f^2*Sinh[2*c + (3*d*x)/2] + 6*d^3*e^2*f*x*Sinh[
2*c + (3*d*x)/2] + 12*d*f^3*x*Sinh[2*c + (3*d*x)/2] + 6*d^3*e*f^2*x^2*Sinh[2*c + (3*d*x)/2] + 2*d^3*f^3*x^3*Si
nh[2*c + (3*d*x)/2])/((Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])))/(4*a*d^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. 698 vs. \(2 (222 ) = 444\).
time = 3.16, size = 699, normalized size = 2.90

method	result	size
risch	\(\frac {f^{2} e \,x^{3}}{a}+\frac {3 f \,e^{2} x^{2}}{2 a}+\frac {e^{3} x}{a}+\frac {f^{3} x^{4}}{4 a}+\frac {e^{4}}{4 a f}+\frac {4 f^{3} c^{3}}{a \,d^{4}}-\frac {2 f^{3} x^{3}}{a d}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 a \,d^{4}}-\frac {2 i \left (f^{3} x^{3}+3 e \,f^{2} x^{2}+3 e^{2} f x +e^{3}\right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {12 f^{3} \polylog \left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {12 f^{2} e \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {6 f \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}-\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 a \,d^{4}}+\frac {12 f^{2} c e \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {12 f^{2} e c x}{a \,d^{2}}-\frac {12 f^{2} c e \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {12 f^{3} \polylog \left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {6 f \ln \left ({\mathrm e}^{d x +c}\right ) e^{2}}{a \,d^{2}}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 f^{3} c^{2} x}{a \,d^{3}}-\frac {6 f^{2} e \,x^{2}}{a d}-\frac {6 f^{2} e \,c^{2}}{a \,d^{3}}\)	\(699\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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. 697 vs. \(2 (221) = 442\).
time = 0.44, size = 697, normalized size = 2.89 \begin {gather*} -\frac {3}{2} \, f {\left (\frac {2 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} + \frac {i \, d^{2} x^{2} e^{c} + i \, d x e^{c} - {\left (-i \, d x e^{\left (3 \, c\right )} + i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} - {\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + {\left (d x + 1\right )} e^{\left (-d x\right )} + i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}} - \frac {4 \, \log \left ({\left (e^{\left (d x + c\right )} - i\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} e^{2} + \frac {1}{2} \, {\left (\frac {2 \, {\left (d x + c\right )}}{a d} + \frac {-5 i \, e^{\left (-d x - c\right )} + 1}{{\left (i \, a e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {i \, e^{\left (-d x - c\right )}}{a d}\right )} e^{3} + \frac {-i \, d^{4} f^{3} x^{4} + 2 \, {\left (-2 i \, d^{4} f^{2} e - 5 i \, d^{3} f^{3}\right )} x^{3} - 12 i \, d f^{2} e - 12 i \, f^{3} + 6 \, {\left (-5 i \, d^{3} f^{2} e - i \, d^{2} f^{3}\right )} x^{2} + 12 \, {\left (-i \, d^{2} f^{2} e - i \, d f^{3}\right )} x + 2 \, {\left (-i \, d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + 6 i \, f^{3} e^{\left (2 \, c\right )} - 6 i \, d f^{2} e^{\left (2 \, c + 1\right )} + 3 \, {\left (i \, d^{2} f^{3} e^{\left (2 \, c\right )} - i \, d^{3} f^{2} e^{\left (2 \, c + 1\right )}\right )} x^{2} + 6 \, {\left (-i \, d f^{3} e^{\left (2 \, c\right )} + i \, d^{2} f^{2} e^{\left (2 \, c + 1\right )}\right )} x\right )} e^{\left (2 \, d x\right )} + {\left (d^{4} f^{3} x^{4} e^{c} + 2 \, {\left (2 \, d^{4} f^{2} e^{\left (c + 1\right )} - d^{3} f^{3} e^{c}\right )} x^{3} - 12 \, d f^{2} e^{\left (c + 1\right )} + 12 \, f^{3} e^{c} - 6 \, {\left (d^{3} f^{2} e^{\left (c + 1\right )} - d^{2} f^{3} e^{c}\right )} x^{2} + 12 \, {\left (d^{2} f^{2} e^{\left (c + 1\right )} - d f^{3} e^{c}\right )} x\right )} e^{\left (d x\right )}}{4 \, {\left (a d^{4} e^{\left (d x + c\right )} - i \, a d^{4}\right )}} + \frac {12 \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} f^{2} e}{a d^{3}} + \frac {6 \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} f^{2} x^{2} e\right )}}{a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*f*(2*x*e^(d*x + c)/(a*d*e^(d*x + c) - I*a*d) + (I*d^2*x^2*e^c + I*d*x*e^c - (-I*d*x*e^(3*c) + I*e^(3*c))*
e^(2*d*x) - (d^2*x^2*e^(2*c) - 3*d*x*e^(2*c) + e^(2*c))*e^(d*x) + (d*x + 1)*e^(-d*x) + I*e^c)/(a*d^2*e^(d*x +
2*c) - I*a*d^2*e^c) - 4*log((e^(d*x + c) - I)*e^(-c))/(a*d^2))*e^2 + 1/2*(2*(d*x + c)/(a*d) + (-5*I*e^(-d*x -
c) + 1)/((I*a*e^(-d*x - c) + a*e^(-2*d*x - 2*c))*d) - I*e^(-d*x - c)/(a*d))*e^3 + 1/4*(-I*d^4*f^3*x^4 + 2*(-2*
I*d^4*f^2*e - 5*I*d^3*f^3)*x^3 - 12*I*d*f^2*e - 12*I*f^3 + 6*(-5*I*d^3*f^2*e - I*d^2*f^3)*x^2 + 12*(-I*d^2*f^2
*e - I*d*f^3)*x + 2*(-I*d^3*f^3*x^3*e^(2*c) + 6*I*f^3*e^(2*c) - 6*I*d*f^2*e^(2*c + 1) + 3*(I*d^2*f^3*e^(2*c) -
I*d^3*f^2*e^(2*c + 1))*x^2 + 6*(-I*d*f^3*e^(2*c) + I*d^2*f^2*e^(2*c + 1))*x)*e^(2*d*x) + (d^4*f^3*x^4*e^c + 2
*(2*d^4*f^2*e^(c + 1) - d^3*f^3*e^c)*x^3 - 12*d*f^2*e^(c + 1) + 12*f^3*e^c - 6*(d^3*f^2*e^(c + 1) - d^2*f^3*e^
c)*x^2 + 12*(d^2*f^2*e^(c + 1) - d*f^3*e^c)*x)*e^(d*x))/(a*d^4*e^(d*x + c) - I*a*d^4) + 12*(d*x*log(I*e^(d*x +
c) + 1) + dilog(-I*e^(d*x + c)))*f^2*e/(a*d^3) + 6*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2*d*x*dilog(-I*e^(d*x +
c)) - 2*polylog(3, -I*e^(d*x + c)))*f^3/(a*d^4) - 2*(d^3*f^3*x^3 + 3*d^3*f^2*x^2*e)/(a*d^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. 825 vs. \(2 (221) = 442\).
time = 0.41, size = 825, normalized size = 3.42 \begin {gather*} -\frac {2 \, d^{3} f^{3} x^{3} + 6 \, d^{2} f^{3} x^{2} + 12 \, d f^{3} x + 2 \, d^{3} e^{3} + 12 \, f^{3} - 48 \, {\left ({\left (d f^{3} x + d f^{2} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d f^{3} x + i \, d f^{2} e\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 6 \, {\left (d^{3} f x + d^{2} f\right )} e^{2} + 6 \, {\left (d^{3} f^{2} x^{2} + 2 \, d^{2} f^{2} x + 2 \, d f^{2}\right )} e + 2 \, {\left (i \, d^{3} f^{3} x^{3} - 3 i \, d^{2} f^{3} x^{2} + 6 i \, d f^{3} x + i \, d^{3} e^{3} - 6 i \, f^{3} + 3 \, {\left (i \, d^{3} f x - i \, d^{2} f\right )} e^{2} + 3 \, {\left (i \, d^{3} f^{2} x^{2} - 2 i \, d^{2} f^{2} x + 2 i \, d f^{2}\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{4} f^{3} x^{4} - 10 \, d^{3} f^{3} x^{3} + 6 \, d^{2} f^{3} x^{2} - 12 \, d f^{3} x - 4 \, {\left (2 \, c^{3} - 3\right )} f^{3} + 2 \, {\left (2 \, d^{4} x - d^{3}\right )} e^{3} + 6 \, {\left (d^{4} f x^{2} - 5 \, d^{3} f x - {\left (4 \, c - 1\right )} d^{2} f\right )} e^{2} + 2 \, {\left (2 \, d^{4} f^{2} x^{3} - 15 \, d^{3} f^{2} x^{2} + 6 \, d^{2} f^{2} x + 6 \, {\left (2 \, c^{2} - 1\right )} d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d^{4} f^{3} x^{4} - 2 i \, d^{3} f^{3} x^{3} - 6 i \, d^{2} f^{3} x^{2} - 12 i \, d f^{3} x - 4 \, {\left (-2 i \, c^{3} + 3 i\right )} f^{3} - 2 \, {\left (2 i \, d^{4} x + 5 i \, d^{3}\right )} e^{3} - 6 \, {\left (i \, d^{4} f x^{2} + i \, d^{3} f x + {\left (-4 i \, c + i\right )} d^{2} f\right )} e^{2} - 2 \, {\left (2 i \, d^{4} f^{2} x^{3} + 3 i \, d^{3} f^{2} x^{2} + 6 i \, d^{2} f^{2} x + 6 \, {\left (2 i \, c^{2} + i\right )} d f^{2}\right )} e\right )} e^{\left (d x + c\right )} - 24 \, {\left ({\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, c^{2} f^{3} - 2 i \, c d f^{2} e + i \, d^{2} f e^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 24 \, {\left ({\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d^{2} f^{3} x^{2} - i \, c^{2} f^{3} + 2 \, {\left (i \, d^{2} f^{2} x + i \, c d f^{2}\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 48 \, {\left (f^{3} e^{\left (2 \, d x + 2 \, c\right )} - i \, f^{3} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{4 \, {\left (a d^{4} e^{\left (2 \, d x + 2 \, c\right )} - i \, a d^{4} e^{\left (d x + c\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*d^3*f^3*x^3 + 6*d^2*f^3*x^2 + 12*d*f^3*x + 2*d^3*e^3 + 12*f^3 - 48*((d*f^3*x + d*f^2*e)*e^(2*d*x + 2*c
) - (I*d*f^3*x + I*d*f^2*e)*e^(d*x + c))*dilog(-I*e^(d*x + c)) + 6*(d^3*f*x + d^2*f)*e^2 + 6*(d^3*f^2*x^2 + 2*
d^2*f^2*x + 2*d*f^2)*e + 2*(I*d^3*f^3*x^3 - 3*I*d^2*f^3*x^2 + 6*I*d*f^3*x + I*d^3*e^3 - 6*I*f^3 + 3*(I*d^3*f*x
- I*d^2*f)*e^2 + 3*(I*d^3*f^2*x^2 - 2*I*d^2*f^2*x + 2*I*d*f^2)*e)*e^(3*d*x + 3*c) - (d^4*f^3*x^4 - 10*d^3*f^3
*x^3 + 6*d^2*f^3*x^2 - 12*d*f^3*x - 4*(2*c^3 - 3)*f^3 + 2*(2*d^4*x - d^3)*e^3 + 6*(d^4*f*x^2 - 5*d^3*f*x - (4*
c - 1)*d^2*f)*e^2 + 2*(2*d^4*f^2*x^3 - 15*d^3*f^2*x^2 + 6*d^2*f^2*x + 6*(2*c^2 - 1)*d*f^2)*e)*e^(2*d*x + 2*c)
- (-I*d^4*f^3*x^4 - 2*I*d^3*f^3*x^3 - 6*I*d^2*f^3*x^2 - 12*I*d*f^3*x - 4*(-2*I*c^3 + 3*I)*f^3 - 2*(2*I*d^4*x +
5*I*d^3)*e^3 - 6*(I*d^4*f*x^2 + I*d^3*f*x + (-4*I*c + I)*d^2*f)*e^2 - 2*(2*I*d^4*f^2*x^3 + 3*I*d^3*f^2*x^2 +
6*I*d^2*f^2*x + 6*(2*I*c^2 + I)*d*f^2)*e)*e^(d*x + c) - 24*((c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*e^(2*d*x + 2*c
) - (I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e^2)*e^(d*x + c))*log(e^(d*x + c) - I) - 24*((d^2*f^3*x^2 - c^2*f^3 +
2*(d^2*f^2*x + c*d*f^2)*e)*e^(2*d*x + 2*c) - (I*d^2*f^3*x^2 - I*c^2*f^3 + 2*(I*d^2*f^2*x + I*c*d*f^2)*e)*e^(d
*x + c))*log(I*e^(d*x + c) + 1) + 48*(f^3*e^(2*d*x + 2*c) - I*f^3*e^(d*x + c))*polylog(3, -I*e^(d*x + c)))/(a*
d^4*e^(2*d*x + 2*c) - I*a*d^4*e^(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*I*e**3 - 6*I*e**2*f*x - 6*I*e*f**2*x**2 - 2*I*f**3*x**3)/(a*d*exp(c)*exp(d*x) - I*a*d) - I*(Integral(I*d*e
**3/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*f**3*x**3/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integ
ral(d*e**3*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*e**3*exp(3*c)*exp(3*d*x)/(exp(c)*
exp(2*d*x) - I*exp(d*x)), x) + Integral(-12*e**2*f*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Inte
gral(-12*f**3*x**2*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*I*d*e*f**2*x**2/(exp(c)*e
xp(2*d*x) - I*exp(d*x)), x) + Integral(3*I*d*e**2*f*x/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*e**3
*exp(2*c)*exp(2*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(d*f**3*x**3*exp(c)*exp(d*x)/(exp(c)*exp(2
*d*x) - I*exp(d*x)), x) + Integral(d*f**3*x**3*exp(3*c)*exp(3*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Inte
gral(-24*e*f**2*x*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(I*d*f**3*x**3*exp(2*c)*exp(2
*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*d*e*f**2*x**2*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*e
xp(d*x)), x) + Integral(3*d*e*f**2*x**2*exp(3*c)*exp(3*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*
d*e**2*f*x*exp(c)*exp(d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*d*e**2*f*x*exp(3*c)*exp(3*d*x)/(e
xp(c)*exp(2*d*x) - I*exp(d*x)), x) + Integral(3*I*d*e*f**2*x**2*exp(2*c)*exp(2*d*x)/(exp(c)*exp(2*d*x) - I*exp
(d*x)), x) + Integral(3*I*d*e**2*f*x*exp(2*c)*exp(2*d*x)/(exp(c)*exp(2*d*x) - I*exp(d*x)), x))*exp(-c)/(2*a*d)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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