3.2.100 \(\int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [200]

Optimal. Leaf size=287 \[ \frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {4 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]

[Out]

1/4*I*f^2*x/a/d^2-I*(f*x+e)^2/a/d+1/2*I*(f*x+e)^3/a/f+2*f^2*cosh(d*x+c)/a/d^3+(f*x+e)^2*cosh(d*x+c)/a/d+4*I*f*
(f*x+e)*ln(1+I*exp(d*x+c))/a/d^2+4*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3-2*f*(f*x+e)*sinh(d*x+c)/a/d^2-1/4*I*f^
2*cosh(d*x+c)*sinh(d*x+c)/a/d^3-1/2*I*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/a/d+1/2*I*f*(f*x+e)*sinh(d*x+c)^2/a/d^
2-I*(f*x+e)^2*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 = 287, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5676, 3392, 32, 2715, 8, 3377, 2718, 3399, 4269, 3797, 2221, 2317, 2438} \begin {gather*} \frac {4 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \cosh (c+d x)}{a d^3}-\frac {i f^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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]

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 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 2715

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

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 3392

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

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]

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{a}\\ &=-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}+\frac {i \int (e+f x)^2 \, dx}{2 a}+\frac {\int (e+f x)^2 \sinh (c+d x) \, dx}{a}-\frac {\left (i f^2\right ) \int \sinh ^2(c+d x) \, dx}{2 a d^2}-\int \frac {(e+f x)^2 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {i (e+f x)^3}{6 a f}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-i \int \frac {(e+f x)^2}{a+i a \sinh (c+d x)} \, dx+\frac {i \int (e+f x)^2 \, dx}{a}-\frac {(2 f) \int (e+f x) \cosh (c+d x) \, dx}{a d}+\frac {\left (i f^2\right ) \int 1 \, dx}{4 a d^2}\\ &=\frac {i f^2 x}{4 a d^2}+\frac {i (e+f x)^3}{2 a f}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i \int (e+f x)^2 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}+\frac {\left (2 f^2\right ) \int \sinh (c+d x) \, dx}{a d^2}\\ &=\frac {i f^2 x}{4 a d^2}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(2 i f) \int (e+f x) \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}\\ &=\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(4 f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}\\ &=\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (4 i f^2\right ) \int \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (4 i f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^3}\\ &=\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {4 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \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. \(831\) vs. \(2(287)=574\).
time = 4.24, size = 831, normalized size = 2.90 \begin {gather*} \frac {\frac {32 f \left (d \left (-i d e^c x (2 e+f x)+2 \left (1+i e^c\right ) (e+f x) \log \left (1+i e^{c+d x}\right )\right )+2 \left (1+i e^c\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )\right )}{-i+e^c}-\frac {-8 i d \left (3 d^2 e^2 x+f^2 x \left (2+d^2 x^2\right )+e f \left (2+3 d^2 x^2\right )\right ) \cosh \left (\frac {d x}{2}\right )-8 \left (2 f^2+d^2 (e+f x)^2\right ) \cosh \left (c+\frac {d x}{2}\right )-6 d^2 e^2 \cosh \left (c+\frac {3 d x}{2}\right )-15 f^2 \cosh \left (c+\frac {3 d x}{2}\right )-12 d^2 e f x \cosh \left (c+\frac {3 d x}{2}\right )-6 d^2 f^2 x^2 \cosh \left (c+\frac {3 d x}{2}\right )+14 i d e f \cosh \left (2 c+\frac {3 d x}{2}\right )+14 i d f^2 x \cosh \left (2 c+\frac {3 d x}{2}\right )-2 i d e f \cosh \left (2 c+\frac {5 d x}{2}\right )-2 i d f^2 x \cosh \left (2 c+\frac {5 d x}{2}\right )-2 d^2 e^2 \cosh \left (3 c+\frac {5 d x}{2}\right )-f^2 \cosh \left (3 c+\frac {5 d x}{2}\right )-4 d^2 e f x \cosh \left (3 c+\frac {5 d x}{2}\right )-2 d^2 f^2 x^2 \cosh \left (3 c+\frac {5 d x}{2}\right )+40 i d^2 e^2 \sinh \left (\frac {d x}{2}\right )+16 i f^2 \sinh \left (\frac {d x}{2}\right )+80 i d^2 e f x \sinh \left (\frac {d x}{2}\right )+40 i d^2 f^2 x^2 \sinh \left (\frac {d x}{2}\right )+16 d e f \sinh \left (c+\frac {d x}{2}\right )+24 d^3 e^2 x \sinh \left (c+\frac {d x}{2}\right )+16 d f^2 x \sinh \left (c+\frac {d x}{2}\right )+24 d^3 e f x^2 \sinh \left (c+\frac {d x}{2}\right )+8 d^3 f^2 x^3 \sinh \left (c+\frac {d x}{2}\right )+14 d e f \sinh \left (c+\frac {3 d x}{2}\right )+14 d f^2 x \sinh \left (c+\frac {3 d x}{2}\right )-6 i d^2 e^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-15 i f^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-12 i d^2 e f x \sinh \left (2 c+\frac {3 d x}{2}\right )-6 i d^2 f^2 x^2 \sinh \left (2 c+\frac {3 d x}{2}\right )+2 i d^2 e^2 \sinh \left (2 c+\frac {5 d x}{2}\right )+i f^2 \sinh \left (2 c+\frac {5 d x}{2}\right )+4 i d^2 e f x \sinh \left (2 c+\frac {5 d x}{2}\right )+2 i d^2 f^2 x^2 \sinh \left (2 c+\frac {5 d x}{2}\right )+2 d e f \sinh \left (3 c+\frac {5 d x}{2}\right )+2 d f^2 x \sinh \left (3 c+\frac {5 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 )}}{16 a d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((32*f*(d*((-I)*d*E^c*x*(2*e + f*x) + 2*(1 + I*E^c)*(e + f*x)*Log[1 + I*E^(c + d*x)]) + 2*(1 + I*E^c)*f*PolyLo
g[2, (-I)*E^(c + d*x)]))/(-I + E^c) - ((-8*I)*d*(3*d^2*e^2*x + f^2*x*(2 + d^2*x^2) + e*f*(2 + 3*d^2*x^2))*Cosh
[(d*x)/2] - 8*(2*f^2 + d^2*(e + f*x)^2)*Cosh[c + (d*x)/2] - 6*d^2*e^2*Cosh[c + (3*d*x)/2] - 15*f^2*Cosh[c + (3
*d*x)/2] - 12*d^2*e*f*x*Cosh[c + (3*d*x)/2] - 6*d^2*f^2*x^2*Cosh[c + (3*d*x)/2] + (14*I)*d*e*f*Cosh[2*c + (3*d
*x)/2] + (14*I)*d*f^2*x*Cosh[2*c + (3*d*x)/2] - (2*I)*d*e*f*Cosh[2*c + (5*d*x)/2] - (2*I)*d*f^2*x*Cosh[2*c + (
5*d*x)/2] - 2*d^2*e^2*Cosh[3*c + (5*d*x)/2] - f^2*Cosh[3*c + (5*d*x)/2] - 4*d^2*e*f*x*Cosh[3*c + (5*d*x)/2] -
2*d^2*f^2*x^2*Cosh[3*c + (5*d*x)/2] + (40*I)*d^2*e^2*Sinh[(d*x)/2] + (16*I)*f^2*Sinh[(d*x)/2] + (80*I)*d^2*e*f
*x*Sinh[(d*x)/2] + (40*I)*d^2*f^2*x^2*Sinh[(d*x)/2] + 16*d*e*f*Sinh[c + (d*x)/2] + 24*d^3*e^2*x*Sinh[c + (d*x)
/2] + 16*d*f^2*x*Sinh[c + (d*x)/2] + 24*d^3*e*f*x^2*Sinh[c + (d*x)/2] + 8*d^3*f^2*x^3*Sinh[c + (d*x)/2] + 14*d
*e*f*Sinh[c + (3*d*x)/2] + 14*d*f^2*x*Sinh[c + (3*d*x)/2] - (6*I)*d^2*e^2*Sinh[2*c + (3*d*x)/2] - (15*I)*f^2*S
inh[2*c + (3*d*x)/2] - (12*I)*d^2*e*f*x*Sinh[2*c + (3*d*x)/2] - (6*I)*d^2*f^2*x^2*Sinh[2*c + (3*d*x)/2] + (2*I
)*d^2*e^2*Sinh[2*c + (5*d*x)/2] + I*f^2*Sinh[2*c + (5*d*x)/2] + (4*I)*d^2*e*f*x*Sinh[2*c + (5*d*x)/2] + (2*I)*
d^2*f^2*x^2*Sinh[2*c + (5*d*x)/2] + 2*d*e*f*Sinh[3*c + (5*d*x)/2] + 2*d*f^2*x*Sinh[3*c + (5*d*x)/2])/((Cosh[c/
2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])))/(16*a*d^3)

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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. 519 vs. \(2 (257 ) = 514\).
time = 2.60, size = 520, normalized size = 1.81

method	result	size
risch	\(\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {2 i f^{2} x^{2}}{a d}+\frac {4 i f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {2 i f^{2} c^{2}}{a \,d^{3}}+\frac {i f^{2} x^{3}}{2 a}+\frac {\left (f^{2} x^{2} d^{2}+2 d^{2} e f x +d^{2} e^{2}-2 d \,f^{2} x -2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 a \,d^{3}}+\frac {\left (f^{2} x^{2} d^{2}+2 d^{2} e f x +d^{2} e^{2}+2 d \,f^{2} x +2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 a \,d^{3}}+\frac {i e^{3}}{2 a f}+\frac {2 x^{2} f^{2}+4 e f x +2 e^{2}}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {4 i f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {i \left (2 f^{2} x^{2} d^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 d \,f^{2} x +2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{3}}-\frac {4 i f^{2} c x}{a \,d^{2}}-\frac {i \left (2 f^{2} x^{2} d^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 d \,f^{2} x -2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{3}}-\frac {4 i f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}-\frac {4 i \ln \left ({\mathrm e}^{d x +c}\right ) e f}{a \,d^{2}}+\frac {3 i e^{2} x}{2 a}+\frac {4 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e f}{a \,d^{2}}+\frac {3 i f e \,x^{2}}{2 a}\)	\(520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*I/a/d^2*f^2*ln(1+I*exp(d*x+c))*x-2*I/a/d*f^2*x^2-2*I/a/d^3*f^2*c^2+4*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3+1/
2*I/a*f^2*x^3+1/2*(d^2*f^2*x^2+2*d^2*e*f*x+d^2*e^2-2*d*f^2*x-2*d*e*f+2*f^2)/a/d^3*exp(d*x+c)+1/2*(d^2*f^2*x^2+
2*d^2*e*f*x+d^2*e^2+2*d*f^2*x+2*d*e*f+2*f^2)/a/d^3*exp(-d*x-c)+1/2*I/a/f*e^3+2*(f^2*x^2+2*e*f*x+e^2)/d/a/(exp(
d*x+c)-I)+4*I/a/d^3*f^2*c*ln(exp(d*x+c))+4*I/a/d^3*f^2*ln(1+I*exp(d*x+c))*c+1/16*I*(2*d^2*f^2*x^2+4*d^2*e*f*x+
2*d^2*e^2+2*d*f^2*x+2*d*e*f+f^2)/a/d^3*exp(-2*d*x-2*c)-4*I/a/d^2*f^2*c*x-1/16*I*(2*d^2*f^2*x^2+4*d^2*e*f*x+2*d
^2*e^2-2*d*f^2*x-2*d*e*f+f^2)/a/d^3*exp(2*d*x+2*c)-4*I/a/d^3*f^2*c*ln(exp(d*x+c)-I)-4*I/a/d^2*ln(exp(d*x+c))*e
*f+3/2*I/a*e^2*x+4*I/a/d^2*ln(exp(d*x+c)-I)*e*f+3/2*I/a*f*e*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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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. 596 vs. \(2 (252) = 504\).
time = 0.41, size = 596, normalized size = 2.08 \begin {gather*} \frac {2 \, d^{2} f^{2} x^{2} + 2 \, d f^{2} x + 2 \, d^{2} e^{2} + f^{2} - 64 \, {\left (-i \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} - f^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 2 \, {\left (2 \, d^{2} f x + d f\right )} e + {\left (-2 i \, d^{2} f^{2} x^{2} + 2 i \, d f^{2} x - 2 i \, d^{2} e^{2} - i \, f^{2} - 2 \, {\left (2 i \, d^{2} f x - i \, d f\right )} e\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (6 \, d^{2} f^{2} x^{2} - 14 \, d f^{2} x + 6 \, d^{2} e^{2} + 15 \, f^{2} + 2 \, {\left (6 \, d^{2} f x - 7 \, d f\right )} e\right )} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, {\left (-i \, d^{3} f^{2} x^{3} + 5 i \, d^{2} f^{2} x^{2} - 2 i \, d f^{2} x + 2 \, {\left (-2 i \, c^{2} + i\right )} f^{2} + {\left (-3 i \, d^{3} x + i \, d^{2}\right )} e^{2} + {\left (-3 i \, d^{3} f x^{2} + 10 i \, d^{2} f x + 2 \, {\left (4 i \, c - i\right )} d f\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} + 8 \, {\left (d^{3} f^{2} x^{3} + d^{2} f^{2} x^{2} + 2 \, d f^{2} x + 2 \, {\left (2 \, c^{2} + 1\right )} f^{2} + {\left (3 \, d^{3} x + 5 \, d^{2}\right )} e^{2} + {\left (3 \, d^{3} f x^{2} + 2 \, d^{2} f x - 2 \, {\left (4 \, c - 1\right )} d f\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-6 i \, d^{2} f^{2} x^{2} - 14 i \, d f^{2} x - 6 i \, d^{2} e^{2} - 15 i \, f^{2} - 2 \, {\left (6 i \, d^{2} f x + 7 i \, d f\right )} e\right )} e^{\left (d x + c\right )} - 64 \, {\left ({\left (i \, c f^{2} - i \, d f e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (c f^{2} - d f e\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 64 \, {\left ({\left (-i \, d f^{2} x - i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f^{2} x + c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{16 \, {\left (a d^{3} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{3} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*d^2*f^2*x^2 + 2*d*f^2*x + 2*d^2*e^2 + f^2 - 64*(-I*f^2*e^(3*d*x + 3*c) - f^2*e^(2*d*x + 2*c))*dilog(-I
*e^(d*x + c)) + 2*(2*d^2*f*x + d*f)*e + (-2*I*d^2*f^2*x^2 + 2*I*d*f^2*x - 2*I*d^2*e^2 - I*f^2 - 2*(2*I*d^2*f*x
- I*d*f)*e)*e^(5*d*x + 5*c) + (6*d^2*f^2*x^2 - 14*d*f^2*x + 6*d^2*e^2 + 15*f^2 + 2*(6*d^2*f*x - 7*d*f)*e)*e^(
4*d*x + 4*c) - 8*(-I*d^3*f^2*x^3 + 5*I*d^2*f^2*x^2 - 2*I*d*f^2*x + 2*(-2*I*c^2 + I)*f^2 + (-3*I*d^3*x + I*d^2)
*e^2 + (-3*I*d^3*f*x^2 + 10*I*d^2*f*x + 2*(4*I*c - I)*d*f)*e)*e^(3*d*x + 3*c) + 8*(d^3*f^2*x^3 + d^2*f^2*x^2 +
2*d*f^2*x + 2*(2*c^2 + 1)*f^2 + (3*d^3*x + 5*d^2)*e^2 + (3*d^3*f*x^2 + 2*d^2*f*x - 2*(4*c - 1)*d*f)*e)*e^(2*d
*x + 2*c) + (-6*I*d^2*f^2*x^2 - 14*I*d*f^2*x - 6*I*d^2*e^2 - 15*I*f^2 - 2*(6*I*d^2*f*x + 7*I*d*f)*e)*e^(d*x +
c) - 64*((I*c*f^2 - I*d*f*e)*e^(3*d*x + 3*c) + (c*f^2 - d*f*e)*e^(2*d*x + 2*c))*log(e^(d*x + c) - I) - 64*((-I
*d*f^2*x - I*c*f^2)*e^(3*d*x + 3*c) - (d*f^2*x + c*f^2)*e^(2*d*x + 2*c))*log(I*e^(d*x + c) + 1))/(a*d^3*e^(3*d
*x + 3*c) - I*a*d^3*e^(2*d*x + 2*c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 e^{2} + 4 e f x + 2 f^{2} x^{2}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \left (- \frac {i d e^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f^{2} x^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d e^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d e^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 i d e f x}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {4 i d e^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d e^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {16 i e f e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {16 i f^{2} x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d f^{2} x^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {4 i d f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d f^{2} x^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 d e f x e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {8 d e f x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {2 d e f x e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {8 i d e f x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {2 i d e f x e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx\right ) e^{- 2 c}}{4 a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*e**2 + 4*e*f*x + 2*f**2*x**2)/(a*d*exp(c)*exp(d*x) - I*a*d) - I*(Integral(-I*d*e**2/(exp(c)*exp(3*d*x) - I*
exp(2*d*x)), x) + Integral(-I*d*f**2*x**2/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-d*e**2*exp(c)*exp
(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-4*d*e**2*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*e
xp(2*d*x)), x) + Integral(d*e**2*exp(5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-2*I*d*
e*f*x/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*e**2*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*
exp(2*d*x)), x) + Integral(I*d*e**2*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-16*
I*e*f*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-16*I*f**2*x*exp(2*c)*exp(2*d*x)/(
exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-d*f**2*x**2*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x
)), x) + Integral(-4*d*f**2*x**2*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(d*f**2*
x**2*exp(5*c)*exp(5*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(4*I*d*f**2*x**2*exp(2*c)*exp(2*d*x)
/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(I*d*f**2*x**2*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*ex
p(2*d*x)), x) + Integral(-2*d*e*f*x*exp(c)*exp(d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(-8*d*e*f
*x*exp(3*c)*exp(3*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(2*d*e*f*x*exp(5*c)*exp(5*d*x)/(exp(c)
*exp(3*d*x) - I*exp(2*d*x)), x) + Integral(8*I*d*e*f*x*exp(2*c)*exp(2*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)),
x) + Integral(2*I*d*e*f*x*exp(4*c)*exp(4*d*x)/(exp(c)*exp(3*d*x) - I*exp(2*d*x)), x))*exp(-2*c)/(4*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)^2*sinh(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*sinh(d*x + c)^3/(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 )}^3\,{\left (e+f\,x\right )}^2}{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)^3*(e + f*x)^2)/(a + a*sinh(c + d*x)*1i),x)

[Out]

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

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