3.114 \(\int \frac{(c+d x)^2}{(a+i a \sinh (e+f x))^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.277598, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3318, 4186, 3767, 8, 4184, 3716, 2190, 2279, 2391} \[ -\frac{4 d^2 \text{PolyLog}\left (2,-i e^{e+f x}\right )}{3 a^2 f^3}-\frac{4 d (c+d x) \log \left (1+i e^{e+f x}\right )}{3 a^2 f^2}+\frac{d (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f^2}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{6 a^2 f}+\frac{(c+d x)^2}{3 a^2 f}-\frac{2 d^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right )}{3 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3318

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*(e + (Pi*a)/(2*b)))/2 + (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 4186

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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rule 4184

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 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [A]  time = 3.55659, size = 269, normalized size = 1.12 \[ \frac{4 d^2 \text{PolyLog}\left (2,i e^{-e-f x}\right )+\frac{i \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \cosh \left (e+\frac{3 f x}{2}\right )+\sinh \left (\frac{f x}{2}\right ) \left (3 c^2 f^2+6 c d f^2 x+d^2 \left (3 f^2 x^2-4\right )\right )+2 i d f (c+d x) \sinh \left (e+\frac{f x}{2}\right )+2 d f (c+d x) \cosh \left (\frac{f x}{2}\right )+2 i d^2 \cosh \left (e+\frac{f x}{2}\right )}{\left (\cosh \left (\frac{e}{2}\right )+i \sinh \left (\frac{e}{2}\right )\right ) \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{2 i f (c+d x) \left (f (c+d x)+2 d \left (1+i e^e\right ) \log \left (1-i e^{-e-f x}\right )\right )}{e^e-i}}{3 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.113, size = 374, normalized size = 1.6 \begin{align*}{\frac{-2\,i{f}^{2}{d}^{2}{x}^{2}-4\,f{d}^{2}x{{\rm e}^{fx+e}}-4\,fcd{{\rm e}^{fx+e}}-8\,{d}^{2}{{\rm e}^{fx+e}}+4\,i{d}^{2}-4\,i{d}^{2}{{\rm e}^{2\,fx+2\,e}}-2\,i{f}^{2}{c}^{2}-4\,i{f}^{2}cdx-4\,if{d}^{2}x{{\rm e}^{2\,fx+2\,e}}-4\,ifcd{{\rm e}^{2\,fx+2\,e}}+6\,{f}^{2}{d}^{2}{x}^{2}{{\rm e}^{fx+e}}+12\,{f}^{2}cdx{{\rm e}^{fx+e}}+6\,{f}^{2}{c}^{2}{{\rm e}^{fx+e}}}{3\, \left ({{\rm e}^{fx+e}}-i \right ) ^{3}{f}^{3}{a}^{2}}}-{\frac{4\,d\ln \left ({{\rm e}^{fx+e}}-i \right ) c}{3\,{a}^{2}{f}^{2}}}+{\frac{4\,d\ln \left ({{\rm e}^{fx+e}} \right ) c}{3\,{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{2}{x}^{2}}{3\,f{a}^{2}}}+{\frac{4\,{d}^{2}ex}{3\,{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{2}{e}^{2}}{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) x}{3\,{a}^{2}{f}^{2}}}-{\frac{4\,{d}^{2}\ln \left ( 1+i{{\rm e}^{fx+e}} \right ) e}{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{fx+e}} \right ) }{3\,{f}^{3}{a}^{2}}}+{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{fx+e}}-i \right ) }{3\,{f}^{3}{a}^{2}}}-{\frac{4\,{d}^{2}e\ln \left ({{\rm e}^{fx+e}} \right ) }{3\,{f}^{3}{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{2}{\left (\frac{-2 i \, f^{2} x^{2} -{\left (4 i \, f x e^{\left (2 \, e\right )} + 4 i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 2 \,{\left (3 \, f^{2} x^{2} e^{e} - 2 \, f x e^{e} - 4 \, e^{e}\right )} e^{\left (f x\right )} + 4 i}{3 \, a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{3} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{3}} - 4 i \, \int \frac{x}{3 \,{\left (a^{2} f e^{\left (f x + e\right )} - i \, a^{2} f\right )}}\,{d x}\right )} + \frac{2}{3} \, c d{\left (\frac{3 \,{\left (2 \, f x e^{\left (3 \, f x + 3 \, e\right )} +{\left (-6 i \, f x e^{\left (2 \, e\right )} - 2 i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 2 \, e^{\left (f x + e\right )}\right )}}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} - \frac{2 \, \log \left (-i \,{\left (i \, e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + c^{2}{\left (\frac{6 \, e^{\left (-f x - e\right )}}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f} + \frac{2 i}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*((-2*I*f^2*x^2 - (4*I*f*x*e^(2*e) + 4*I*e^(2*e))*e^(2*f*x) + 2*(3*f^2*x^2*e^e - 2*f*x*e^e - 4*e^e)*e^(f*x)
+ 4*I)/(3*a^2*f^3*e^(3*f*x + 3*e) - 9*I*a^2*f^3*e^(2*f*x + 2*e) - 9*a^2*f^3*e^(f*x + e) + 3*I*a^2*f^3) - 4*I*
integrate(1/3*x/(a^2*f*e^(f*x + e) - I*a^2*f), x)) + 2/3*c*d*(3*(2*f*x*e^(3*f*x + 3*e) + (-6*I*f*x*e^(2*e) - 2
*I*e^(2*e))*e^(2*f*x) - 2*e^(f*x + e))/(3*a^2*f^2*e^(3*f*x + 3*e) - 9*I*a^2*f^2*e^(2*f*x + 2*e) - 9*a^2*f^2*e^
(f*x + e) + 3*I*a^2*f^2) - 2*log(-I*(I*e^(f*x + e) + 1)*e^(-e))/(a^2*f^2)) + c^2*(6*e^(-f*x - e)/((9*a^2*e^(-f
*x - e) - 9*I*a^2*e^(-2*f*x - 2*e) - 3*a^2*e^(-3*f*x - 3*e) + 3*I*a^2)*f) + 2*I/((9*a^2*e^(-f*x - e) - 9*I*a^2
*e^(-2*f*x - 2*e) - 3*a^2*e^(-3*f*x - 3*e) + 3*I*a^2)*f))

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Fricas [B]  time = 2.50375, size = 1162, normalized size = 4.82 \begin{align*} \frac{-2 i \, d^{2} e^{2} + 4 i \, c d e f - 2 i \, c^{2} f^{2} + 4 i \, d^{2} -{\left (4 \, d^{2} e^{\left (3 \, f x + 3 \, e\right )} - 12 i \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 12 \, d^{2} e^{\left (f x + e\right )} + 4 i \, d^{2}\right )}{\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) + 2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x - d^{2} e^{2} + 2 \, c d e f\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (-6 i \, d^{2} f^{2} x^{2} + 6 i \, d^{2} e^{2} - 4 i \, d^{2} +{\left (-12 i \, c d e - 4 i \, c d\right )} f +{\left (-12 i \, c d f^{2} - 4 i \, d^{2} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \,{\left (3 \, d^{2} e^{2} + 3 \, c^{2} f^{2} - 2 \, d^{2} f x - 4 \, d^{2} - 2 \,{\left (3 \, c d e + c d\right )} f\right )} e^{\left (f x + e\right )} +{\left (4 i \, d^{2} e - 4 i \, c d f + 4 \,{\left (d^{2} e - c d f\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (-12 i \, d^{2} e + 12 i \, c d f\right )} e^{\left (2 \, f x + 2 \, e\right )} - 12 \,{\left (d^{2} e - c d f\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) +{\left (-4 i \, d^{2} f x - 4 i \, d^{2} e - 4 \,{\left (d^{2} f x + d^{2} e\right )} e^{\left (3 \, f x + 3 \, e\right )} +{\left (12 i \, d^{2} f x + 12 i \, d^{2} e\right )} e^{\left (2 \, f x + 2 \, e\right )} + 12 \,{\left (d^{2} f x + d^{2} e\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right )}{3 \, a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{3} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*I*d^2*e^2 + 4*I*c*d*e*f - 2*I*c^2*f^2 + 4*I*d^2 - (4*d^2*e^(3*f*x + 3*e) - 12*I*d^2*e^(2*f*x + 2*e) - 12*d
^2*e^(f*x + e) + 4*I*d^2)*dilog(-I*e^(f*x + e)) + 2*(d^2*f^2*x^2 + 2*c*d*f^2*x - d^2*e^2 + 2*c*d*e*f)*e^(3*f*x
+ 3*e) + (-6*I*d^2*f^2*x^2 + 6*I*d^2*e^2 - 4*I*d^2 + (-12*I*c*d*e - 4*I*c*d)*f + (-12*I*c*d*f^2 - 4*I*d^2*f)*
x)*e^(2*f*x + 2*e) + 2*(3*d^2*e^2 + 3*c^2*f^2 - 2*d^2*f*x - 4*d^2 - 2*(3*c*d*e + c*d)*f)*e^(f*x + e) + (4*I*d^
2*e - 4*I*c*d*f + 4*(d^2*e - c*d*f)*e^(3*f*x + 3*e) + (-12*I*d^2*e + 12*I*c*d*f)*e^(2*f*x + 2*e) - 12*(d^2*e -
c*d*f)*e^(f*x + e))*log(e^(f*x + e) - I) + (-4*I*d^2*f*x - 4*I*d^2*e - 4*(d^2*f*x + d^2*e)*e^(3*f*x + 3*e) +
(12*I*d^2*f*x + 12*I*d^2*e)*e^(2*f*x + 2*e) + 12*(d^2*f*x + d^2*e)*e^(f*x + e))*log(I*e^(f*x + e) + 1))/(3*a^2
*f^3*e^(3*f*x + 3*e) - 9*I*a^2*f^3*e^(2*f*x + 2*e) - 9*a^2*f^3*e^(f*x + e) + 3*I*a^2*f^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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