3.64 \(\int \frac{\sinh ^2(x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=36 \[ \frac{3 i x}{2}+2 \cosh (x)-\frac{3}{2} i \sinh (x) \cosh (x)-\frac{\sinh (x) \cosh (x)}{\text{csch}(x)+i} \]

[Out]

((3*I)/2)*x + 2*Cosh[x] - ((3*I)/2)*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x])/(I + Csch[x])

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Rubi [A]  time = 0.0598771, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2638} \[ \frac{3 i x}{2}+2 \cosh (x)-\frac{3}{2} i \sinh (x) \cosh (x)-\frac{\sinh (x) \cosh (x)}{\text{csch}(x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^2/(I + Csch[x]),x]

[Out]

((3*I)/2)*x + 2*Cosh[x] - ((3*I)/2)*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x])/(I + Csch[x])

Rule 3819

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2638

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

Rubi steps

\begin{align*} \int \frac{\sinh ^2(x)}{i+\text{csch}(x)} \, dx &=-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}+\int (-3 i+2 \text{csch}(x)) \sinh ^2(x) \, dx\\ &=-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}-3 i \int \sinh ^2(x) \, dx+2 \int \sinh (x) \, dx\\ &=2 \cosh (x)-\frac{3}{2} i \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}+\frac{3}{2} i \int 1 \, dx\\ &=\frac{3 i x}{2}+2 \cosh (x)-\frac{3}{2} i \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh (x)}{i+\text{csch}(x)}\\ \end{align*}

Mathematica [A]  time = 0.124247, size = 46, normalized size = 1.28 \[ \cosh (x)+\frac{1}{4} i \left (6 x-\sinh (2 x)-\frac{8 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^2/(I + Csch[x]),x]

[Out]

Cosh[x] + (I/4)*(6*x - (8*Sinh[x/2])/(Cosh[x/2] + I*Sinh[x/2]) - Sinh[2*x])

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Maple [B]  time = 0.042, size = 96, normalized size = 2.7 \begin{align*}{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) + \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}-{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{2\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}-{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}- \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}-{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2/(I+csch(x)),x)

[Out]

1/2*I/(tanh(1/2*x)+1)^2+3/2*I*ln(tanh(1/2*x)+1)+1/(tanh(1/2*x)+1)-1/2*I/(tanh(1/2*x)+1)-2*I/(tanh(1/2*x)-I)-3/
2*I*ln(tanh(1/2*x)-1)-1/2*I/(tanh(1/2*x)-1)^2-1/(tanh(1/2*x)-1)-1/2*I/(tanh(1/2*x)-1)

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Maxima [A]  time = 1.00302, size = 63, normalized size = 1.75 \begin{align*} \frac{3}{2} i \, x + \frac{3 i \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} + 1}{4 \,{\left (2 i \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )}\right )}} + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{8} i \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(I+csch(x)),x, algorithm="maxima")

[Out]

3/2*I*x + 1/4*(3*I*e^(-x) + 20*e^(-2*x) + 1)/(2*I*e^(-2*x) + 2*e^(-3*x)) + 1/2*e^(-x) + 1/8*I*e^(-2*x)

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Fricas [B]  time = 1.59214, size = 151, normalized size = 4.19 \begin{align*} \frac{{\left (12 i \, x - 4 i\right )} e^{\left (3 \, x\right )} + 4 \,{\left (3 \, x + 5\right )} e^{\left (2 \, x\right )} - i \, e^{\left (5 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 3 i \, e^{x} + 1}{8 \,{\left (e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(I+csch(x)),x, algorithm="fricas")

[Out]

1/8*((12*I*x - 4*I)*e^(3*x) + 4*(3*x + 5)*e^(2*x) - I*e^(5*x) + 3*e^(4*x) - 3*I*e^x + 1)/(e^(3*x) - I*e^(2*x))

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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(sinh(x)**2/(I+csch(x)),x)

[Out]

Timed out

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Giac [A]  time = 1.17289, size = 54, normalized size = 1.5 \begin{align*} \frac{3}{2} i \, x + \frac{{\left (-20 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-2 \, x\right )}}{8 \,{\left (-i \, e^{x} - 1\right )}} - \frac{1}{8} i \, e^{\left (2 \, x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(I+csch(x)),x, algorithm="giac")

[Out]

3/2*I*x + 1/8*(-20*I*e^(2*x) - 3*e^x - I)*e^(-2*x)/(-I*e^x - 1) - 1/8*I*e^(2*x) + 1/2*e^x