3.214 \(\int \frac{\text{csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 \coth (c+d x)}{a d}+\frac{i \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\coth (c+d x)}{d (a+i a \sinh (c+d x))} \]

[Out]

(I*ArcTanh[Cosh[c + d*x]])/(a*d) - (2*Coth[c + d*x])/(a*d) + Coth[c + d*x]/(d*(a + I*a*Sinh[c + d*x]))

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Rubi [A]  time = 0.0856202, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2768, 2748, 3767, 8, 3770} \[ -\frac{2 \coth (c+d x)}{a d}+\frac{i \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\coth (c+d x)}{d (a+i a \sinh (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*ArcTanh[Cosh[c + d*x]])/(a*d) - (2*Coth[c + d*x])/(a*d) + Coth[c + d*x]/(d*(a + I*a*Sinh[c + d*x]))

Rule 2768

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

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 3770

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

Rubi steps

\begin{align*} \int \frac{\text{csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=\frac{\coth (c+d x)}{d (a+i a \sinh (c+d x))}-\frac{\int \text{csch}^2(c+d x) (-2 a+i a \sinh (c+d x)) \, dx}{a^2}\\ &=\frac{\coth (c+d x)}{d (a+i a \sinh (c+d x))}-\frac{i \int \text{csch}(c+d x) \, dx}{a}+\frac{2 \int \text{csch}^2(c+d x) \, dx}{a}\\ &=\frac{i \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\coth (c+d x)}{d (a+i a \sinh (c+d x))}-\frac{(2 i) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{a d}\\ &=\frac{i \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{2 \coth (c+d x)}{a d}+\frac{\coth (c+d x)}{d (a+i a \sinh (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.205343, size = 61, normalized size = 1.07 \[ -\frac{\text{sech}(c+d x) \left (2 \sinh (c+d x)+\text{csch}(c+d x)-i \sqrt{\cosh ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\cosh ^2(c+d x)}\right )+i\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sech[c + d*x]*(I - I*ArcTanh[Sqrt[Cosh[c + d*x]^2]]*Sqrt[Cosh[c + d*x]^2] + Csch[c + d*x] + 2*Sinh[c + d*x]
))/(a*d))

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Maple [A]  time = 0.04, size = 79, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{i}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-2\,{\frac{1}{da \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d/a*tanh(1/2*d*x+1/2*c)-1/2/d/a/tanh(1/2*d*x+1/2*c)-I/d/a*ln(tanh(1/2*d*x+1/2*c))-2/d/a/(-I+tanh(1/2*d*x+
1/2*c))

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Maxima [B]  time = 1.07214, size = 149, normalized size = 2.61 \begin{align*} -\frac{4 \,{\left (e^{\left (-d x - c\right )} - i \, e^{\left (-2 \, d x - 2 \, c\right )} + 2 i\right )}}{{\left (2 \, a e^{\left (-d x - c\right )} - 2 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} + 2 i \, a\right )} d} + \frac{i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac{i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(e^(-d*x - c) - I*e^(-2*d*x - 2*c) + 2*I)/((2*a*e^(-d*x - c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e^(-3*d*x - 3*c
) + 2*I*a)*d) + I*log(e^(-d*x - c) + 1)/(a*d) - I*log(e^(-d*x - c) - 1)/(a*d)

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Fricas [B]  time = 2.56622, size = 378, normalized size = 6.63 \begin{align*} \frac{{\left (i \, e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )} - i \, e^{\left (d x + c\right )} - 1\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) +{\left (-i \, e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (2 \, d x + 2 \, c\right )} + i \, e^{\left (d x + c\right )} + 1\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 i}{a d e^{\left (3 \, d x + 3 \, c\right )} - i \, a d e^{\left (2 \, d x + 2 \, c\right )} - a d e^{\left (d x + c\right )} + i \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((I*e^(3*d*x + 3*c) + e^(2*d*x + 2*c) - I*e^(d*x + c) - 1)*log(e^(d*x + c) + 1) + (-I*e^(3*d*x + 3*c) - e^(2*d
*x + 2*c) + I*e^(d*x + c) + 1)*log(e^(d*x + c) - 1) - 2*I*e^(2*d*x + 2*c) - 2*e^(d*x + c) + 4*I)/(a*d*e^(3*d*x
+ 3*c) - I*a*d*e^(2*d*x + 2*c) - a*d*e^(d*x + c) + I*a*d)

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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(csch(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.2282, size = 128, normalized size = 2.25 \begin{align*} \frac{i \, \log \left (e^{\left (d x + c\right )} + 1\right )}{a d} - \frac{i \, \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a d} + \frac{2 \,{\left (e^{\left (2 \, d x + 2 \, c\right )} - i \, e^{\left (d x + c\right )} - 2\right )}}{a d{\left (i \, e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )} - i \, e^{\left (d x + c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*log(e^(d*x + c) + 1)/(a*d) - I*log(abs(e^(d*x + c) - 1))/(a*d) + 2*(e^(2*d*x + 2*c) - I*e^(d*x + c) - 2)/(a*
d*(I*e^(3*d*x + 3*c) + e^(2*d*x + 2*c) - I*e^(d*x + c) - 1))