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

Optimal. Leaf size=41 \[ -\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))} \]

[Out]

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

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Rubi [A]  time = 0.0599134, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2747, 3770, 2648} \[ -\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\cosh (c+d x)}{d (a+i a \sinh (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2747

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[b/(
b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; Fre
eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3770

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

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0642838, size = 52, normalized size = 1.27 \[ -\frac{\text{sech}(c+d x) \left (i \sinh (c+d x)+\sqrt{\cosh ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\cosh ^2(c+d x)}\right )-1\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sech[c + d*x]*(-1 + ArcTanh[Sqrt[Cosh[c + d*x]^2]]*Sqrt[Cosh[c + d*x]^2] + I*Sinh[c + d*x]))/(a*d))

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Maple [A]  time = 0.037, size = 42, normalized size = 1. \begin{align*}{\frac{-2\,i}{da} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08649, size = 84, normalized size = 2.05 \begin{align*} -\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} + \frac{2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.49623, size = 154, normalized size = 3.76 \begin{align*} -\frac{{\left (e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) -{\left (e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 2}{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)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{csch}{\left (c + d x \right )}}{i \sinh{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(c + d*x)/(I*sinh(c + d*x) + 1), x)/a

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Giac [A]  time = 1.24345, size = 72, normalized size = 1.76 \begin{align*} -\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{a d} + \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a d} + \frac{2}{a d{\left (e^{\left (d x + c\right )} - i\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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