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

Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\frac{\text{csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable[Csch[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]

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Rubi [A]  time = 0.0501509, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\text{csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=\int \frac{\text{csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\\ \end{align*}

Mathematica [A]  time = 46.1088, size = 0, normalized size = 0. \[ \int \frac{\text{csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.742, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm csch} \left (dx+c\right )}{ \left ( fx+e \right ) \left ( a+ia\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*f*integrate(1/(-I*a*d*f^2*x^2 - 2*I*a*d*e*f*x - I*a*d*e^2 + (a*d*f^2*x^2*e^c + 2*a*d*e*f*x*e^c + a*d*e^2*e^c
)*e^(d*x)), x) + 2/(-I*a*d*f*x - I*a*d*e + (a*d*f*x*e^c + a*d*e*e^c)*e^(d*x)) + 2*integrate(1/2/(a*f*x + a*e +
(a*f*x*e^c + a*e*e^c)*e^(d*x)), x) + 2*integrate(-1/2/(a*f*x + a*e - (a*f*x*e^c + a*e*e^c)*e^(d*x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (-i \, a d f x - i \, a d e +{\left (a d f x + a d e\right )} e^{\left (d x + c\right )}\right )}{\rm integral}\left (\frac{2 \,{\left (d f x + d e + f\right )} e^{\left (2 \, d x + 2 \, c\right )} +{\left (-2 i \, d f x - 2 i \, d e\right )} e^{\left (d x + c\right )} - 2 \, f}{i \, a d f^{2} x^{2} + 2 i \, a d e f x + i \, a d e^{2} +{\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} +{\left (-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} -{\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (d x + c\right )}}, x\right ) + 2}{-i \, a d f x - i \, a d e +{\left (a d f x + a d e\right )} e^{\left (d x + c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-I*a*d*f*x - I*a*d*e + (a*d*f*x + a*d*e)*e^(d*x + c))*integral((2*(d*f*x + d*e + f)*e^(2*d*x + 2*c) + (-2*I*
d*f*x - 2*I*d*e)*e^(d*x + c) - 2*f)/(I*a*d*f^2*x^2 + 2*I*a*d*e*f*x + I*a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x +
a*d*e^2)*e^(3*d*x + 3*c) + (-I*a*d*f^2*x^2 - 2*I*a*d*e*f*x - I*a*d*e^2)*e^(2*d*x + 2*c) - (a*d*f^2*x^2 + 2*a*d
*e*f*x + a*d*e^2)*e^(d*x + c)), x) + 2)/(-I*a*d*f*x - I*a*d*e + (a*d*f*x + a*d*e)*e^(d*x + c))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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