3.3.9 \(\int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [209]

Optimal. Leaf size=32 \[ \text {Int}\left (\frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 32.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {csch}\left (d x +c \right )}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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) - (I*d*f
*x + I*d*e)*e^(d*x + c) - f)/(I*a*d*f^2*x^2 + 2*I*a*d*f*x*e + I*a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*f*x*e + a*d*e^2
)*e^(3*d*x + 3*c) + (-I*a*d*f^2*x^2 - 2*I*a*d*f*x*e - I*a*d*e^2)*e^(2*d*x + 2*c) - (a*d*f^2*x^2 + 2*a*d*f*x*e
+ 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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \int \frac {\operatorname {csch}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*(e + f*x)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(1/(sinh(c + d*x)*(e + f*x)*(a + a*sinh(c + d*x)*1i)), x)

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