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

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

[Out]

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

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Rubi [A]  time = 0.0504119, 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 = {} \[ \int \frac{\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 48.143, size = 0, normalized size = 0. \[ \int \frac{\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.091, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sinh \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(sinh(d*x+c)/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

[Out]

int(sinh(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 )}} - \frac{i \, \log \left (f x + e\right )}{a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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)) - I*log(f*x + e)/(a*f)

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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{d f x + d e -{\left (-i \, d f x - 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 (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(sinh(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(-(d*f*x + d*e - (-I*d*f*x - 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^(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(sinh(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{\sinh \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(sinh(d*x+c)/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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