3.192 \(\int \frac{\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\)

Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\frac{\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable[Sinh[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]

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Rubi [A]  time = 0.0511999, 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 = {} \[ \int \frac{\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx &=\int \frac{\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\\ \end{align*}

Mathematica [A]  time = 41.715, size = 0, normalized size = 0. \[ \int \frac{\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sinh \left ( dx+c \right ) }{ \left ( fx+e \right ) ^{2} \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)^2/(a+I*a*sinh(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -4 \, f \int \frac{1}{-i \, a d f^{3} x^{3} - 3 i \, a d e f^{2} x^{2} - 3 i \, a d e^{2} f x - i \, a d e^{3} +{\left (a d f^{3} x^{3} e^{c} + 3 \, a d e f^{2} x^{2} e^{c} + 3 \, a d e^{2} f x e^{c} + a d e^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} + \frac{2 \, d f x + 2 \, d e +{\left (2 i \, d f x e^{c} + 2 i \, d e e^{c}\right )} e^{\left (d x\right )} - 4 \, f}{2 \,{\left (-i \, a d f^{3} x^{2} - 2 i \, a d e f^{2} x - i \, a d e^{2} f +{\left (a d f^{3} x^{2} e^{c} + 2 \, a d e f^{2} x e^{c} + a d e^{2} f e^{c}\right )} e^{\left (d x\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (-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 )}\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 )} + 4 \, f}{-i \, a d f^{3} x^{3} - 3 i \, a d e f^{2} x^{2} - 3 i \, a d e^{2} f x - i \, a d e^{3} +{\left (a d f^{3} x^{3} + 3 \, a d e f^{2} x^{2} + 3 \, a d e^{2} f x + a d e^{3}\right )} e^{\left (d x + c\right )}}, x\right ) - 2}{-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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-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))*integral(-(d
*f*x + d*e - (-I*d*f*x - I*d*e)*e^(d*x + c) + 4*f)/(-I*a*d*f^3*x^3 - 3*I*a*d*e*f^2*x^2 - 3*I*a*d*e^2*f*x - I*a
*d*e^3 + (a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*e^(d*x + c)), x) - 2)/(-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))

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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)**2/(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 )}^{2}{\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)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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