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

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\sinh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {\sinh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ \frac {{\left (-i \, a d f x - i \, a d e + {\left (a d f x + a d e\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (\frac {d f x + d e + {\left (-i \, d f x - i \, d e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (d f x + d e\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-i \, d f x - i \, d e - 4 i \, f\right )} e^{\left (d x + c\right )}}{2 \, {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-2 i \, a d f^{2} x^{2} - 4 i \, a d e f x - 2 i \, a d e^{2}\right )} e^{\left (d x + c\right )}}, x\right ) - 2 i}{-i \, a d f x - i \, a d e + {\left (a d f x + a d e\right )} e^{\left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}\left (d x +c \right )}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinh(d*x+c)^2/(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 \[ -2 i \, f \int \frac {1}{-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2} + {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} - \frac {2 i}{-i \, a d f x - i \, a d e + {\left (a d f x e^{c} + a d e e^{c}\right )} e^{\left (d x\right )}} + \frac {\log \left (f x + e\right )}{a f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*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) - 1/2*I*e^(-c + d*e/f)*exp_integral_e(1, (f*x + e)*d/f)/(a*f) + 1/2*I*e^(c - d*e/f)*exp_inte
gral_e(1, -(f*x + e)*d/f)/(a*f) - 2*I/(-I*a*d*f*x - I*a*d*e + (a*d*f*x*e^c + a*d*e*e^c)*e^(d*x)) + log(f*x + e
)/(a*f)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2/((e + f*x)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(sinh(c + d*x)^2/((e + f*x)*(a + a*sinh(c + d*x)*1i)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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