\(\int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [198]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.82 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

\[\int \frac {\sinh \left (d x +c \right )^{2}}{\left (f x +e \right )^{2} \left (a +i a \sinh \left (d x +c \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 9.06 \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 270, normalized size of antiderivative = 8.71 \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\sinh ^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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