\(\int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [237]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sinh(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

Int[Sinh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][Sinh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 11.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

Integrate[Sinh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Integrate[Sinh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.66 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int \frac {\sinh \left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}d x\]

[In]

int(sinh(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(sinh(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sinh(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sinh(d*x + c)^3/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(sinh(d*x+c)**3/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.43 \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sinh(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sinh(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^3/((f*x + e)*(b*sinh(d*x + c) + a)), x)

Mupad [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int(sinh(c + d*x)^3/((e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(sinh(c + d*x)^3/((e + f*x)*(a + b*sinh(c + d*x))), x)