\(\int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx\) [238]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\text {Int}\left (\frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 36.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{{\left (b \cosh \left (d x + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 22.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\sinh ^{3}{\left (c + d x \right )}}{x \left (a + b \cosh {\left (c + d x \right )}\right )}\, dx \]

[In]

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

[Out]

Integral(sinh(c + d*x)**3/(x*(a + b*cosh(c + d*x))), x)

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.96 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{{\left (b \cosh \left (d x + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

1/4*Ei(2*d*x)*e^(2*c)/b + 1/2*a*Ei(-d*x)*e^(-c)/b^2 - 1/4*Ei(-2*d*x)*e^(-2*c)/b - 1/2*a*Ei(d*x)*e^c/b^2 + (a^2
- b^2)*log(x)/b^3 - 1/8*integrate(16*(a^2*b - b^3 + (a^3*e^c - a*b^2*e^c)*e^(d*x))/(b^4*x*e^(2*d*x + 2*c) + 2
*a*b^3*x*e^(d*x + c) + b^4*x), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{{\left (b \cosh \left (d x + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sinh ^3(c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{x\,\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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