\(\int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx\) [226]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\text {Int}\left (\frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.90 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \cosh \left (d x + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.67 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{x \left (a + b \cosh {\left (c + d x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \cosh \left (d x + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (b \cosh \left (d x + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )}{x\,\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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