\(\int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx\) [172]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a+b \sinh (e+f x))},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)/(a+b*sinh(f*x+e)),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx \]

[In]

Int[1/((c + d*x)*(a + b*Sinh[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)*(a + b*Sinh[e + f*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx \]

[In]

Integrate[1/((c + d*x)*(a + b*Sinh[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)*(a + b*Sinh[e + f*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right ) \left (a +b \sinh \left (f x +e \right )\right )}d x\]

[In]

int(1/(d*x+c)/(a+b*sinh(f*x+e)),x)

[Out]

int(1/(d*x+c)/(a+b*sinh(f*x+e)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \sinh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*sinh(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d*x + a*c + (b*d*x + b*c)*sinh(f*x + e)), x)

Sympy [N/A]

Not integrable

Time = 10.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int \frac {1}{\left (a + b \sinh {\left (e + f x \right )}\right ) \left (c + d x\right )}\, dx \]

[In]

integrate(1/(d*x+c)/(a+b*sinh(f*x+e)),x)

[Out]

Integral(1/((a + b*sinh(e + f*x))*(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \sinh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*sinh(f*x+e)),x, algorithm="maxima")

[Out]

integrate(1/((d*x + c)*(b*sinh(f*x + e) + a)), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (b \sinh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a+b*sinh(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(b*sinh(f*x + e) + a)), x)

Mupad [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+b \sinh (e+f x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a + b*sinh(e + f*x))*(c + d*x)),x)

[Out]

int(1/((a + b*sinh(e + f*x))*(c + d*x)), x)