\(\int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx\) [21]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\text {Int}\left (\frac {\cosh (c+d x) \sinh (c+d x)}{(e+f x) (b+a \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 45.95 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

\[\int \frac {\cosh \left (d x +c \right )}{\left (f x +e \right ) \left (a +b \,\operatorname {csch}\left (d x +c \right )\right )}d x\]

[In]

int(cosh(d*x+c)/(f*x+e)/(a+b*csch(d*x+c)),x)

[Out]

int(cosh(d*x+c)/(f*x+e)/(a+b*csch(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \operatorname {csch}\left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(cosh(d*x + c)/(a*f*x + a*e + (b*f*x + b*e)*csch(d*x + c)), x)

Sympy [N/A]

Not integrable

Time = 6.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{\left (a + b \operatorname {csch}{\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

[In]

integrate(cosh(d*x+c)/(f*x+e)/(a+b*csch(d*x+c)),x)

[Out]

Integral(cosh(c + d*x)/((a + b*csch(c + d*x))*(e + f*x)), x)

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 164, normalized size of antiderivative = 6.31 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \operatorname {csch}\left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \operatorname {csch}\left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integrate(cosh(d*x + c)/((f*x + e)*(b*csch(d*x + c) + a)), x)

Mupad [N/A]

Not integrable

Time = 2.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )} \,d x \]

[In]

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

[Out]

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