\(\int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [495]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 112.92 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 37.81 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 912, normalized size of antiderivative = 26.82 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-(a*f - 2*(b*d*f*x*e^(3*c) + b*d*e*e^(3*c))*e^(3*d*x) + (2*a*d*f*x*e^(2*c) + (2*d*e - f)*a*e^(2*c))*e^(2*d*x)
+ 2*(b*d*f*x*e^c + b*d*e*e^c)*e^(d*x))/(a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + a^2*d^2*e^2 + (a^2*d^2*f^2*x^2*e^(
4*c) + 2*a^2*d^2*e*f*x*e^(4*c) + a^2*d^2*e^2*e^(4*c))*e^(4*d*x) - 2*(a^2*d^2*f^2*x^2*e^(2*c) + 2*a^2*d^2*e*f*x
*e^(2*c) + a^2*d^2*e^2*e^(2*c))*e^(2*d*x)) - 16*integrate(1/16*(b^2*d^2*e^2 + a*b*d*e*f - (d^2*e^2 - f^2)*a^2
- (a^2*d^2*f^2 - b^2*d^2*f^2)*x^2 - (2*a^2*d^2*e*f - 2*b^2*d^2*e*f - a*b*d*f^2)*x)/(a^3*d^2*f^3*x^3 + 3*a^3*d^
2*e*f^2*x^2 + 3*a^3*d^2*e^2*f*x + a^3*d^2*e^3 - (a^3*d^2*f^3*x^3*e^c + 3*a^3*d^2*e*f^2*x^2*e^c + 3*a^3*d^2*e^2
*f*x*e^c + a^3*d^2*e^3*e^c)*e^(d*x)), x) + 16*integrate(-1/16*(b^2*d^2*e^2 - a*b*d*e*f - (d^2*e^2 - f^2)*a^2 -
(a^2*d^2*f^2 - b^2*d^2*f^2)*x^2 - (2*a^2*d^2*e*f - 2*b^2*d^2*e*f + a*b*d*f^2)*x)/(a^3*d^2*f^3*x^3 + 3*a^3*d^2
*e*f^2*x^2 + 3*a^3*d^2*e^2*f*x + a^3*d^2*e^3 + (a^3*d^2*f^3*x^3*e^c + 3*a^3*d^2*e*f^2*x^2*e^c + 3*a^3*d^2*e^2*
f*x*e^c + a^3*d^2*e^3*e^c)*e^(d*x)), x) + 16*integrate(-1/8*(a*b^4*e^(d*x + c) - b^5)/(a^5*b*e + a^3*b^3*e + (
a^5*b*f + a^3*b^3*f)*x - (a^5*b*e*e^(2*c) + a^3*b^3*e*e^(2*c) + (a^5*b*f*e^(2*c) + a^3*b^3*f*e^(2*c))*x)*e^(2*
d*x) - 2*(a^6*e*e^c + a^4*b^2*e*e^c + (a^6*f*e^c + a^4*b^2*f*e^c)*x)*e^(d*x)), x) + 16*integrate(1/8*(b*e^(d*x
+ c) - a)/(a^2*e + b^2*e + (a^2*f + b^2*f)*x + (a^2*e*e^(2*c) + b^2*e*e^(2*c) + (a^2*f*e^(2*c) + b^2*f*e^(2*c
))*x)*e^(2*d*x)), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 11.51 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int(1/(cosh(c + d*x)*sinh(c + d*x)^3*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(1/(cosh(c + d*x)*sinh(c + d*x)^3*(e + f*x)*(a + b*sinh(c + d*x))), x)