3.5.0 ∫ csch(c+dx)sech3 (c+dx) (e+fx)(a+b sinh(c+dx)) dx [448]

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

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

Rubi [N/A]

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

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

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

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

Mathematica [N/A]

Time = 102.42 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

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

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

Maple [N/A] (veriﬁed)

Time = 0.91 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

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

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

Fricas [N/A]

Time = 19.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

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

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

Sympy [F(-1)]

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

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

Maxima [N/A]

Time = 2.43 (sec) , antiderivative size = 1214, normalized size of antiderivative = 35.71 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

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

-(a*f + (b*d*f*x*e^(3*c) + (d*e - f)*b*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) - (b*d*f*x*e^c + (d*e + f)*b*e^c)*e^(d*x))/(a^2*d^2*e^2 + b^2*d^2*e^2 + (a^2*d^2*f^2 + b^2*d^2*f^2)*x^2 +

2*(a^2*d^2*e*f + b^2*d^2*e*f)*x + (a^2*d^2*e^2*e^(4*c) + b^2*d^2*e^2*e^(4*c) + (a^2*d^2*f^2*e^(4*c) + b^2*d^2*

f^2*e^(4*c))*x^2 + 2*(a^2*d^2*e*f*e^(4*c) + b^2*d^2*e*f*e^(4*c))*x)*e^(4*d*x) + 2*(a^2*d^2*e^2*e^(2*c) + b^2*d

^2*e^2*e^(2*c) + (a^2*d^2*f^2*e^(2*c) + b^2*d^2*f^2*e^(2*c))*x^2 + 2*(a^2*d^2*e*f*e^(2*c) + b^2*d^2*e*f*e^(2*c

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

*a^3*b^3*f + a*b^5*f)*x - (a^5*b*e*e^(2*c) + 2*a^3*b^3*e*e^(2*c) + a*b^5*e*e^(2*c) + (a^5*b*f*e^(2*c) + 2*a^3*

b^3*f*e^(2*c) + a*b^5*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^6*e*e^c + 2*a^4*b^2*e*e^c + a^2*b^4*e*e^c + (a^6*f*e^c +

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

f^2)*a*b^2 + 2*(a^3*d^2*f^2 + 2*a*b^2*d^2*f^2)*x^2 + 4*(a^3*d^2*e*f + 2*a*b^2*d^2*e*f)*x - ((d^2*e^2 - 2*f^2)*

a^2*b*e^c + (3*d^2*e^2 - 2*f^2)*b^3*e^c + (a^2*b*d^2*f^2*e^c + 3*b^3*d^2*f^2*e^c)*x^2 + 2*(a^2*b*d^2*e*f*e^c +

3*b^3*d^2*e*f*e^c)*x)*e^(d*x))/(a^4*d^2*e^3 + 2*a^2*b^2*d^2*e^3 + b^4*d^2*e^3 + (a^4*d^2*f^3 + 2*a^2*b^2*d^2*

f^3 + b^4*d^2*f^3)*x^3 + 3*(a^4*d^2*e*f^2 + 2*a^2*b^2*d^2*e*f^2 + b^4*d^2*e*f^2)*x^2 + 3*(a^4*d^2*e^2*f + 2*a^

2*b^2*d^2*e^2*f + b^4*d^2*e^2*f)*x + (a^4*d^2*e^3*e^(2*c) + 2*a^2*b^2*d^2*e^3*e^(2*c) + b^4*d^2*e^3*e^(2*c) +

(a^4*d^2*f^3*e^(2*c) + 2*a^2*b^2*d^2*f^3*e^(2*c) + b^4*d^2*f^3*e^(2*c))*x^3 + 3*(a^4*d^2*e*f^2*e^(2*c) + 2*a^2

*b^2*d^2*e*f^2*e^(2*c) + b^4*d^2*e*f^2*e^(2*c))*x^2 + 3*(a^4*d^2*e^2*f*e^(2*c) + 2*a^2*b^2*d^2*e^2*f*e^(2*c) +

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

) + 16*integrate(-1/16/(a*f*x + a*e - (a*f*x*e^c + a*e*e^c)*e^(d*x)), x)

Giac [F(-1)]

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

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

Mupad [N/A]

Time = 12.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

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

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

\(\int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [448]

Optimal result