3.472 \(\int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\)

Optimal. Leaf size=39 \[ \text {Int}\left (\frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(csch(d*x+c)^2*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Csch[c + d*x]^2*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][(Csch[c + d*x]^2*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 153.97, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Csch[c + d*x]^2*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Integrate[(Csch[c + d*x]^2*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]

________________________________________________________________________________________

fricas [A]  time = 1.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (d x + c\right )^{2} \operatorname {sech}\left (d x + c\right )^{2}}{a f x + a e + {\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(csch(d*x + c)^2*sech(d*x + c)^2/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 2.20, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (d x +c \right )^{2} \mathrm {sech}\left (d x +c \right )^{2}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(csch(d*x+c)^2*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 16 \, b^{4} \int -\frac {e^{\left (d x + c\right )}}{8 \, {\left (a^{4} b e + a^{2} b^{3} e + {\left (a^{4} b f + a^{2} b^{3} f\right )} x - {\left (a^{4} b e e^{\left (2 \, c\right )} + a^{2} b^{3} e e^{\left (2 \, c\right )} + {\left (a^{4} b f e^{\left (2 \, c\right )} + a^{2} b^{3} f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{5} e e^{c} + a^{3} b^{2} e e^{c} + {\left (a^{5} f e^{c} + a^{3} b^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}\right )}}\,{d x} + \frac {2 \, {\left (a b e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b e^{\left (d x + c\right )} + 2 \, a^{2} + b^{2}\right )}}{a^{3} d e + a b^{2} d e + {\left (a^{3} d f + a b^{2} d f\right )} x - {\left (a^{3} d e e^{\left (4 \, c\right )} + a b^{2} d e e^{\left (4 \, c\right )} + {\left (a^{3} d f e^{\left (4 \, c\right )} + a b^{2} d f e^{\left (4 \, c\right )}\right )} x\right )} e^{\left (4 \, d x\right )}} - 16 \, \int -\frac {b d f x + b d e + a f}{16 \, {\left (a^{2} d f^{2} x^{2} + 2 \, a^{2} d e f x + a^{2} d e^{2} - {\left (a^{2} d f^{2} x^{2} e^{c} + 2 \, a^{2} d e f x e^{c} + a^{2} d e^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 16 \, \int \frac {b d f x + b d e - a f}{16 \, {\left (a^{2} d f^{2} x^{2} + 2 \, a^{2} d e f x + a^{2} d e^{2} + {\left (a^{2} d f^{2} x^{2} e^{c} + 2 \, a^{2} d e f x e^{c} + a^{2} d e^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 16 \, \int \frac {b f e^{\left (d x + c\right )} - a f}{8 \, {\left (a^{2} d e^{2} + b^{2} d e^{2} + {\left (a^{2} d f^{2} + b^{2} d f^{2}\right )} x^{2} + 2 \, {\left (a^{2} d e f + b^{2} d e f\right )} x + {\left (a^{2} d e^{2} e^{\left (2 \, c\right )} + b^{2} d e^{2} e^{\left (2 \, c\right )} + {\left (a^{2} d f^{2} e^{\left (2 \, c\right )} + b^{2} d f^{2} e^{\left (2 \, c\right )}\right )} x^{2} + 2 \, {\left (a^{2} d e f e^{\left (2 \, c\right )} + b^{2} d e f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^2*sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(1/(cosh(c + d*x)^2*sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**2*sech(d*x+c)**2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________