3.475 \(\int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\)

Optimal. Leaf size=39 \[ \text {Int}\left (\frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(csch(d*x+c)^2*sech(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

________________________________________________________________________________________

Rubi [A]  time = 0.14, 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 = {} \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Csch[c + d*x]^2*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][(Csch[c + d*x]^2*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac {\text {csch}^2(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[(Csch[c + d*x]^2*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

$Aborted

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 4.65, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (d x +c \right )^{2} \mathrm {sech}\left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2*sech(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(csch(d*x+c)^2*sech(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*sech(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(a*b*f + (2*b^2*d*e*e^(5*c) + (3*d*e - f)*a^2*e^(5*c) + (3*a^2*d*f*e^(5*c) + 2*b^2*d*f*e^(5*c))*x)*e^(5*d*x) +
(2*a*b*d*f*x*e^(4*c) + (2*d*e - f)*a*b*e^(4*c))*e^(4*d*x) + 2*(a^2*d*e*e^(3*c) + 2*b^2*d*e*e^(3*c) + (a^2*d*f
*e^(3*c) + 2*b^2*d*f*e^(3*c))*x)*e^(3*d*x) - 2*(a*b*d*f*x*e^(2*c) + a*b*d*e*e^(2*c))*e^(2*d*x) + (2*b^2*d*e*e^
c + (3*d*e + f)*a^2*e^c + (3*a^2*d*f*e^c + 2*b^2*d*f*e^c)*x)*e^(d*x))/(a^3*d^2*e^2 + a*b^2*d^2*e^2 + (a^3*d^2*
f^2 + a*b^2*d^2*f^2)*x^2 + 2*(a^3*d^2*e*f + a*b^2*d^2*e*f)*x - (a^3*d^2*e^2*e^(6*c) + a*b^2*d^2*e^2*e^(6*c) +
(a^3*d^2*f^2*e^(6*c) + a*b^2*d^2*f^2*e^(6*c))*x^2 + 2*(a^3*d^2*e*f*e^(6*c) + a*b^2*d^2*e*f*e^(6*c))*x)*e^(6*d*
x) - (a^3*d^2*e^2*e^(4*c) + a*b^2*d^2*e^2*e^(4*c) + (a^3*d^2*f^2*e^(4*c) + a*b^2*d^2*f^2*e^(4*c))*x^2 + 2*(a^3
*d^2*e*f*e^(4*c) + a*b^2*d^2*e*f*e^(4*c))*x)*e^(4*d*x) + (a^3*d^2*e^2*e^(2*c) + a*b^2*d^2*e^2*e^(2*c) + (a^3*d
^2*f^2*e^(2*c) + a*b^2*d^2*f^2*e^(2*c))*x^2 + 2*(a^3*d^2*e*f*e^(2*c) + a*b^2*d^2*e*f*e^(2*c))*x)*e^(2*d*x)) -
32*integrate(-1/16*(a*b^5*e^(d*x + c) - b^6)/(a^6*b*e + 2*a^4*b^3*e + a^2*b^5*e + (a^6*b*f + 2*a^4*b^3*f + a^2
*b^5*f)*x - (a^6*b*e*e^(2*c) + 2*a^4*b^3*e*e^(2*c) + a^2*b^5*e*e^(2*c) + (a^6*b*f*e^(2*c) + 2*a^4*b^3*f*e^(2*c
) + a^2*b^5*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^7*e*e^c + 2*a^5*b^2*e*e^c + a^3*b^4*e*e^c + (a^7*f*e^c + 2*a^5*b^2*
f*e^c + a^3*b^4*f*e^c)*x)*e^(d*x)), x) - 32*integrate(1/32*(2*(d^2*e^2 - f^2)*a^2*b + 2*(2*d^2*e^2 - f^2)*b^3
+ 2*(a^2*b*d^2*f^2 + 2*b^3*d^2*f^2)*x^2 + 4*(a^2*b*d^2*e*f + 2*b^3*d^2*e*f)*x + ((3*d^2*e^2 - 2*f^2)*a^3*e^c +
(5*d^2*e^2 - 2*f^2)*a*b^2*e^c + (3*a^3*d^2*f^2*e^c + 5*a*b^2*d^2*f^2*e^c)*x^2 + 2*(3*a^3*d^2*e*f*e^c + 5*a*b^
2*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) - 32*integrate(-1/32*(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) + 32*integrate(1/32*(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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^3*sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

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

[Out]

Timed out

________________________________________________________________________________________