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

Optimal. Leaf size=37 \[ \text {Int}\left (\frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Csch[c + d*x]^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*} \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\\ \end {align*}

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Mathematica [A]  time = 158.10, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Csch[c + d*x]^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]

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fricas [A]  time = 83.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )}{a f x + a e + {\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^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)

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

[Out]

Timed out

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maple [A]  time = 4.13, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (d x +c \right )^{3} \mathrm {sech}\left (d x +c \right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a f - 2 \, {\left (b d f x e^{\left (3 \, c\right )} + b d e e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (2 \, a d f x e^{\left (2 \, c\right )} + {\left (2 \, d e - f\right )} a e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \, {\left (b d f x e^{c} + b d e e^{c}\right )} e^{\left (d x\right )}}{a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} d^{2} e f x + a^{2} d^{2} e^{2} + {\left (a^{2} d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \, a^{2} d^{2} e f x e^{\left (4 \, c\right )} + a^{2} d^{2} e^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 2 \, {\left (a^{2} d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, a^{2} d^{2} e f x e^{\left (2 \, c\right )} + a^{2} d^{2} e^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - 16 \, \int \frac {b^{2} d^{2} e^{2} + a b d e f - {\left (d^{2} e^{2} - f^{2}\right )} a^{2} - {\left (a^{2} d^{2} f^{2} - b^{2} d^{2} f^{2}\right )} x^{2} - {\left (2 \, a^{2} d^{2} e f - 2 \, b^{2} d^{2} e f - a b d f^{2}\right )} x}{16 \, {\left (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} - {\left (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}\right )} e^{\left (d x\right )}\right )}}\,{d x} + 16 \, \int -\frac {b^{2} d^{2} e^{2} - a b d e f - {\left (d^{2} e^{2} - f^{2}\right )} a^{2} - {\left (a^{2} d^{2} f^{2} - b^{2} d^{2} f^{2}\right )} x^{2} - {\left (2 \, a^{2} d^{2} e f - 2 \, b^{2} d^{2} e f + a b d f^{2}\right )} x}{16 \, {\left (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} + {\left (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}\right )} e^{\left (d x\right )}\right )}}\,{d x} + 16 \, \int -\frac {a b^{4} e^{\left (d x + c\right )} - b^{5}}{8 \, {\left (a^{5} b e + a^{3} b^{3} e + {\left (a^{5} b f + a^{3} b^{3} f\right )} x - {\left (a^{5} b e e^{\left (2 \, c\right )} + a^{3} b^{3} e e^{\left (2 \, c\right )} + {\left (a^{5} b f e^{\left (2 \, c\right )} + a^{3} b^{3} f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{6} e e^{c} + a^{4} b^{2} e e^{c} + {\left (a^{6} f e^{c} + a^{4} b^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}\right )}}\,{d x} + 16 \, \int \frac {b e^{\left (d x + c\right )} - a}{8 \, {\left (a^{2} e + b^{2} e + {\left (a^{2} f + b^{2} f\right )} x + {\left (a^{2} e e^{\left (2 \, c\right )} + b^{2} e e^{\left (2 \, c\right )} + {\left (a^{2} f e^{\left (2 \, c\right )} + b^{2} 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)^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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