3.499 \(\int \frac{\text{csch}^3(c+d x) \text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\)
Optimal. Leaf size=38 \[ \text{Unintegrable}\left (\frac{\text{csch}^3(c+d x) \text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]
[Out]
Unintegrable[(Csch[c + d*x]^3*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]
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Rubi [A] time = 0.134132, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{\text{csch}^3(c+d x) \text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(Csch[c + d*x]^3*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]
[Out]
Defer[Int][(Csch[c + d*x]^3*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]
Rubi steps
\begin{align*} \int \frac{\text{csch}^3(c+d x) \text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac{\text{csch}^3(c+d x) \text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end{align*}
Mathematica [F] time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
Integrate[(Csch[c + d*x]^3*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]
[Out]
$Aborted
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Maple [A] time = 3.536, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{3} \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) \left ( a+b\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)
[Out]
int(csch(d*x+c)^3*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-32*b^5*integrate(-1/16*e^(d*x + c)/(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) - (4*a^2*b*d*e + 2*b^3*d*e + 2*(2*a^2*b*d*f + b^3*d*f)*x + ((3*d*e - f)*a^3*e^
(5*c) + (d*e - f)*a*b^2*e^(5*c) + (3*a^3*d*f*e^(5*c) + a*b^2*d*f*e^(5*c))*x)*e^(5*d*x) - 2*(b^3*d*f*x*e^(4*c)
+ b^3*d*e*e^(4*c))*e^(4*d*x) - 2*(a^3*d*e*e^(3*c) - a*b^2*d*e*e^(3*c) + (a^3*d*f*e^(3*c) - a*b^2*d*f*e^(3*c))*
x)*e^(3*d*x) - 4*(a^2*b*d*f*x*e^(2*c) + a^2*b*d*e*e^(2*c))*e^(2*d*x) + ((3*d*e + f)*a^3*e^c + (d*e + f)*a*b^2*
e^c + (3*a^3*d*f*e^c + a*b^2*d*f*e^c)*x)*e^(d*x))/(a^4*d^2*e^2 + a^2*b^2*d^2*e^2 + (a^4*d^2*f^2 + a^2*b^2*d^2*
f^2)*x^2 + 2*(a^4*d^2*e*f + a^2*b^2*d^2*e*f)*x + (a^4*d^2*e^2*e^(6*c) + a^2*b^2*d^2*e^2*e^(6*c) + (a^4*d^2*f^2
*e^(6*c) + a^2*b^2*d^2*f^2*e^(6*c))*x^2 + 2*(a^4*d^2*e*f*e^(6*c) + a^2*b^2*d^2*e*f*e^(6*c))*x)*e^(6*d*x) - (a^
4*d^2*e^2*e^(4*c) + a^2*b^2*d^2*e^2*e^(4*c) + (a^4*d^2*f^2*e^(4*c) + a^2*b^2*d^2*f^2*e^(4*c))*x^2 + 2*(a^4*d^2
*e*f*e^(4*c) + a^2*b^2*d^2*e*f*e^(4*c))*x)*e^(4*d*x) - (a^4*d^2*e^2*e^(2*c) + a^2*b^2*d^2*e^2*e^(2*c) + (a^4*d
^2*f^2*e^(2*c) + a^2*b^2*d^2*f^2*e^(2*c))*x^2 + 2*(a^4*d^2*e*f*e^(2*c) + a^2*b^2*d^2*e*f*e^(2*c))*x)*e^(2*d*x)
) - 32*integrate(1/64*(2*b^2*d^2*e^2 + 2*a*b*d*e*f - (3*d^2*e^2 - 2*f^2)*a^2 - (3*a^2*d^2*f^2 - 2*b^2*d^2*f^2)
*x^2 - 2*(3*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) - 32*integrate(-1/64*(2*b^2*d^2*e^2 - 2*a*b*d*e*f - (3*d^2*e^2 - 2*f^2)*a^2 - (3*a^2*d^2*f^2 - 2
*b^2*d^2*f^2)*x^2 - 2*(3*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) - 32*integrate(1/16*(a*f*e^(d*x + c) + b*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)
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{csch}\left (d x + c\right )^{3} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")
[Out]
integral(csch(d*x + c)^3*sech(d*x + c)^2/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**3*sech(d*x+c)**2/(f*x+e)/(a+b*sinh(d*x+c)),x)
[Out]
Timed out
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*sech(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out