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

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

[Out]

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

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Rubi [A]
time = 0.06, 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 = {} \begin {gather*} \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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*} \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\\ \end {align*}

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Mathematica [A]
time = 123.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(csch(d*x+c)*sech(d*x+c)^3/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*f + (b*d*f*x*e^(3*c) - b*f*e^(3*c) + b*d*e^(3*c + 1))*e^(3*d*x) - (2*a*d*f*x*e^(2*c) - a*f*e^(2*c) + 2*a*d
*e^(2*c + 1))*e^(2*d*x) - (b*d*f*x*e^c + b*d*e^(c + 1) + b*f*e^c)*e^(d*x))/((a^2*d^2*f^2 + b^2*d^2*f^2)*x^2 +
2*(a^2*d^2*f + b^2*d^2*f)*x*e + (a^2*d^2 + b^2*d^2)*e^2 + ((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*f*e^(4*c) + b^2*d^2*f*e^(4*c))*x*e + (a^2*d^2*e^(4*c) + b^2*d^2*e^(4*c))*e^2)*e^(4*d*x) + 2*((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*f*e^(2*c) + b^2*d^2*f*e^(2*c))*x*e + (a^2*d^2*e^(2*c) +
b^2*d^2*e^(2*c))*e^2)*e^(2*d*x)) - 16*integrate(1/16*(2*a^3*f^2 + 2*a*b^2*f^2 - 2*(a^3*d^2*f^2 + 2*a*b^2*d^2*f
^2)*x^2 - 4*(a^3*d^2*f + 2*a*b^2*d^2*f)*x*e - 2*(a^3*d^2 + 2*a*b^2*d^2)*e^2 - (2*a^2*b*f^2*e^c + 2*b^3*f^2*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*f*e^c + 3*b^3*d^2*f*e^c)*x*e - (a^2*b*d^2*e^c +
3*b^3*d^2*e^c)*e^2)*e^(d*x))/((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*f^2 + 2*a^2*b^2
*d^2*f^2 + b^4*d^2*f^2)*x^2*e + 3*(a^4*d^2*f + 2*a^2*b^2*d^2*f + b^4*d^2*f)*x*e^2 + (a^4*d^2 + 2*a^2*b^2*d^2 +
b^4*d^2)*e^3 + ((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*f^2*
e^(2*c) + 2*a^2*b^2*d^2*f^2*e^(2*c) + b^4*d^2*f^2*e^(2*c))*x^2*e + 3*(a^4*d^2*f*e^(2*c) + 2*a^2*b^2*d^2*f*e^(2
*c) + b^4*d^2*f*e^(2*c))*x*e^2 + (a^4*d^2*e^(2*c) + 2*a^2*b^2*d^2*e^(2*c) + b^4*d^2*e^(2*c))*e^3)*e^(2*d*x)),
x) + 16*integrate(-1/8*(a*b^4*e^(d*x + c) - b^5)/((a^5*b*f + 2*a^3*b^3*f + a*b^5*f)*x + (a^5*b + 2*a^3*b^3 + a
*b^5)*e - ((a^5*b*f*e^(2*c) + 2*a^3*b^3*f*e^(2*c) + a*b^5*f*e^(2*c))*x + (a^5*b*e^(2*c) + 2*a^3*b^3*e^(2*c) +
a*b^5*e^(2*c))*e)*e^(2*d*x) - 2*((a^6*f*e^c + 2*a^4*b^2*f*e^c + a^2*b^4*f*e^c)*x + (a^6*e^c + 2*a^4*b^2*e^c +
a^2*b^4*e^c)*e)*e^(d*x)), x) - 16*integrate(1/16/(a*f*x + a*e + (a*f*x*e^c + a*e^(c + 1))*e^(d*x)), x) + 16*in
tegrate(-1/16/(a*f*x + a*e - (a*f*x*e^c + a*e^(c + 1))*e^(d*x)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3437 deep

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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