3.24 \(\int \frac {x^4}{(a+b \text {csch}(c+d x^2))^2} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {x^4}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(x^4/(a+b*csch(d*x^2+c))^2,x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {x^4}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^4/(a + b*Csch[c + d*x^2])^2,x]

[Out]

Defer[Int][x^4/(a + b*Csch[c + d*x^2])^2, x]

Rubi steps

\begin {align*} \int \frac {x^4}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx &=\int \frac {x^4}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx\\ \end {align*}

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

Verification is Not applicable to the result.

[In]

Integrate[x^4/(a + b*Csch[c + d*x^2])^2,x]

[Out]

$Aborted

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{b^{2} \operatorname {csch}\left (d x^{2} + c\right )^{2} + 2 \, a b \operatorname {csch}\left (d x^{2} + c\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*csch(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(x^4/(b^2*csch(d*x^2 + c)^2 + 2*a*b*csch(d*x^2 + c) + a^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*csch(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(x^4/(b*csch(d*x^2 + c) + a)^2, x)

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maple [A]  time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+b*csch(d*x^2+c))^2,x)

[Out]

int(x^4/(a+b*csch(d*x^2+c))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*csch(d*x^2+c))^2,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x^4}{{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b/sinh(c + d*x^2))^2,x)

[Out]

int(x^4/(a + b/sinh(c + d*x^2))^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b*csch(d*x**2+c))**2,x)

[Out]

Integral(x**4/(a + b*csch(c + d*x**2))**2, x)

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