3.152 \(\int \frac {x^4 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

Unintegrable(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x)

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Defer[Int][(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 8.80, size = 0, normalized size = 0.00 \[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} \operatorname {arcsch}\left (c x\right ) + a x^{4}\right )} \sqrt {e x^{2} + d}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral((b*x^4*arccsch(c*x) + a*x^4)*sqrt(e*x^2 + d)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x)

[Out]

int(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (\frac {x^{3}}{\sqrt {e x^{2} + d} e} + \frac {3 \, d x}{\sqrt {e x^{2} + d} e^{2}} - \frac {3 \, d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {5}{2}}}\right )} a + b \int \frac {x^{4} \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

1/2*(x^3/(sqrt(e*x^2 + d)*e) + 3*d*x/(sqrt(e*x^2 + d)*e^2) - 3*d*arcsinh(e*x/sqrt(d*e))/e^(5/2))*a + b*integra
te(x^4*log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/(e*x^2 + d)^(3/2), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(3/2),x)

[Out]

int((x^4*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(3/2), 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(x**4*(a+b*acsch(c*x))/(e*x**2+d)**(3/2),x)

[Out]

Timed out

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