3.2.51 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^3 (d+e x^2)^{3/2}} \, dx\) [151]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 31.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^3 \left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/2*a*(3*arcsinh(sqrt(d)*e^(-1/2)/abs(x))*e/d^(5/2) - 3*e/(sqrt(x^2*e + d)*d^2) - 1/(sqrt(x^2*e + d)*d*x^2)) +
b*integrate(log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/((x^2*e + d)^(3/2)*x^3), 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((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^2*e + d)*(b*arccsch(c*x) + a)/(x^7*e^2 + 2*d*x^5*e + d^2*x^3), x)

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Sympy [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((a+b*acsch(c*x))/x**3/(e*x**2+d)**(3/2),x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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