3.1.68 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx\) [68]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right ) \]

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 16.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \left (e x +d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/2*((2*(3*(x*e + d)*e^2 - 2*d*e^2)/((x*e + d)^(3/2)*d^2 - sqrt(x*e + d)*d^3) + 3*e^2*log((sqrt(x*e + d) - sqr
t(d))/(sqrt(x*e + d) + sqrt(d)))/d^(5/2))*e^(-1)*log(c) - 2*integrate(log(x)/(sqrt(x*e + d)*x^3*e + sqrt(x*e +
d)*d*x^2), x) + 2*integrate(log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(x*e + d)*x^3*e + sqrt(x*e + d)*d*x^2), x))*b - 1
/2*a*(2*(3*(x*e + d)*e - 2*d*e)/((x*e + d)^(3/2)*d^2 - sqrt(x*e + d)*d^3) + 3*e*log((sqrt(x*e + d) - sqrt(d))/
(sqrt(x*e + d) + sqrt(d)))/d^(5/2))

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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^2/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

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

[Out]

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

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