3.1.67 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x)^{3/2}} \, dx\) [67]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 14.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x \left (e x +d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*acsch(c*x))/(x*(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/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

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

[Out]

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

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