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

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 = 0.09, 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 = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \]

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 = 12.17, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \]

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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fricas [A]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}, x\right ) \]

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

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

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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maple [A]  time = 6.99, 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 \[ -b {\left (\frac {{\left (\frac {e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, e}{\sqrt {e x + d} d}\right )} \log \relax (c)}{e} + \int \frac {\log \relax (x)}{\sqrt {e x + d} e x^{2} + \sqrt {e x + d} d x}\,{d x} - \int \frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e x^{2} + \sqrt {e x + d} d x}\,{d x}\right )} + a {\left (\frac {\log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2}{\sqrt {e x + d} d}\right )} \]

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]

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

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

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

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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