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

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

[Out]

Unintegrable((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x)

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 7.97, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 5.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e x +d}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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