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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 9.21, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e x^{3} + d x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 5.26, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \sqrt {e x +d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left ({\left (\frac {2 \, \sqrt {e x + d} e}{{\left (e x + d\right )} d - d^{2}} + \frac {e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {3}{2}}}\right )} \log \relax (c) - 2 \, \int \frac {\log \relax (x)}{\sqrt {e x + d} x^{2}}\,{d x} + 2 \, \int \frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} x^{2}}\,{d x}\right )} b - \frac {1}{2} \, a {\left (\frac {2 \, \sqrt {e x + d} e}{{\left (e x + d\right )} d - d^{2}} + \frac {e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {3}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**2/(e*x+d)**(1/2),x)

[Out]

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

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