3.1.62 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx\) [62]

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 = 0.05, 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 \sqrt {d+e x}} \, dx \end {gather*}

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 = 7.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \end {gather*}

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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Maple [A]
time = 0.12, 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 \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)^(1/2),x, algorithm="maxima")

[Out]

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

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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)^(1/2),x, algorithm="fricas")

[Out]

integral((b*arccsch(c*x) + a)*sqrt(x*e + d)/(x^3*e + d*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} \sqrt {d + e x}}\, 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)**(1/2),x)

[Out]

Integral((a + b*acsch(c*x))/(x**2*sqrt(d + e*x)), 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)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/(sqrt(e*x + d)*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\,\sqrt {d+e\,x}} \,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)^(1/2)),x)

[Out]

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

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