3.1.54 \(\int \frac {\sqrt {d+e x} (a+b \text {csch}^{-1}(c x))}{x} \, dx\) [54]

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

[Out]

Unintegrable((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x,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 {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e x +d}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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))*(e*x+d)^(1/2)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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