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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x,x]

[Out]

Defer[Int][((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x, x]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx &=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 6.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x,x]

[Out]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x, x]

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fricas [A]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsch}\left (c x\right )\right )} \sqrt {e x^{2} + d}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^(3/2)*(a + b*asinh(1/(c*x))))/x,x)

[Out]

int(((d + e*x^2)^(3/2)*(a + b*asinh(1/(c*x))))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*acsch(c*x))/x,x)

[Out]

Integral((a + b*acsch(c*x))*(d + e*x**2)**(3/2)/x, x)

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