3.2.62 \(\int \frac {a+b \text {sech}^{-1}(c x)}{x (d+e x^2)^{3/2}} \, dx\) [162]

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

[Out]

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

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Rubi [A]
time = 0.07, 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 {sech}^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 21.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arcsech}\left (c x \right )}{x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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*arcsech(c*x))/x/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^2*e + d)*(b*arcsech(c*x) + a)/(x^5*e^2 + 2*d*x^3*e + d^2*x), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))/(x*(d + e*x**2)**(3/2)), 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*arcsech(c*x))/x/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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