3.1.57 \(\int \frac {x}{(a+b \text {sech}^{-1}(c x))^2} \, dx\) [57]

Optimal. Leaf size=15 \[ \text {Int}\left (\frac {x}{\left (a+b \text {sech}^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, 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 {x}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x/(a + b*ArcSech[c*x])^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 12.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x/(a + b*ArcSech[c*x])^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-((c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x + (c^2*x^3 - x)*x)/((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c)
- (b^2*log(c) + b^2*log(x) - a*b)*sqrt(c*x + 1)*sqrt(-c*x + 1) + a*b - (b^2*c^2*x^2 - sqrt(c*x + 1)*sqrt(-c*x
+ 1)*b^2 - b^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b^2*c^2*x^2 - b^2)*log(x)) + integrate((2*(2*c^2*x^2
- 1)*(c*x + 1)*(c*x - 1)*x + (3*c^4*x^4 - 8*c^2*x^2 + 4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x + 2*(c^4*x^4 - 2*c^2*
x^2 + 1)*x)/((b^2*c^4*log(c) - a*b*c^4)*x^4 - (b^2*log(c) + b^2*log(x) - a*b)*(c*x + 1)*(c*x - 1) - 2*(b^2*c^2
*log(c) - a*b*c^2)*x^2 + b^2*log(c) - 2*((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b + (b^2*c^2*x^2 - b^
2)*log(x))*sqrt(c*x + 1)*sqrt(-c*x + 1) - a*b - (b^2*c^4*x^4 - 2*b^2*c^2*x^2 - (c*x + 1)*(c*x - 1)*b^2 - 2*(b^
2*c^2*x^2 - b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) + b^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b^2*c^4*x^4 - 2*
b^2*c^2*x^2 + b^2)*log(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(x/(a+b*arcsech(c*x))^2,x, algorithm="fricas")

[Out]

integral(x/(b^2*arcsech(c*x)^2 + 2*a*b*arcsech(c*x) + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*acosh(1/(c*x)))^2,x)

[Out]

int(x/(a + b*acosh(1/(c*x)))^2, x)

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