3.39 \(\int \frac{x}{(a+b \sec ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=14 \[ \text{Unintegrable}\left (\frac{x}{\left (a+b \sec ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[x/(a + b*ArcSec[c*x])^2, x]

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Rubi [A]  time = 0.0145601, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x/(a + b*ArcSec[c*x])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 9.87738, size = 0, normalized size = 0. \[ \int \frac{x}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x/(a + b*ArcSec[c*x])^2,x]

[Out]

Integrate[x/(a + b*ArcSec[c*x])^2, x]

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Maple [A]  time = 1.09, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( a+b{\rm arcsec} \left (cx\right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \,{\left (b x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + a x^{2}\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 4 \,{\left (4 \, b^{3} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} + b^{3} \log \left (c^{2} x^{2}\right )^{2} + 4 \, b^{3} \log \left (c\right )^{2} + 8 \, b^{3} \log \left (c\right ) \log \left (x\right ) + 4 \, b^{3} \log \left (x\right )^{2} + 8 \, a b^{2} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + 4 \, a^{2} b - 4 \,{\left (b^{3} \log \left (c\right ) + b^{3} \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right )\right )} \int \frac{{\left (3 \, a c^{2} x^{3} - 2 \, a x +{\left (3 \, b c^{2} x^{3} - 2 \, b x\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{4 \, b^{3} \log \left (c\right )^{2} + 4 \, a^{2} b - 4 \,{\left (b^{3} c^{2} \log \left (c\right )^{2} + a^{2} b c^{2}\right )} x^{2} - 4 \,{\left (b^{3} c^{2} x^{2} - b^{3}\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} -{\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \left (c^{2} x^{2}\right )^{2} - 4 \,{\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \left (x\right )^{2} - 8 \,{\left (a b^{2} c^{2} x^{2} - a b^{2}\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + 4 \,{\left (b^{3} c^{2} x^{2} \log \left (c\right ) - b^{3} \log \left (c\right ) +{\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right ) - 8 \,{\left (b^{3} c^{2} x^{2} \log \left (c\right ) - b^{3} \log \left (c\right )\right )} \log \left (x\right )}\,{d x}}{4 \, b^{3} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} + b^{3} \log \left (c^{2} x^{2}\right )^{2} + 4 \, b^{3} \log \left (c\right )^{2} + 8 \, b^{3} \log \left (c\right ) \log \left (x\right ) + 4 \, b^{3} \log \left (x\right )^{2} + 8 \, a b^{2} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + 4 \, a^{2} b - 4 \,{\left (b^{3} \log \left (c\right ) + b^{3} \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*(b*x^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + a*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1) - (4*b^3*arctan(sqrt(c*x +
1)*sqrt(c*x - 1))^2 + b^3*log(c^2*x^2)^2 + 4*b^3*log(c)^2 + 8*b^3*log(c)*log(x) + 4*b^3*log(x)^2 + 8*a*b^2*ar
ctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*a^2*b - 4*(b^3*log(c) + b^3*log(x))*log(c^2*x^2))*integrate(-4*(3*a*c^2*
x^3 - 2*a*x + (3*b*c^2*x^3 - 2*b*x)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)))*sqrt(c*x + 1)*sqrt(c*x - 1)/(4*b^3*lo
g(c)^2 + 4*a^2*b - 4*(b^3*c^2*log(c)^2 + a^2*b*c^2)*x^2 - 4*(b^3*c^2*x^2 - b^3)*arctan(sqrt(c*x + 1)*sqrt(c*x
- 1))^2 - (b^3*c^2*x^2 - b^3)*log(c^2*x^2)^2 - 4*(b^3*c^2*x^2 - b^3)*log(x)^2 - 8*(a*b^2*c^2*x^2 - a*b^2)*arct
an(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*(b^3*c^2*x^2*log(c) - b^3*log(c) + (b^3*c^2*x^2 - b^3)*log(x))*log(c^2*x^2
) - 8*(b^3*c^2*x^2*log(c) - b^3*log(c))*log(x)), x))/(4*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + b^3*log(c^
2*x^2)^2 + 4*b^3*log(c)^2 + 8*b^3*log(c)*log(x) + 4*b^3*log(x)^2 + 8*a*b^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))
+ 4*a^2*b - 4*(b^3*log(c) + b^3*log(x))*log(c^2*x^2))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{b^{2} \operatorname{arcsec}\left (c x\right )^{2} + 2 \, a b \operatorname{arcsec}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arcsec(c*x))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(a + b*asec(c*x))**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arcsec(c*x))^2,x, algorithm="giac")

[Out]

integrate(x/(b*arcsec(c*x) + a)^2, x)