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

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{\left (a+b \sec ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0056968, 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 = {} \[ \int \frac{1}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSec[c*x])^(-2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 21.5741, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSec[c*x])^(-2),x]

[Out]

Integrate[(a + b*ArcSec[c*x])^(-2), x]

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Maple [A]  time = 0.461, size = 0, normalized size = 0. \begin{align*} \int \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(1/(a+b*arcsec(c*x))^2,x)

[Out]

int(1/(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 \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + a x\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 (2 \, a c^{2} x^{2} +{\left (2 \, b c^{2} x^{2} - b\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) - a\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(1/(a+b*arcsec(c*x))^2,x, algorithm="maxima")

[Out]

-(4*(b*x*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + a*x)*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*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))*integrate(-4*(2*a*c^2*x^2
+ (2*b*c^2*x^2 - b)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - a)*sqrt(c*x + 1)*sqrt(c*x - 1)/(4*b^3*log(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)*arctan(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{1}{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(1/(a+b*arcsec(c*x))^2,x, algorithm="fricas")

[Out]

integral(1/(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{1}{\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(1/(a+b*asec(c*x))**2,x)

[Out]

Integral((a + b*asec(c*x))**(-2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\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(1/(a+b*arcsec(c*x))^2,x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)^(-2), x)