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

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{(d x)^m}{\left (a+b \sec ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0240248, 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 = {} \[ \int \frac{(d x)^m}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m/(a + b*ArcSec[c*x])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.499552, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*x)^m/(a + b*ArcSec[c*x])^2,x]

[Out]

Integrate[(d*x)^m/(a + b*ArcSec[c*x])^2, x]

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Maple [A]  time = 1.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \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((d*x)^m/(a+b*arcsec(c*x))^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*(b*d^m*x*x^m*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + a*d^m*x*x^m)*sqrt(c*x + 1)*sqrt(c*x - 1) - (4*b^3*arcta
n(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*((b*d^m*m - (b*c^2*d^m*m + 2*b*c^2*d^m)*x^2 + b*d^m)*x^m*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + (a*d^m*m - (a
*c^2*d^m*m + 2*a*c^2*d^m)*x^2 + a*d^m)*x^m)*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{\left (d x\right )^{m}}{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((d*x)^m/(a+b*arcsec(c*x))^2,x, algorithm="fricas")

[Out]

integral((d*x)^m/(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{\left (d x\right )^{m}}{\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((d*x)**m/(a+b*asec(c*x))**2,x)

[Out]

Integral((d*x)**m/(a + b*asec(c*x))**2, x)

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

[Out]

integrate((d*x)^m/(b*arcsec(c*x) + a)^2, x)