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

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {(d x)^m}{\left (a+b \sec ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {(d x)^m}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

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*}

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Mathematica [A]  time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} \operatorname {arcsec}\left (c x\right )^{2} + 2 \, a b \operatorname {arcsec}\left (c x\right ) + a^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 2.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (a +b \,\mathrm {arcsec}\left (c x \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {4 \, {\left (b d^{m} x x^{m} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + a d^{m} x x^{m}\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 \relax (c)^{2} + 8 \, b^{3} \log \relax (c) \log \relax (x) + 4 \, b^{3} \log \relax (x)^{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 \relax (c) + b^{3} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right )\right )} \int \frac {{\left ({\left (b d^{m} m - {\left (b c^{2} d^{m} m + 2 \, b c^{2} d^{m}\right )} x^{2} + b d^{m}\right )} x^{m} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (a d^{m} m - {\left (a c^{2} d^{m} m + 2 \, a c^{2} d^{m}\right )} x^{2} + a d^{m}\right )} x^{m}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{4 \, b^{3} \log \relax (c)^{2} + 4 \, a^{2} b - 4 \, {\left (b^{3} c^{2} \log \relax (c)^{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 \relax (x)^{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 \relax (c) - b^{3} \log \relax (c) + {\left (b^{3} c^{2} x^{2} - b^{3}\right )} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right ) - 8 \, {\left (b^{3} c^{2} x^{2} \log \relax (c) - b^{3} \log \relax (c)\right )} \log \relax (x)}\,{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 \relax (c)^{2} + 8 \, b^{3} \log \relax (c) \log \relax (x) + 4 \, b^{3} \log \relax (x)^{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 \relax (c) + b^{3} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right )} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(a + b*acos(1/(c*x)))^2,x)

[Out]

int((d*x)^m/(a + b*acos(1/(c*x)))^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}\, dx \]

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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