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

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m/(a + b*ArcSech[c*x])^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} \operatorname {arsech}\left (c x\right )^{2} + 2 \, a b \operatorname {arsech}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c^{2} d^{m} x^{3} - d^{m} x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{m} + {\left (c^{2} d^{m} x^{3} - d^{m} x\right )} x^{m}}{{\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} - b^{2} \log \relax (c) - {\left (b^{2} \log \relax (c) + b^{2} \log \relax (x) - a b\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + a b - {\left (b^{2} c^{2} x^{2} - \sqrt {c x + 1} \sqrt {-c x + 1} b^{2} - b^{2}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \relax (x)} + \int \frac {{\left (c^{2} d^{m} {\left (m + 3\right )} x^{2} - d^{m} {\left (m + 1\right )}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} x^{m} + {\left (c^{4} d^{m} {\left (m + 2\right )} x^{4} - c^{2} d^{m} {\left (3 \, m + 5\right )} x^{2} + 2 \, d^{m} {\left (m + 1\right )}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{m} + {\left (c^{4} d^{m} {\left (m + 1\right )} x^{4} - 2 \, c^{2} d^{m} {\left (m + 1\right )} x^{2} + d^{m} {\left (m + 1\right )}\right )} x^{m}}{{\left (b^{2} c^{4} \log \relax (c) - a b c^{4}\right )} x^{4} - {\left (b^{2} \log \relax (c) + b^{2} \log \relax (x) - a b\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - 2 \, {\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - 2 \, {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} - b^{2} \log \relax (c) + a b + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \log \relax (x)\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - a b - {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} - {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} - 2 \, {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + b^{2}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) + {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \relax (x)}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*d^m*x^3 - d^m*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m + (c^2*d^m*x^3 - d^m*x)*x^m)/((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 - sqr
t(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)) + int
egrate(((c^2*d^m*(m + 3)*x^2 - d^m*(m + 1))*(c*x + 1)*(c*x - 1)*x^m + (c^4*d^m*(m + 2)*x^4 - c^2*d^m*(3*m + 5)
*x^2 + 2*d^m*(m + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m + (c^4*d^m*(m + 1)*x^4 - 2*c^2*d^m*(m + 1)*x^2 + d^m*(m
+ 1))*x^m)/((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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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