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

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

[Out]

Unintegrable((d*x)^m/(a+b*arccsc(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 = {} \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m/(a + b*ArcCsc[c*x])^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.78, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b \operatorname {arccsc}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (a +b \,\mathrm {arccsc}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

(4*sqrt(c*x + 1)*sqrt(c*x - 1)*(b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a)*d^m*x*x^m - (4*b^3*arctan2(1, s
qrt(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*arctan2(1, 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*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a)*d^m*m - ((b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a)*c
^2*d^m*m + 2*(b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a)*c^2*d^m)*x^2 + (b*arctan2(1, sqrt(c*x + 1)*sqrt(c
*x - 1)) + a)*d^m)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^m/(4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 4*b^3*lo
g(c)^2 + 8*a*b^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*a^2*b - 4*(b^3*c^2*log(c)^2 + (b^3*arctan2(1, sqr
t(c*x + 1)*sqrt(c*x - 1))^2 + 2*a*b^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a^2*b)*c^2)*x^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 + 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*arctan2(1, sqrt(c*x + 1)*s
qrt(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*arctan2
(1, 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 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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