\(\int \frac {(d x)^m}{(a+b \csc ^{-1}(c x))^2} \, dx\) [43]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\text {Int}\left (\frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 1.39 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\left (d x \right )^{m}}{\left (a +b \,\operatorname {arccsc}\left (c x \right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 5.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.74 (sec) , antiderivative size = 684, normalized size of antiderivative = 42.75 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[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))

Giac [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

[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)