\(\int (d x)^m (a+b \csc ^{-1}(c x))^2 \, dx\) [40]

Optimal result

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

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

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

Mathematica [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx \]

[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 = 0.83 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[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 = 30, normalized size of antiderivative = 1.88 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.78 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \]

[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 = 6.87 (sec) , antiderivative size = 551, normalized size of antiderivative = 34.44 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

(d*x)^(m + 1)*a^2/(d*(m + 1)) + 1/4*(4*b^2*d^m*x*x^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 - b^2*d^m*x*x^m
*log(c^2*x^2)^2 + 4*(m + 1)*integrate((2*sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*d^m*x^m*arctan2(1, sqrt(c*x + 1)*sqrt
(c*x - 1)) + (b^2*d^m*m + b^2*d^m - (b^2*c^2*d^m*m + b^2*c^2*d^m)*x^2)*x^m*log(x)^2 + 2*(b^2*d^m*m*log(c) + b^
2*d^m*log(c) - (b^2*c^2*d^m*m*log(c) + b^2*c^2*d^m*log(c))*x^2)*x^m*log(x) + ((b^2*log(c)^2 - 2*a*b*arctan2(1,
sqrt(c*x + 1)*sqrt(c*x - 1)))*d^m*m - ((b^2*c^2*log(c)^2 - 2*a*b*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))
*d^m*m + (b^2*c^2*log(c)^2 - 2*a*b*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*d^m)*x^2 + (b^2*log(c)^2 - 2*a
*b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*d^m)*x^m - ((b^2*d^m*m + b^2*d^m - (b^2*c^2*d^m*m + b^2*c^2*d^m)*x
^2)*x^m*log(x) + (b^2*d^m*m*log(c) - (b^2*c^2*d^m*m*log(c) + (b^2*c^2*log(c) + b^2*c^2)*d^m)*x^2 + (b^2*log(c)
+ b^2)*d^m)*x^m)*log(c^2*x^2))/((c^2*m + c^2)*x^2 - m - 1), x))/(m + 1)

Giac [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2} \left (d x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]

[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)