\(\int (d+e x^2)^{3/2} (a+b \csc ^{-1}(c x)) \, dx\) [133]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Int}\left (\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ),x\right ) \]

[Out]

Unintegrable((e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]

[In]

Int[(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]

[Out]

Defer[Int][(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 18.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]

[In]

Integrate[(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]

[Out]

Integrate[(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 1.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )d x\]

[In]

int((e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((e*x**2+d)**(3/2)*(a+b*acsc(c*x)),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arccsc(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int {\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

int((d + e*x^2)^(3/2)*(a + b*asin(1/(c*x))),x)

[Out]

int((d + e*x^2)^(3/2)*(a + b*asin(1/(c*x))), x)