\(\int \frac {\sqrt {d+e x^2} (a+b \csc ^{-1}(c x))}{x^2} \, dx\) [125]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\text {Int}\left (\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2},x\right ) \]

[Out]

Unintegrable((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{2}}\, dx \]

[In]

integrate((a+b*acsc(c*x))*(e*x**2+d)**(1/2)/x**2,x)

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^2,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 = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^2} \,d x \]

[In]

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

[Out]

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