\(\int \frac {\sqrt {d+e x} (a+b \csc ^{-1}(c x))}{x^2} \, dx\) [55]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\text {Int}\left (\frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 22.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.19 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

1/2*(a*e*x*log(e*x/(e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)) + 2*b*sqrt(d)*x*integrate(sqrt(e*x + d)*arctan2(1, s
qrt(c*x + 1)*sqrt(c*x - 1))/x^2, x) - 2*sqrt(e*x + d)*a*sqrt(d))/(sqrt(d)*x)

Giac [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x}}{x^2} \,d x \]

[In]

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

[Out]

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