\(\int \frac {\sqrt {d+e x} (a+b \csc ^{-1}(c x))}{x} \, dx\) [54]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 38.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.72 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.48 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x}}{x} \,d x \]

[In]

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

[Out]

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