\(\int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx\) [62]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.47 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx=\int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.63 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x + d} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 19.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{2} \sqrt {d + e x}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 89, normalized size of antiderivative = 4.24 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x + d} x^{2}} \,d x } \]

[In]

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

[Out]

1/2*(2*b*d^(3/2)*x*integrate(arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(sqrt(e*x + d)*x^2), x) - a*e*x*log(e*x/(
e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)) - 2*sqrt(e*x + d)*a*sqrt(d))/(d^(3/2)*x)

Giac [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x + d} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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