\(\int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx\) [67]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 11.99 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.78 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.62 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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