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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x \sqrt {d+e x^2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e x^{3} + d x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 4.18, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x \sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {e x^{2} + d} x}\,{d x} - \frac {a \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{\sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x \sqrt {d + e x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________