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

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

[Out]

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

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 16.60, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {arccsc}\left (c x\right ) + a x^{2}}{\sqrt {e x^{2} + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 7.45, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {\sqrt {e x^{2} + d} x}{e} - \frac {d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {3}{2}}}\right )} + b \int \frac {x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {e x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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