3.2.43 \(\int \frac {x^2 (a+b \csc ^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [143]

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 = 0.06, 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 = {} \begin {gather*} \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \end {gather*}

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 = 69.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \end {gather*}

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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Maple [A]
time = 180.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x^2*e + d), x)

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

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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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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