3.2.62 \(\int \frac {x^4 (a+b \csc ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\) [162]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x)

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Rubi [A]
time = 0.07, 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^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

Defer[Int][(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end {align*}

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Mathematica [A]
time = 9.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x)

[Out]

int(x^4*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),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^4*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/3*((3*x^2*e^(-1)/(x^2*e + d)^(3/2) + 2*d*e^(-2)/(x^2*e + d)^(3/2))*x - 3*arcsinh(x*e^(1/2)/sqrt(d))*e^(-5/2
) + x*e^(-2)/sqrt(x^2*e + d))*a + b*integrate(x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/((x^4*e^2 + 2*d*x^2*
e + d^2)*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^4*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x^4*arccsc(c*x) + a*x^4)*sqrt(x^2*e + d)/(x^6*e^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e + d^3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acsc(c*x))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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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^4*(a+b*arccsc(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)*x^4/(e*x^2 + d)^(5/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*asin(1/(c*x))))/(d + e*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*asin(1/(c*x))))/(d + e*x^2)^(5/2), x)

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