3.2.72 \(\int \frac {1}{x^2 (a+b \text {ArcCos}(c x))^3} \, dx\) [172]

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{x^2 (a+b \text {ArcCos}(c x))^3},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*arccos(c*x))^3,x)

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Rubi [A]
time = 0.02, 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 {1}{x^2 (a+b \text {ArcCos}(c x))^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x^2*(a + b*ArcCos[c*x])^3),x]

[Out]

Defer[Int][1/(x^2*(a + b*ArcCos[c*x])^3), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^3} \, dx &=\int \frac {1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 23.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 (a+b \text {ArcCos}(c x))^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x^2*(a + b*ArcCos[c*x])^3),x]

[Out]

Integrate[1/(x^2*(a + b*ArcCos[c*x])^3), x]

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Maple [A]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \arccos \left (c x \right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*arccos(c*x))^3,x)

[Out]

int(1/x^2/(a+b*arccos(c*x))^3,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(1/x^2/(a+b*arccos(c*x))^3,x, algorithm="maxima")

[Out]

-1/2*(a*c^2*x^2 - sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x + (b*c^2*x^2 - 2*b)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1),
c*x) + 2*(b^4*c^2*x^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*x^3*arctan2(sqrt(c*x + 1)*sq
rt(-c*x + 1), c*x) + a^2*b^2*c^2*x^3)*integrate(1/2*(c^2*x^2 - 6)/(b^3*c^2*x^4*arctan2(sqrt(c*x + 1)*sqrt(-c*x
+ 1), c*x) + a*b^2*c^2*x^4), x) - 2*a)/(b^4*c^2*x^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^
2*x^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a^2*b^2*c^2*x^3)

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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(1/x^2/(a+b*arccos(c*x))^3,x, algorithm="fricas")

[Out]

integral(1/(b^3*x^2*arccos(c*x)^3 + 3*a*b^2*x^2*arccos(c*x)^2 + 3*a^2*b*x^2*arccos(c*x) + a^3*x^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*acos(c*x))**3,x)

[Out]

Integral(1/(x**2*(a + b*acos(c*x))**3), 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(1/x^2/(a+b*arccos(c*x))^3,x, algorithm="giac")

[Out]

integrate(1/((b*arccos(c*x) + a)^3*x^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*acos(c*x))^3),x)

[Out]

int(1/(x^2*(a + b*acos(c*x))^3), x)

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