3.2.67 \(\int \frac {1}{x^2 (a+b \text {ArcCos}(c x))^2} \, dx\) [167]

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

[Out]

Unintegrable(1/x^2/(a+b*arccos(c*x))^2,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))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x^2*(a + b*ArcCos[c*x])^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 38.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 (a+b \text {ArcCos}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.48, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \arccos \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

((b^2*c*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*x^2)*integrate((c^2*x^2 - 2)*sqrt(c*x + 1)*sqrt
(-c*x + 1)/(a*b*c^3*x^5 - a*b*c*x^3 + (b^2*c^3*x^5 - b^2*c*x^3)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)), x
) + sqrt(c*x + 1)*sqrt(-c*x + 1))/(b^2*c*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*x^2)

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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))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^2*arccos(c*x)^2 + 2*a*b*x^2*arccos(c*x) + a^2*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 )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((b*arccos(c*x) + a)^2*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 )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*acos(c*x))^2),x)

[Out]

int(1/(x^2*(a + b*acos(c*x))^2), x)

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