3.197 \(\int \frac {1}{x^2 (a+b \cos ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 13.94, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a +b \arccos \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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