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

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b \cos ^{-1}(c x)\right )^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0226184, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (a+b \cos ^{-1}(c x)\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*ArcCos[c*x])^3),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.37358, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a+b \cos ^{-1}(c x)\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.317, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\arccos \left ( cx \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{c x + 1} \sqrt{-c x + 1} b c x + b \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a + \frac{2 \,{\left (b^{4} c^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )^{2} + 2 \, a b^{3} c^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a^{2} b^{2} c^{2} x^{2}\right )} \int \frac{1}{{\left (b \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a\right )} x^{3}}\,{d x}}{b^{2} c^{2}}}{2 \,{\left (b^{4} c^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )^{2} + 2 \, a b^{3} c^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a^{2} b^{2} c^{2} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} x \arccos \left (c x\right )^{3} + 3 \, a b^{2} x \arccos \left (c x\right )^{2} + 3 \, a^{2} b x \arccos \left (c x\right ) + a^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \arccos \left (c x\right ) + a\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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