3.139 \(\int \frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}},x\right ) \]

[Out]

Unintegrable[ArcCos[a*x]^n/(b*x)^(3/2), x]

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Rubi [A]  time = 0.0220214, 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 = {} \[ \int \frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcCos[a*x]^n/(b*x)^(3/2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.80819, size = 0, normalized size = 0. \[ \int \frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcCos[a*x]^n/(b*x)^(3/2),x]

[Out]

Integrate[ArcCos[a*x]^n/(b*x)^(3/2), x]

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Maple [A]  time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arccos \left ( ax \right ) \right ) ^{n} \left ( bx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^n/(b*x)^(3/2),x)

[Out]

int(arccos(a*x)^n/(b*x)^(3/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x)*arccos(a*x)^n/(b^2*x^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}^{n}{\left (a x \right )}}{\left (b x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**n/(b*x)**(3/2),x)

[Out]

Integral(acos(a*x)**n/(b*x)**(3/2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x\right )^{n}}{\left (b x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x)^n/(b*x)^(3/2), x)