3.2.20 \(\int (b x)^m \text {ArcCos}(a x)^3 \, dx\) [120]

Optimal. Leaf size=65 \[ \frac {(b x)^{1+m} \text {ArcCos}(a x)^3}{b (1+m)}+\frac {3 a \text {Int}\left (\frac {(b x)^{1+m} \text {ArcCos}(a x)^2}{\sqrt {1-a^2 x^2}},x\right )}{b (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.08, 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 (b x)^m \text {ArcCos}(a x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(b*x)^m*ArcCos[a*x]^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.91, size = 0, normalized size = 0.00 \begin {gather*} \int (b x)^m \text {ArcCos}(a x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(b*x)^m*ArcCos[a*x]^3,x]

[Out]

Integrate[(b*x)^m*ArcCos[a*x]^3, x]

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Maple [A]
time = 0.61, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \arccos \left (a x \right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*arccos(a*x)^3,x)

[Out]

int((b*x)^m*arccos(a*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((b*x)^m*arccos(a*x)^3,x, algorithm="maxima")

[Out]

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

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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((b*x)^m*arccos(a*x)^3,x, algorithm="fricas")

[Out]

integral((b*x)^m*arccos(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*acos(a*x)**3,x)

[Out]

Integral((b*x)**m*acos(a*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((b*x)^m*arccos(a*x)^3,x, algorithm="giac")

[Out]

integrate((b*x)^m*arccos(a*x)^3, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {acos}\left (a\,x\right )}^3\,{\left (b\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)^3*(b*x)^m,x)

[Out]

int(acos(a*x)^3*(b*x)^m, x)

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