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