Optimal. Leaf size=150 \[ \frac {(b x)^{1+m} \text {ArcCos}(a x)^2}{b (1+m)}+\frac {2 a (b x)^{2+m} \text {ArcCos}(a x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{b^2 (1+m) (2+m)}+\frac {2 a^2 (b x)^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{b^3 (1+m) (2+m) (3+m)} \]
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Rubi [A]
time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4724, 4806}
\begin {gather*} \frac {2 a^2 (b x)^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};a^2 x^2\right )}{b^3 (m+1) (m+2) (m+3)}+\frac {2 a \text {ArcCos}(a x) (b x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{b^2 (m+1) (m+2)}+\frac {\text {ArcCos}(a x)^2 (b x)^{m+1}}{b (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 4724
Rule 4806
Rubi steps
\begin {align*} \int (b x)^m \cos ^{-1}(a x)^2 \, dx &=\frac {(b x)^{1+m} \cos ^{-1}(a x)^2}{b (1+m)}+\frac {(2 a) \int \frac {(b x)^{1+m} \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{b (1+m)}\\ &=\frac {(b x)^{1+m} \cos ^{-1}(a x)^2}{b (1+m)}+\frac {2 a (b x)^{2+m} \cos ^{-1}(a x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{b^2 (1+m) (2+m)}+\frac {2 a^2 (b x)^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};a^2 x^2\right )}{b^3 (1+m) (2+m) (3+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 1.72, size = 132, normalized size = 0.88 \begin {gather*} \frac {x (b x)^m \left (4 \text {ArcCos}(a x)^2+a x \left (\frac {8 \sqrt {1-a^2 x^2} \text {ArcCos}(a x) \, _2F_1\left (1,\frac {3+m}{2};\frac {4+m}{2};a^2 x^2\right )}{2+m}+2^{-m} a \sqrt {\pi } x \text {Gamma}(2+m) \, _3\tilde {F}_2\left (1,\frac {3+m}{2},\frac {3+m}{2};\frac {4+m}{2},\frac {5+m}{2};a^2 x^2\right )\right )\right )}{4 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.56, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \arccos \left (a x \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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 [F]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x\right )^{m} \operatorname {acos}^{2}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {acos}\left (a\,x\right )}^2\,{\left (b\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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