Optimal. Leaf size=150 \[ \frac {2 a^2 (b x)^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};a^2 x^2\right )}{b^3 (m+1) (m+2) (m+3)}+\frac {2 a \cos ^{-1}(a x) (b x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{b^2 (m+1) (m+2)}+\frac {\cos ^{-1}(a x)^2 (b x)^{m+1}}{b (m+1)} \]
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Rubi [A] time = 0.11, 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 = {4628, 4712} \[ \frac {2 a^2 (b x)^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};a^2 x^2\right )}{b^3 (m+1) (m+2) (m+3)}+\frac {2 a \cos ^{-1}(a x) (b x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{b^2 (m+1) (m+2)}+\frac {\cos ^{-1}(a x)^2 (b x)^{m+1}}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4712
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] time = 2.38, size = 132, normalized size = 0.88 \[ \frac {x (b x)^m \left (a x \left (\sqrt {\pi } a 2^{-m} x \Gamma (m+2) \, _3\tilde {F}_2\left (1,\frac {m+3}{2},\frac {m+3}{2};\frac {m+4}{2},\frac {m+5}{2};a^2 x^2\right )+\frac {8 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \, _2F_1\left (1,\frac {m+3}{2};\frac {m+4}{2};a^2 x^2\right )}{m+2}\right )+4 \cos ^{-1}(a x)^2\right )}{4 (m+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b x\right )^{m} \arccos \left (a x\right )^{2}, x\right ) \]
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 \[ \int \left (b x\right )^{m} \arccos \left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.48, 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(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\mathrm {acos}\left (a\,x\right )}^2\,{\left (b\,x\right )}^m \,d x \]
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 \[ \int \left (b x\right )^{m} \operatorname {acos}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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