Optimal. Leaf size=68 \[ \frac {(b x)^{1+m} \text {ArcCos}(a x)}{b (1+m)}+\frac {a (b x)^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{b^2 (1+m) (2+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4724, 371}
\begin {gather*} \frac {a (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) (b x)^{m+1}}{b (m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 4724
Rubi steps
\begin {align*} \int (b x)^m \cos ^{-1}(a x) \, dx &=\frac {(b x)^{1+m} \cos ^{-1}(a x)}{b (1+m)}+\frac {a \int \frac {(b x)^{1+m}}{\sqrt {1-a^2 x^2}} \, dx}{b (1+m)}\\ &=\frac {(b x)^{1+m} \cos ^{-1}(a x)}{b (1+m)}+\frac {a (b x)^{2+m} \, _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)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 54, normalized size = 0.79 \begin {gather*} \frac {x (b x)^m \left ((2+m) \text {ArcCos}(a x)+a x \, _2F_1\left (\frac {1}{2},1+\frac {m}{2};2+\frac {m}{2};a^2 x^2\right )\right )}{(1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.82, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \arccos \left (a x \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x\right )^{m} \operatorname {acos}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {acos}\left (a\,x\right )\,{\left (b\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________