Optimal. Leaf size=160 \[ \frac {x^3 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )}+\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )}+\frac {2 b^2 n^2 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{b^4 n^4+10 b^2 n^2+9}+\frac {2 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{3 \left (b^4 n^4+10 b^2 n^2+9\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4488, 4486} \[ \frac {2 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{3 \left (b^4 n^4+10 b^2 n^2+9\right )}+\frac {x^3 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )}+\frac {2 b^2 n^2 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{b^4 n^4+10 b^2 n^2+9}+\frac {b n x^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4486
Rule 4488
Rubi steps
\begin {align*} \int x^2 \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {x^3 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}+\frac {b n x^3 \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}+\frac {\left (2 b^2 n^2\right ) \int x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 \left (1+b^2 n^2\right )}\\ &=\frac {2 b^2 n^2 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{9+10 b^2 n^2+b^4 n^4}+\frac {x^3 \cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}+\frac {2 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{3 \left (9+10 b^2 n^2+b^4 n^4\right )}+\frac {b n x^3 \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.56, size = 120, normalized size = 0.75 \[ \frac {x^3 \left (27 \left (b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )+\left (b^2 n^2+9\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+2 b n \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+9\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+5 b^2 n^2+9\right )\right )}{12 \left (b^4 n^4+10 b^2 n^2+9\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 127, normalized size = 0.79 \[ \frac {6 \, b^{2} n^{2} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + {\left (b^{2} n^{2} + 9\right )} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} + {\left (2 \, b^{3} n^{3} x^{3} + {\left (b^{3} n^{3} + 9 \, b n\right )} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2}\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{3 \, {\left (b^{4} n^{4} + 10 \, b^{2} n^{2} + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\cos ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 1007, normalized size = 6.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.06, size = 122, normalized size = 0.76 \[ \frac {x^3\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}\,3{}\mathrm {i}}{8\,b\,n+24{}\mathrm {i}}+\frac {3\,x^3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{24+b\,n\,8{}\mathrm {i}}+\frac {x^3\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{24\,b\,n+24{}\mathrm {i}}+\frac {x^3\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}{24+b\,n\,24{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________