Optimal. Leaf size=47 \[ \frac {3}{5} \cos ^3(x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}+\frac {3}{11} \cos (x) \sin ^3(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1986, 15, 14}
\begin {gather*} \frac {3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac {3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 15
Rule 1986
Rubi steps
\begin {align*} \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt [3]{x^2 \left (1+x^2\right )^6}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \text {Subst}\left (\int x^{2/3} \left (1+x^2\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac {2}{3}}(x)}\\ &=\frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \text {Subst}\left (\int \left (x^{2/3}+x^{8/3}\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac {2}{3}}(x)}\\ &=\frac {3}{5} \cos ^3(x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}+\frac {3}{11} \cos (x) \sin ^3(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 63, normalized size = 1.34 \begin {gather*} \frac {3 \cos (x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \left (-3+8 \left (-\tan ^2(x)\right )^{5/6}+3 \cos (2 x) \left (-1+\left (-\tan ^2(x)\right )^{5/6}\right )\right )}{55 \left (-\tan ^2(x)\right )^{5/6}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \left (\frac {\sin ^{2}\left (x \right )}{\cos \left (x \right )^{14}}\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 1.17, size = 13, normalized size = 0.28 \begin {gather*} \frac {3}{11} \, \tan \left (x\right )^{\frac {11}{3}} + \frac {3}{5} \, \tan \left (x\right )^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.38, size = 29, normalized size = 0.62 \begin {gather*} \frac {3}{55} \, {\left (6 \, \cos \left (x\right )^{3} + 5 \, \cos \left (x\right )\right )} \left (-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )^{14}}\right )^{\frac {1}{3}} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 3.94, size = 32, normalized size = 0.68 \begin {gather*} \frac {6\,\sin \left (2\,x\right )\,{\left (1-\cos \left (2\,x\right )\right )}^{1/3}\,\left (3\,\cos \left (2\,x\right )+8\right )}{55\,{\left (\cos \left (2\,x\right )+1\right )}^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________