Optimal. Leaf size=133 \[ \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [6]{1-3 \sec ^2(x)}}{\sqrt {3}}\right )+\frac {1}{4} \log \left (\sec ^2(x)\right )-\frac {3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {1-3 \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}+\frac {1}{2 \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 3.19, antiderivative size = 174, normalized size of antiderivative = 1.31, number
of steps used = 29, number of rules used = 16, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules
used = {4446, 6874, 6816, 267, 6829, 348, 59, 632, 210, 31, 6820, 272, 43, 65, 212, 25}
\begin {gather*} \frac {\cos ^2(x)}{6}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\sqrt [6]{1-3 \sec ^2(x)}-\frac {3}{2} \log \left (1-\sqrt [6]{1-3 \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {1-3 \sec ^2(x)}\right )+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}+\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{1-3 \sec ^2(x)}+1}{\sqrt {3}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-3 \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (\left (3-\cos ^2(x)\right ) \sec ^2(x)\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 25
Rule 31
Rule 43
Rule 59
Rule 65
Rule 210
Rule 212
Rule 267
Rule 272
Rule 348
Rule 632
Rule 4446
Rule 6816
Rule 6820
Rule 6829
Rule 6874
Rubi steps
\begin {align*} \int \frac {\sec ^2(x) \tan (x) \left (\sqrt [3]{1-3 \sec ^2(x)} \sin ^2(x)+3 \tan ^2(x)\right )}{\left (1-3 \sec ^2(x)\right )^{5/6} \left (1-\sqrt {1-3 \sec ^2(x)}\right )} \, dx &=-\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (3+\sqrt [3]{1-\frac {3}{x^2}} x^2\right )}{\left (1-\sqrt {1-\frac {3}{x^2}}\right ) \left (1-\frac {3}{x^2}\right )^{5/6} x^5} \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {-3-x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}+\frac {3+x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}\right ) \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \frac {-3-x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {3+x^2 \sqrt [3]{\frac {-3+x^2}{x^2}}}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{\sqrt {1-\frac {3}{x^2}} x^3 \left (1-\sqrt {\frac {-3+x^2}{x^2}}\right )}-\frac {3}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}\right ) \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \left (\frac {3}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}+\frac {1}{\sqrt {1-\frac {3}{x^2}} x \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )}\right ) \, dx,x,\cos (x)\right )\\ &=3 \text {Subst}\left (\int \frac {1}{\left (1-\frac {3}{x^2}\right )^{5/6} x^5 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-3 \text {Subst}\left (\int \frac {1}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3 \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {3}{x^2}} x^3 \left (1-\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {3}{x^2}} x \left (-1+\sqrt {\frac {-3+x^2}{x^2}}\right )} \, dx,x,\cos (x)\right )\\ &=\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (-1+\sqrt {x}\right ) x^{5/6}} \, dx,x,\left (-3+\cos ^2(x)\right ) \sec ^2(x)\right )+3 \text {Subst}\left (\int \left (-\frac {1}{3 \left (1-\frac {3}{x^2}\right )^{5/6} x^3}-\frac {1}{3 \sqrt [3]{1-\frac {3}{x^2}} x^3}\right ) \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{-\frac {3}{x}+x-\sqrt {1-\frac {3}{x^2}} x} \, dx,x,\cos (x)\right )\\ &=\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )-\text {Subst}\left (\int \frac {1}{\left (1-\frac {3}{x^2}\right )^{5/6} x^3} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{\sqrt [3]{1-\frac {3}{x^2}} x^3} \, dx,x,\cos (x)\right )-\text {Subst}\left (\int \frac {1}{(-1+x) x^{2/3}} \, dx,x,\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\text {Subst}\left (\int \left (-\frac {x}{3}-\frac {1}{3} \sqrt {1-\frac {3}{x^2}} x+\frac {\sqrt {1-\frac {3}{x^2}} x}{3-x^2}\right ) \, dx,x,\cos (x)\right )\\ &=\frac {\cos ^2(x)}{6}+\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}+\frac {1}{3} \text {Subst}\left (\int \sqrt {1-\frac {3}{x^2}} x \, dx,x,\cos (x)\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\text {Subst}\left (\int \frac {\sqrt {1-\frac {3}{x^2}} x}{3-x^2} \, dx,x,\cos (x)\right )\\ &=\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {1-3 x}}{x^2} \, dx,x,\sec ^2(x)\right )-3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [6]{\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )+\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {3}{x^2}} x} \, dx,x,\cos (x)\right )\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}}{\sqrt {3}}\right )+\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-3 x} x} \, dx,x,\sec ^2(x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-3 x} x} \, dx,x,\sec ^2(x)\right )\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}}{\sqrt {3}}\right )+\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {x^2}{3}} \, dx,x,\sqrt {1-3 \sec ^2(x)}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\frac {1}{3}-\frac {x^2}{3}} \, dx,x,\sqrt {1-3 \sec ^2(x)}\right )\\ &=\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}}{\sqrt {3}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-3 \sec ^2(x)}\right )+\frac {\cos ^2(x)}{6}-\frac {3}{2} \log \left (1-\sqrt [6]{-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{3} \log \left (1-\sqrt {-\left (3-\cos ^2(x)\right ) \sec ^2(x)}\right )+\frac {1}{2} \log \left (1-\sqrt {\left (-3+\cos ^2(x)\right ) \sec ^2(x)}\right )-\sqrt [6]{1-3 \sec ^2(x)}+\frac {1}{6} \cos ^2(x) \sqrt {1-3 \sec ^2(x)}-\frac {1}{4} \left (1-3 \sec ^2(x)\right )^{2/3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(370\) vs. \(2(133)=266\).
time = 54.12, size = 370, normalized size = 2.78 \begin {gather*} -\frac {\left (3+\cos ^2(x) \sqrt [3]{1-3 \sec ^2(x)}\right ) \left (6-12 \log \left (-1+\sqrt [6]{-2-3 \tan ^2(x)}\right )+6 \log \left (1+\sqrt [6]{-2-3 \tan ^2(x)}+\sqrt [3]{-2-3 \tan ^2(x)}\right )+\log \left (-1+\sqrt {-2-3 \tan ^2(x)}\right )+3 \log \left (1+\sqrt {-2-3 \tan ^2(x)}\right )-18 \sqrt [6]{-2-3 \tan ^2(x)}-9 \tan ^2(x) \sqrt [6]{-2-3 \tan ^2(x)}+12 \log \left (-1+\sqrt [6]{-2-3 \tan ^2(x)}\right ) \sqrt {-2-3 \tan ^2(x)}-6 \log \left (1+\sqrt [6]{-2-3 \tan ^2(x)}+\sqrt [3]{-2-3 \tan ^2(x)}\right ) \sqrt {-2-3 \tan ^2(x)}-\log \left (-1+\sqrt {-2-3 \tan ^2(x)}\right ) \sqrt {-2-3 \tan ^2(x)}-3 \log \left (1+\sqrt {-2-3 \tan ^2(x)}\right ) \sqrt {-2-3 \tan ^2(x)}+9 \left (-2-3 \tan ^2(x)\right )^{2/3}-12 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [6]{-2-3 \tan ^2(x)}}{\sqrt {3}}\right ) \left (-1+\sqrt {-2-3 \tan ^2(x)}\right )\right )}{6 \left (6+\sqrt [3]{-2-3 \tan ^2(x)}+\cos (2 x) \sqrt [3]{-2-3 \tan ^2(x)}\right ) \left (-1+\sqrt {-2-3 \tan ^2(x)}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\tan \left (x \right ) \left (\left (1-3 \left (\sec ^{2}\left (x \right )\right )\right )^{\frac {1}{3}} \left (\sin ^{2}\left (x \right )\right )+3 \left (\tan ^{2}\left (x \right )\right )\right )}{\cos \left (x \right )^{2} \left (1-3 \left (\sec ^{2}\left (x \right )\right )\right )^{\frac {5}{6}} \left (1-\sqrt {1-3 \left (\sec ^{2}\left (x \right )\right )}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)]
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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Limit: Max order reached or unable to make series expansion Error: Bad Argument Value} \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 \frac {\mathrm {tan}\left (x\right )\,\left ({\sin \left (x\right )}^2\,{\left (1-\frac {3}{{\cos \left (x\right )}^2}\right )}^{1/3}+3\,{\mathrm {tan}\left (x\right )}^2\right )}{{\cos \left (x\right )}^2\,\left (\sqrt {1-\frac {3}{{\cos \left (x\right )}^2}}-1\right )\,{\left (1-\frac {3}{{\cos \left (x\right )}^2}\right )}^{5/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________