Optimal. Leaf size=88 \[ \frac {1}{3} a^2 \tan ^3(x)+a^2 \tan (x)-\frac {1}{2} a^2 \cot ^2(x)+\left (a^2+4\right ) \log (\sin (x))+4 a x+4 a \cot (x)+(4-a) a \log (\cos (x))+\frac {x}{2}+\cos ^4(x)+2 \cos ^2(x)-\sin (x) \cos ^3(x)+\frac {1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.55, antiderivative size = 84, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1805, 1802, 635, 203, 260} \[ \frac {1}{3} a^2 \tan ^3(x)+a^2 \tan (x)-\frac {1}{2} a^2 \cot ^2(x)+\left (a^2+4\right ) \log (\tan (x))+\frac {1}{2} (8 a+1) x+4 a \cot (x)+4 (a+1) \log (\cos (x))+\cos ^4(x) (1-\tan (x))+\frac {1}{2} \cos ^2(x) (\tan (x)+4) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1802
Rule 1805
Rubi steps
\begin {align*} \int \left (1+\cot ^3(x)\right ) \left (a \sec ^2(x)-\sin (2 x)\right )^2 \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^3\right ) \left (a-2 x+2 a x^2+a x^4\right )^2}{x^3 \left (1+x^2\right )^3} \, dx,x,\tan (x)\right )\\ &=\cos ^4(x) (1-\tan (x))-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-4 a^2+16 a x-4 \left (4+3 a^2\right ) x^2-4 \left (1-4 a+a^2\right ) x^3+4 (4-3 a) a x^4-12 a^2 x^5+4 (4-a) a x^6-12 a^2 x^7-4 a^2 x^9}{x^3 \left (1+x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=\cos ^4(x) (1-\tan (x))+\frac {1}{2} \cos ^2(x) (4+\tan (x))+\frac {1}{8} \operatorname {Subst}\left (\int \frac {8 a^2-32 a x+16 \left (2+a^2\right ) x^2+4 \left (1+2 a^2\right ) x^3-8 (4-a) a x^4+16 a^2 x^5+8 a^2 x^7}{x^3 \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\cos ^4(x) (1-\tan (x))+\frac {1}{2} \cos ^2(x) (4+\tan (x))+\frac {1}{8} \operatorname {Subst}\left (\int \left (8 a^2+\frac {8 a^2}{x^3}-\frac {32 a}{x^2}+\frac {8 \left (4+a^2\right )}{x}+8 a^2 x^2+\frac {4 (1+8 a-8 (1+a) x)}{1+x^2}\right ) \, dx,x,\tan (x)\right )\\ &=4 a \cot (x)-\frac {1}{2} a^2 \cot ^2(x)+\left (4+a^2\right ) \log (\tan (x))+\cos ^4(x) (1-\tan (x))+a^2 \tan (x)+\frac {1}{3} a^2 \tan ^3(x)+\frac {1}{2} \cos ^2(x) (4+\tan (x))+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+8 a-8 (1+a) x}{1+x^2} \, dx,x,\tan (x)\right )\\ &=4 a \cot (x)-\frac {1}{2} a^2 \cot ^2(x)+\left (4+a^2\right ) \log (\tan (x))+\cos ^4(x) (1-\tan (x))+a^2 \tan (x)+\frac {1}{3} a^2 \tan ^3(x)+\frac {1}{2} \cos ^2(x) (4+\tan (x))-(4 (1+a)) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (x)\right )+\frac {1}{2} (1+8 a) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} (1+8 a) x+4 a \cot (x)-\frac {1}{2} a^2 \cot ^2(x)+4 (1+a) \log (\cos (x))+\left (4+a^2\right ) \log (\tan (x))+\cos ^4(x) (1-\tan (x))+a^2 \tan (x)+\frac {1}{3} a^2 \tan ^3(x)+\frac {1}{2} \cos ^2(x) (4+\tan (x))\\ \end {align*}
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Mathematica [A] time = 2.02, size = 127, normalized size = 1.44 \[ -\frac {2 \sin (x) \cos ^3(x) \left (\sin (2 x)-a \sec ^2(x)\right )^2 \left (-8 a^2 (\cos (2 x)+2) \sec ^2(x)-3 \cot (x) \left (-4 a^2 \csc ^2(x)+8 a^2 \log (\sin (x))-8 a^2 \log (\cos (x))+32 a x+32 a \log (\cos (x))+4 x-\sin (4 x)+12 \cos (2 x)+\cos (4 x)+32 \log (\sin (x))\right )-96 a \cot ^2(x)\right )}{3 (-4 a+2 \sin (2 x)+\sin (4 x))^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+\cot ^3(x)\right ) \left (a \sec ^2(x)-\sin (2 x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.95, size = 178, normalized size = 2.02 \[ \frac {24 \, \cos \relax (x)^{9} + 24 \, \cos \relax (x)^{7} + 3 \, {\left (4 \, {\left (8 \, a + 1\right )} x - 27\right )} \cos \relax (x)^{5} + 3 \, {\left (4 \, a^{2} - 4 \, {\left (8 \, a + 1\right )} x + 11\right )} \cos \relax (x)^{3} - 12 \, {\left ({\left (a^{2} - 4 \, a\right )} \cos \relax (x)^{5} - {\left (a^{2} - 4 \, a\right )} \cos \relax (x)^{3}\right )} \log \left (\cos \relax (x)^{2}\right ) + 12 \, {\left ({\left (a^{2} + 4\right )} \cos \relax (x)^{5} - {\left (a^{2} + 4\right )} \cos \relax (x)^{3}\right )} \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right ) - 4 \, {\left (6 \, \cos \relax (x)^{8} - 9 \, \cos \relax (x)^{6} - {\left (4 \, a^{2} - 24 \, a - 3\right )} \cos \relax (x)^{4} + 2 \, a^{2} \cos \relax (x)^{2} + 2 \, a^{2}\right )} \sin \relax (x)}{24 \, {\left (\cos \relax (x)^{5} - \cos \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 149, normalized size = 1.69 \[ \frac {1}{3} \, a^{2} \tan \relax (x)^{3} + a^{2} \tan \relax (x) + \frac {1}{2} \, {\left (8 \, a + 1\right )} x - 2 \, {\left (a + 1\right )} \log \left (\tan \relax (x)^{2} + 1\right ) + {\left (a^{2} + 4\right )} \log \left ({\left | \tan \relax (x) \right |}\right ) - \frac {a^{2} \tan \relax (x)^{6} - 4 \, a \tan \relax (x)^{6} + 3 \, a^{2} \tan \relax (x)^{4} - 8 \, a \tan \relax (x)^{5} - 8 \, a \tan \relax (x)^{4} - \tan \relax (x)^{5} + 3 \, a^{2} \tan \relax (x)^{2} - 16 \, a \tan \relax (x)^{3} - 4 \, \tan \relax (x)^{4} - 4 \, a \tan \relax (x)^{2} + \tan \relax (x)^{3} + a^{2} - 8 \, a \tan \relax (x) - 6 \, \tan \relax (x)^{2}}{2 \, {\left (\tan \relax (x)^{3} + \tan \relax (x)\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.56, size = 210, normalized size = 2.39
method | result | size |
default | \(\frac {2 \left (\cos ^{8}\relax (x )\right )}{\sin \relax (x )^{2}}+2 \left (\cos ^{6}\relax (x )\right )+\cos ^{4}\relax (x )+2 \left (\cos ^{2}\relax (x )\right )+4 \ln \left (\sin \relax (x )\right )-\frac {4 \left (\cos ^{7}\relax (x )\right )}{\sin \relax (x )}-4 \left (\cos ^{5}\relax (x )+\frac {5 \left (\cos ^{3}\relax (x )\right )}{4}+\frac {15 \cos \relax (x )}{8}\right ) \sin \relax (x )+\frac {x}{2}-\frac {2 \left (\cos ^{6}\relax (x )\right )}{\sin \relax (x )^{2}}+\frac {8 \left (\cos ^{5}\relax (x )\right )}{\sin \relax (x )}+8 \left (\cos ^{3}\relax (x )+\frac {3 \cos \relax (x )}{2}\right ) \sin \relax (x )-4 a \left (-\frac {\left (\cot ^{2}\relax (x )\right )}{2}-\ln \left (\sin \relax (x )\right )\right )-4 a \left (-\cot \relax (x )-x \right )-4 \cot \relax (x )-\frac {4 a}{\sin \relax (x )^{2}}+a^{2} \left (-\frac {1}{2 \sin \relax (x )^{2}}+\ln \left (\tan \relax (x )\right )\right )-a^{2} \left (\frac {1}{\sin \relax (x ) \cos \relax (x )}-2 \cot \relax (x )\right )-4 a \left (-\frac {1}{2 \sin \relax (x )^{2}}+\ln \left (\tan \relax (x )\right )\right )+a^{2} \left (\frac {1}{3 \sin \relax (x ) \cos \relax (x )^{3}}+\frac {4}{3 \sin \relax (x ) \cos \relax (x )}-\frac {8 \cot \relax (x )}{3}\right )\) | \(210\) |
risch | \(\frac {x}{2}-4 i x a +4 a x +\frac {i {\mathrm e}^{4 i x}}{16}+\frac {{\mathrm e}^{4 i x}}{16}-\frac {i {\mathrm e}^{-4 i x}}{16}+\frac {3 \,{\mathrm e}^{2 i x}}{4}+\frac {3 \,{\mathrm e}^{-2 i x}}{4}+\frac {{\mathrm e}^{-4 i x}}{16}-4 i x +\frac {2 a \left (12 i {\mathrm e}^{8 i x}+3 a \,{\mathrm e}^{8 i x}+6 i a \,{\mathrm e}^{6 i x}+24 i {\mathrm e}^{6 i x}+9 a \,{\mathrm e}^{6 i x}-10 i a \,{\mathrm e}^{4 i x}+9 a \,{\mathrm e}^{4 i x}+2 i a \,{\mathrm e}^{2 i x}-24 i {\mathrm e}^{2 i x}+3 a \,{\mathrm e}^{2 i x}+2 i a -12 i\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{2} \left (1+{\mathrm e}^{2 i x}\right )^{3}}+\ln \left ({\mathrm e}^{2 i x}-1\right ) a^{2}+4 \ln \left ({\mathrm e}^{2 i x}-1\right )-\ln \left (1+{\mathrm e}^{2 i x}\right ) a^{2}+4 \ln \left (1+{\mathrm e}^{2 i x}\right ) a\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 115, normalized size = 1.31 \[ \frac {1}{3} \, {\left (\tan \relax (x)^{3} + 3 \, \tan \relax (x)\right )} a^{2} - \frac {1}{2} \, a^{2} {\left (\frac {1}{\sin \relax (x)^{2}} + \log \left (\sin \relax (x)^{2} - 1\right ) - \log \left (\sin \relax (x)^{2}\right )\right )} + 4 \, a {\left (x + \frac {1}{\tan \relax (x)}\right )} + 2 \, a \log \left (-\sin \relax (x)^{2} + 1\right ) + \frac {1}{2} \, x + \frac {1}{8} \, \cos \left (4 \, x\right ) + \frac {3}{2} \, \cos \left (2 \, x\right ) + 2 \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + 2 \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - \frac {1}{8} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 133, normalized size = 1.51 \[ a^2\,\mathrm {tan}\relax (x)-\frac {{\mathrm {tan}\relax (x)}^4\,\left (\frac {a^2}{2}-2\right )-4\,a\,\mathrm {tan}\relax (x)+\frac {a^2}{2}-{\mathrm {tan}\relax (x)}^5\,\left (4\,a+\frac {1}{2}\right )-{\mathrm {tan}\relax (x)}^3\,\left (8\,a-\frac {1}{2}\right )+{\mathrm {tan}\relax (x)}^2\,\left (a^2-3\right )}{{\mathrm {tan}\relax (x)}^6+2\,{\mathrm {tan}\relax (x)}^4+{\mathrm {tan}\relax (x)}^2}-\ln \left (\mathrm {tan}\relax (x)-\mathrm {i}\right )\,\left (a\,\left (2+2{}\mathrm {i}\right )+2+\frac {1}{4}{}\mathrm {i}\right )-\ln \left (\mathrm {tan}\relax (x)+1{}\mathrm {i}\right )\,\left (a\,\left (2-2{}\mathrm {i}\right )+2-\frac {1}{4}{}\mathrm {i}\right )+\frac {a^2\,{\mathrm {tan}\relax (x)}^3}{3}+\ln \left (\mathrm {tan}\relax (x)\right )\,\left (a^2+4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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