Optimal. Leaf size=57 \[ -\frac {7 \tan (x)}{8 \sqrt {9 \tan ^2(x)+4}}+\frac {3}{8} \log \left (9 \tan ^2(x)+4\right )-\frac {3}{4} \log (\tan (x))-\frac {\cot (x)}{4 \sqrt {9 \tan ^2(x)+4}} \]
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Rubi [A] time = 0.85, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {6742, 191, 271, 266, 36, 29, 31} \[ -\frac {7 \tan (x)}{8 \sqrt {9 \tan ^2(x)+4}}+\frac {3}{8} \log \left (9 \tan ^2(x)+4\right )-\frac {3}{4} \log (\tan (x))-\frac {\cot (x)}{4 \sqrt {9 \tan ^2(x)+4}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 191
Rule 266
Rule 271
Rule 6742
Rubi steps
\begin {align*} \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2-3 x \sqrt {4+9 x^2}}{x^2 \left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{\left (4+9 x^2\right )^{3/2}}+\frac {1}{x^2 \left (4+9 x^2\right )^{3/2}}-\frac {3}{x \left (4+9 x^2\right )}\right ) \, dx,x,\tan (x)\right )\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{x \left (4+9 x^2\right )} \, dx,x,\tan (x)\right )\right )+\operatorname {Subst}\left (\int \frac {1}{\left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right )+\operatorname {Subst}\left (\int \frac {1}{x^2 \left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=-\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}+\frac {\tan (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{x (4+9 x)} \, dx,x,\tan ^2(x)\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{\left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=-\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {7 \tan (x)}{8 \sqrt {4+9 \tan ^2(x)}}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\tan ^2(x)\right )+\frac {27}{8} \operatorname {Subst}\left (\int \frac {1}{4+9 x} \, dx,x,\tan ^2(x)\right )\\ &=-\frac {3}{4} \log (\tan (x))+\frac {3}{8} \log \left (4+9 \tan ^2(x)\right )-\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {7 \tan (x)}{8 \sqrt {4+9 \tan ^2(x)}}\\ \end {align*}
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Mathematica [B] time = 0.92, size = 116, normalized size = 2.04 \[ \frac {-5 \tan (x)+5 \cot (x)-9 \csc (x) \sec (x)-6 \sqrt {2} \log \left (\tan \left (\frac {x}{2}\right )\right ) \sqrt {5 \tan ^2(x)+13 \sec ^2(x)-5}+6 \sqrt {\frac {13-5 \cos (2 x)}{\cos (2 x)+1}} \log \left (\tan ^4\left (\frac {x}{2}\right )+7 \tan ^2\left (\frac {x}{2}\right )+1\right )}{16 \sqrt {\frac {13-5 \cos (2 x)}{\cos (2 x)+1}}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.58, size = 84, normalized size = 1.47 \[ \frac {3 \, {\left (5 \, \cos \relax (x)^{2} - 9\right )} \log \left (-\frac {5}{4} \, \cos \relax (x)^{2} + \frac {9}{4}\right ) \sin \relax (x) - 6 \, {\left (5 \, \cos \relax (x)^{2} - 9\right )} \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x) - {\left (5 \, \cos \relax (x)^{3} - 7 \, \cos \relax (x)\right )} \sqrt {-\frac {5 \, \cos \relax (x)^{2} - 9}{\cos \relax (x)^{2}}}}{8 \, {\left (5 \, \cos \relax (x)^{2} - 9\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \relax (x)^{2} - 3 \, \sqrt {4 \, \sec \relax (x)^{2} + 5 \, \tan \relax (x)^{2}} \tan \relax (x)}{{\left (4 \, \sec \relax (x)^{2} + 5 \, \tan \relax (x)^{2}\right )}^{\frac {3}{2}} \sin \relax (x)^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.97, size = 117, normalized size = 2.05
method | result | size |
default | \(-\frac {6 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x ) \left (-\frac {5 \left (\cos ^{2}\relax (x )\right )-9}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}} \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-3 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x ) \left (-\frac {5 \left (\cos ^{2}\relax (x )\right )-9}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}} \ln \left (-\frac {5 \left (\cos ^{2}\relax (x )\right )-9}{\left (1+\cos \relax (x )\right )^{2}}\right )+25 \left (\cos ^{4}\relax (x )\right )-80 \left (\cos ^{2}\relax (x )\right )+63}{8 \cos \relax (x )^{3} \sin \relax (x ) \left (-\frac {5 \left (\cos ^{2}\relax (x )\right )-9}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 47, normalized size = 0.82 \[ -\frac {7 \, \tan \relax (x)}{8 \, \sqrt {9 \, \tan \relax (x)^{2} + 4}} - \frac {1}{4 \, \sqrt {9 \, \tan \relax (x)^{2} + 4} \tan \relax (x)} + \frac {3}{8} \, \log \left (9 \, \tan \relax (x)^{2} + 4\right ) - \frac {3}{4} \, \log \left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 113, normalized size = 1.98 \[ \frac {3\,\ln \left (\left (\cos \left (2\,x\right )+\sin \left (2\,x\right )\,1{}\mathrm {i}\right )\,\left (5\,\cos \left (2\,x\right )-13\right )\right )}{8}-\frac {3\,\ln \left (\cos \left (2\,x\right )\,852930{}\mathrm {i}-852930\,\sin \left (2\,x\right )-852930{}\mathrm {i}\right )}{4}-\frac {\frac {18\,\sin \left (2\,x\right )\,\sqrt {13-5\,\cos \left (2\,x\right )}}{\sqrt {\cos \left (2\,x\right )+1}}-\frac {5\,\sin \left (4\,x\right )\,\sqrt {13-5\,\cos \left (2\,x\right )}}{\sqrt {\cos \left (2\,x\right )+1}}}{80\,{\cos \left (2\,x\right )}^2-288\,\cos \left (2\,x\right )+208} \]
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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