Integrand size = 15, antiderivative size = 54 \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {1}{32} \arctan \left (\frac {1}{2} \sqrt {1+5 \tan ^2(x)}\right )-\frac {1}{12 \left (1+5 \tan ^2(x)\right )^{3/2}}+\frac {1}{16 \sqrt {1+5 \tan ^2(x)}} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3751, 455, 53, 65, 209} \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {1}{32} \arctan \left (\frac {1}{2} \sqrt {5 \tan ^2(x)+1}\right )+\frac {1}{16 \sqrt {5 \tan ^2(x)+1}}-\frac {1}{12 \left (5 \tan ^2(x)+1\right )^{3/2}} \]
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Rule 53
Rule 65
Rule 209
Rule 455
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (1+5 x^2\right )^{5/2}} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) (1+5 x)^{5/2}} \, dx,x,\tan ^2(x)\right ) \\ & = -\frac {1}{12 \left (1+5 \tan ^2(x)\right )^{3/2}}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{(1+x) (1+5 x)^{3/2}} \, dx,x,\tan ^2(x)\right ) \\ & = -\frac {1}{12 \left (1+5 \tan ^2(x)\right )^{3/2}}+\frac {1}{16 \sqrt {1+5 \tan ^2(x)}}+\frac {1}{32} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+5 x}} \, dx,x,\tan ^2(x)\right ) \\ & = -\frac {1}{12 \left (1+5 \tan ^2(x)\right )^{3/2}}+\frac {1}{16 \sqrt {1+5 \tan ^2(x)}}+\frac {1}{80} \text {Subst}\left (\int \frac {1}{\frac {4}{5}+\frac {x^2}{5}} \, dx,x,\sqrt {1+5 \tan ^2(x)}\right ) \\ & = \frac {1}{32} \arctan \left (\frac {1}{2} \sqrt {1+5 \tan ^2(x)}\right )-\frac {1}{12 \left (1+5 \tan ^2(x)\right )^{3/2}}+\frac {1}{16 \sqrt {1+5 \tan ^2(x)}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.31 \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {(-3+2 \cos (2 x)) \left (-6 \cos (x)+8 \cos (3 x)-3 (-3+2 \cos (2 x))^{3/2} \log \left (2 \cos (x)+\sqrt {-3+2 \cos (2 x)}\right )\right ) \sec ^5(x)}{96 \left (1+5 \tan ^2(x)\right )^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {\sqrt {1+5 \left (\tan ^{2}\left (x \right )\right )}}{2}\right )}{32}+\frac {1}{16 \sqrt {1+5 \left (\tan ^{2}\left (x \right )\right )}}-\frac {1}{12 {\left (1+5 \left (\tan ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}\) | \(41\) |
default | \(\frac {\arctan \left (\frac {\sqrt {1+5 \left (\tan ^{2}\left (x \right )\right )}}{2}\right )}{32}+\frac {1}{16 \sqrt {1+5 \left (\tan ^{2}\left (x \right )\right )}}-\frac {1}{12 {\left (1+5 \left (\tan ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}\) | \(41\) |
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {3 \, {\left (25 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 1\right )} \arctan \left (\frac {5 \, \tan \left (x\right )^{2} - 3}{4 \, \sqrt {5 \, \tan \left (x\right )^{2} + 1}}\right ) + 4 \, {\left (15 \, \tan \left (x\right )^{2} - 1\right )} \sqrt {5 \, \tan \left (x\right )^{2} + 1}}{192 \, {\left (25 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 1\right )}} \]
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Time = 2.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {\operatorname {atan}{\left (\frac {\sqrt {5 \tan ^{2}{\left (x \right )} + 1}}{2} \right )}}{32} + \frac {1}{16 \sqrt {5 \tan ^{2}{\left (x \right )} + 1}} - \frac {1}{12 \left (5 \tan ^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}} \]
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\[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (5 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {15 \, \tan \left (x\right )^{2} - 1}{48 \, {\left (5 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {3}{2}}} + \frac {1}{32} \, \arctan \left (\frac {1}{2} \, \sqrt {5 \, \tan \left (x\right )^{2} + 1}\right ) \]
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Time = 0.53 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.19 \[ \int \frac {\tan (x)}{\left (1+5 \tan ^2(x)\right )^{5/2}} \, dx=\frac {\ln \left (\mathrm {tan}\left (x\right )-\frac {2\,\sqrt {5}\,\sqrt {{\mathrm {tan}\left (x\right )}^2+\frac {1}{5}}}{5}+\frac {1}{5}{}\mathrm {i}\right )\,1{}\mathrm {i}}{64}+\frac {\ln \left (\mathrm {tan}\left (x\right )+\frac {2\,\sqrt {5}\,\sqrt {{\mathrm {tan}\left (x\right )}^2+\frac {1}{5}}}{5}-\frac {1}{5}{}\mathrm {i}\right )\,1{}\mathrm {i}}{64}-\frac {\ln \left (\mathrm {tan}\left (x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{64}-\frac {\ln \left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64}-\frac {\sqrt {{\mathrm {tan}\left (x\right )}^2+\frac {1}{5}}\,1{}\mathrm {i}}{96\,\left (\mathrm {tan}\left (x\right )-\frac {\sqrt {5}\,1{}\mathrm {i}}{5}\right )}+\frac {\sqrt {{\mathrm {tan}\left (x\right )}^2+\frac {1}{5}}\,1{}\mathrm {i}}{96\,\left (\mathrm {tan}\left (x\right )+\frac {\sqrt {5}\,1{}\mathrm {i}}{5}\right )}+\frac {\sqrt {5}\,\sqrt {{\mathrm {tan}\left (x\right )}^2+\frac {1}{5}}}{240\,\left ({\mathrm {tan}\left (x\right )}^2+\frac {2{}\mathrm {i}\,\sqrt {5}\,\mathrm {tan}\left (x\right )}{5}-\frac {1}{5}\right )}-\frac {\sqrt {5}\,\sqrt {{\mathrm {tan}\left (x\right )}^2+\frac {1}{5}}}{240\,\left (-{\mathrm {tan}\left (x\right )}^2+\frac {2{}\mathrm {i}\,\sqrt {5}\,\mathrm {tan}\left (x\right )}{5}+\frac {1}{5}\right )} \]
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