Integrand size = 25, antiderivative size = 222 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}+\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}+\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}} \]
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Time = 0.26 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3623, 12, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}+\frac {\sqrt {2} a^2 \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{d^{3/2} f}-\frac {a^2 \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} d^{3/2} f}+\frac {a^2 \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}} \]
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Rule 12
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rule 3623
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {2 a^2 d}{\sqrt {d \tan (e+f x)}} \, dx}{d^2} \\ & = -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt {d \tan (e+f x)}} \, dx}{d} \\ & = -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \tan (e+f x)\right )}{f} \\ & = -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f} \\ & = -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{d f}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{d f} \\ & = -\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}-\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}-\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{d f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{d f} \\ & = -\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}+\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}}+\frac {\left (\sqrt {2} a^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {\left (\sqrt {2} a^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f} \\ & = -\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}+\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{3/2} f}-\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}+\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{3/2} f}-\frac {2 a^2}{d f \sqrt {d \tan (e+f x)}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.83 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {a^2 \left (4+2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {\tan (e+f x)}-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {\tan (e+f x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {\tan (e+f x)}-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {\tan (e+f x)}\right )}{2 d f \sqrt {d \tan (e+f x)}} \]
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Time = 0.91 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {1}{\sqrt {d \tan \left (f x +e \right )}}\right )}{f d}\) | \(159\) |
default | \(\frac {2 a^{2} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {1}{\sqrt {d \tan \left (f x +e \right )}}\right )}{f d}\) | \(159\) |
parts | \(\frac {2 a^{2} d \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}-\frac {1}{d^{2} \sqrt {d \tan \left (f x +e \right )}}\right )}{f}+\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f d \left (d^{2}\right )^{\frac {1}{4}}}+\frac {a^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{2 f \,d^{2}}\) | \(441\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.21 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=\frac {d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{2}\right ) \tan \left (f x + e\right ) + i \, d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{2}\right ) \tan \left (f x + e\right ) - i \, d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{2}\right ) \tan \left (f x + e\right ) - d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-d^{2} f \left (-\frac {a^{8}}{d^{6} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{2}\right ) \tan \left (f x + e\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )} a^{2}}{d^{2} f \tan \left (f x + e\right )} \]
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\[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=a^{2} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
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Time = 0.34 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.79 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=\frac {a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {4 \, a^{2}}{\sqrt {d \tan \left (f x + e\right )}}}{2 \, d f} \]
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Timed out. \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 4.98 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.39 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2\,a^2}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,2{}\mathrm {i}}{d^{3/2}\,f}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,2{}\mathrm {i}}{d^{3/2}\,f} \]
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