Integrand size = 25, antiderivative size = 247 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} f}-\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} f}-\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{5/2} f}+\frac {a^2 \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} d^{5/2} f}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}} \]
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Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3623, 12, 3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} f}-\frac {\sqrt {2} a^2 \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{d^{5/2} f}-\frac {a^2 \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} d^{5/2} f}+\frac {a^2 \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} d^{5/2} f}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}} \]
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Rule 12
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rule 3623
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}+\frac {\int \frac {2 a^2 d}{(d \tan (e+f x))^{3/2}} \, dx}{d^2} \\ & = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}+\frac {\left (2 a^2\right ) \int \frac {1}{(d \tan (e+f x))^{3/2}} \, dx}{d} \\ & = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}}-\frac {\left (2 a^2\right ) \int \sqrt {d \tan (e+f x)} \, dx}{d^3} \\ & = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{d^2 f} \\ & = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}}-\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{d^2 f} \\ & = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 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^2 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^2 f} \\ & = -\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 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^{5/2} 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^{5/2} 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^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^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^{5/2} 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^{5/2} f}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 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^{5/2} 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^{5/2} f} \\ & = \frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} f}-\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{d^{5/2} 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^{5/2} 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^{5/2} f}-\frac {2 a^2}{3 d f (d \tan (e+f x))^{3/2}}-\frac {4 a^2}{d^2 f \sqrt {d \tan (e+f x)}} \\ \end{align*}
Time = 5.55 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.62 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {a^2 (1+\cot (e+f x))^2 \left (-48 \sin ^2(e+f x)-4 \sin (2 (e+f x))-6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)+6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)-3 \sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {5}{2}}(e+f x)+3 \sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {5}{2}}(e+f x)+12 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \left (\cos ^2(e+f x) (-\tan (e+f x))^{3/4} \tan ^{\frac {7}{4}}(e+f x)+2 \sin ^2(e+f x) \sqrt [4]{-\tan ^2(e+f x)}\right )+12 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) \left ((-\tan (e+f x))^{3/4} \tan ^{\frac {7}{4}}(e+f x)+2 \left (-\tan ^2(e+f x)\right )^{5/4}\right )\right )}{12 d^2 f (\cos (e+f x)+\sin (e+f x))^2 \sqrt {d \tan (e+f x)}} \]
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Time = 0.93 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (-\frac {1}{3 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f d}\) | \(174\) |
default | \(\frac {2 a^{2} \left (-\frac {1}{3 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f d}\) | \(174\) |
parts | \(\frac {2 a^{2} d \left (-\frac {1}{3 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\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 )}{8 d^{4}}\right )}{f}+\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 )}{4 f \,d^{3}}+\frac {2 a^{2} \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 )}{4 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}-\frac {2}{d^{2} \sqrt {d \tan \left (f x +e \right )}}\right )}{f}\) | \(459\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.21 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {3 \, d^{3} f \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (d^{8} f^{3} \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{6}\right ) \tan \left (f x + e\right )^{2} - 3 i \, d^{3} f \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{8} f^{3} \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{6}\right ) \tan \left (f x + e\right )^{2} + 3 i \, d^{3} f \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{8} f^{3} \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{6}\right ) \tan \left (f x + e\right )^{2} - 3 \, d^{3} f \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (-d^{8} f^{3} \left (-\frac {a^{8}}{d^{10} f^{4}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (f x + e\right )} a^{6}\right ) \tan \left (f x + e\right )^{2} + 2 \, {\left (6 \, a^{2} \tan \left (f x + e\right ) + a^{2}\right )} \sqrt {d \tan \left (f x + e\right )}}{3 \, d^{3} f \tan \left (f x + e\right )^{2}} \]
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\[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=a^{2} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {2 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \]
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Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=-\frac {\frac {3 \, a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )}}{d} + \frac {4 \, {\left (6 \, a^{2} d \tan \left (f x + e\right ) + a^{2} d\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d}}{6 \, d f} \]
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Timed out. \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Time = 5.62 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.40 \[ \int \frac {(a+a \tan (e+f x))^2}{(d \tan (e+f x))^{5/2}} \, dx=\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{5/2}\,f}-\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{d^{5/2}\,f}-\frac {4\,a^2\,\mathrm {tan}\left (e+f\,x\right )+\frac {2\,a^2}{3}}{d\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \]
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