Integrand size = 25, antiderivative size = 141 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}} \]
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Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3646, 3709, 3610, 3613, 211} \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {2 \left (a^3 \tan (e+f x)+a^3\right )}{5 d f (d \tan (e+f x))^{5/2}} \]
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Rule 211
Rule 3610
Rule 3613
Rule 3646
Rule 3709
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}+\frac {2 \int \frac {6 a^3 d^2+5 a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{5/2}} \, dx}{5 d^3} \\ & = -\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}+\frac {2 \int \frac {5 a^3 d^3-5 a^3 d^3 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{5 d^5} \\ & = -\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}+\frac {2 \int \frac {-5 a^3 d^4-5 a^3 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{5 d^7} \\ & = -\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}}-\frac {\left (20 a^6 d\right ) \text {Subst}\left (\int \frac {1}{50 a^6 d^8+d x^2} \, dx,x,\frac {-5 a^3 d^4+5 a^3 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f} \\ & = \frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {8 a^3}{5 d^2 f (d \tan (e+f x))^{3/2}}-\frac {4 a^3}{d^3 f \sqrt {d \tan (e+f x)}}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{5 d f (d \tan (e+f x))^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(415\) vs. \(2(141)=282\).
Time = 5.64 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.94 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {a^3 (1+\cot (e+f x))^3 \left (8 \cos ^2(e+f x) \sin (e+f x)+40 \cos (e+f x) \sin ^2(e+f x)+80 \sin ^3(e+f x)+10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^3(e+f x) \tan ^{\frac {7}{2}}(e+f x)-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^3(e+f x) \tan ^{\frac {7}{2}}(e+f x)+5 \sqrt {2} \cos ^3(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)-5 \sqrt {2} \cos ^3(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)-20 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^3(e+f x) \left (2 \sqrt [4]{-\tan (e+f x)} \tan ^{\frac {13}{4}}(e+f x)-3 \left (-\tan ^2(e+f x)\right )^{7/4}\right )+20 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^3(e+f x) \left (2 \sqrt [4]{-\tan (e+f x)} \tan ^{\frac {13}{4}}(e+f x)+3 \left (-\tan ^2(e+f x)\right )^{7/4}\right )\right )}{20 d^3 f (\cos (e+f x)+\sin (e+f x))^3 \sqrt {d \tan (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(120)=240\).
Time = 1.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.29
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d}-\frac {d}{5 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}\right )}{f \,d^{2}}\) | \(323\) |
default | \(\frac {2 a^{3} \left (\frac {-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d}-\frac {d}{5 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}\right )}{f \,d^{2}}\) | \(323\) |
parts | \(\frac {2 a^{3} d \left (-\frac {1}{5 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {1}{d^{4} \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 )}{8 d^{4} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}+\frac {a^{3} \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^{4}}+\frac {3 a^{3} \left (-\frac {2}{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 )}{4 d^{4}}\right )}{f}+\frac {6 a^{3} \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 d}\) | \(634\) |
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Time = 0.27 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.82 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=\left [\frac {5 \, \sqrt {2} a^{3} d \sqrt {-\frac {1}{d}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) - 1\right )} - \tan \left (f x + e\right )^{2} + 4 \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{3} - 2 \, {\left (10 \, a^{3} \tan \left (f x + e\right )^{2} + 5 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{5 \, d^{4} f \tan \left (f x + e\right )^{3}}, -\frac {2 \, {\left (5 \, \sqrt {2} a^{3} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{3} + {\left (10 \, a^{3} \tan \left (f x + e\right )^{2} + 5 \, a^{3} \tan \left (f x + e\right ) + a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{5 \, d^{4} f \tan \left (f x + e\right )^{3}}\right ] \]
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\[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=a^{3} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx\right ) \]
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Time = 0.48 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left (\frac {5 \, a^{3} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )}}{d^{2}} + \frac {10 \, a^{3} d^{2} \tan \left (f x + e\right )^{2} + 5 \, a^{3} d^{2} \tan \left (f x + e\right ) + a^{3} d^{2}}{\left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} d^{2}}\right )}}{5 \, d f} \]
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Timed out. \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 6.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{7/2}} \, dx=-\frac {4\,d\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2+2\,d\,a^3\,\mathrm {tan}\left (e+f\,x\right )+\frac {2\,d\,a^3}{5}}{d^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}-\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{d^{7/2}\,f} \]
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