Integrand size = 25, antiderivative size = 114 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt {d \tan (e+f x)}} \]
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Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3646, 3711, 3613, 211} \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}+\frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3 \tan (e+f x)+a^3\right )}{d f \sqrt {d \tan (e+f x)}} \]
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Rule 211
Rule 3613
Rule 3646
Rule 3711
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt {d \tan (e+f x)}}+\frac {2 \int \frac {2 a^3 d^2+a^3 d^2 \tan (e+f x)+a^3 d^2 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{d^3} \\ & = \frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt {d \tan (e+f x)}}+\frac {2 \int \frac {a^3 d^2+a^3 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{d^3} \\ & = \frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt {d \tan (e+f x)}}-\frac {\left (4 a^6 d\right ) \text {Subst}\left (\int \frac {1}{2 a^6 d^4+d x^2} \, dx,x,\frac {a^3 d^2-a^3 d^2 \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^{3/2} f}+\frac {4 a^3 \sqrt {d \tan (e+f x)}}{d^2 f}-\frac {2 \left (a^3+a^3 \tan (e+f x)\right )}{d f \sqrt {d \tan (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(357\) vs. \(2(114)=228\).
Time = 3.09 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.13 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=-\frac {a^3 \cos (e+f x) (1+\tan (e+f x))^3 \left (-4 \sin ^2(e+f x)+2 \sin (2 (e+f x))+4 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) (-\tan (e+f x))^{5/4} \sqrt [4]{\tan (e+f x)}+4 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^2(e+f x) \sqrt [4]{-\tan (e+f x)} \tan ^{\frac {5}{4}}(e+f x)+2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)+\sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)-\sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)\right )}{2 f (\cos (e+f x)+\sin (e+f x))^3 (d \tan (e+f x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(99)=198\).
Time = 0.93 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.68
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {d}{\sqrt {d \tan \left (f x +e \right )}}+2 d \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(305\) |
default | \(\frac {2 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\frac {d}{\sqrt {d \tan \left (f x +e \right )}}+2 d \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(305\) |
parts | \(\frac {2 a^{3} 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 {2 a^{3} \left (\sqrt {d \tan \left (f x +e \right )}-\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}\right )}{f \,d^{2}}+\frac {3 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^{2}}+\frac {3 a^{3} \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}}}\) | \(594\) |
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Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.96 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\left [\frac {\sqrt {2} a^{3} d \sqrt {-\frac {1}{d}} \log \left (\frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) - 1\right )} + \tan \left (f x + e\right )^{2} - 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) + 2 \, {\left (a^{3} \tan \left (f x + e\right ) - a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}}{d^{2} f \tan \left (f x + e\right )}, \frac {2 \, {\left (\sqrt {2} a^{3} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + {\left (a^{3} \tan \left (f x + e\right ) - a^{3}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{d^{2} f \tan \left (f x + e\right )}\right ] \]
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\[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=a^{3} \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {3 \tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {3 \tan ^{2}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
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Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\frac {2 \, {\left (a^{3} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )} - \frac {a^{3}}{\sqrt {d \tan \left (f x + e\right )}} + \frac {\sqrt {d \tan \left (f x + e\right )} a^{3}}{d}\right )}}{d f} \]
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Timed out. \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 5.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \tan (e+f x))^3}{(d \tan (e+f x))^{3/2}} \, dx=\frac {2\,a^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{d^2\,f}-\frac {2\,a^3}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}+\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{d^{3/2}\,f} \]
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