Integrand size = 28, antiderivative size = 113 \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=\frac {4 \sqrt [4]{-1} a^2 d^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {4 a^2 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3624, 3609, 3614, 211} \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=\frac {4 \sqrt [4]{-1} a^2 d^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac {4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a^2 d \sqrt {d \tan (e+f x)}}{f} \]
[In]
[Out]
Rule 211
Rule 3609
Rule 3614
Rule 3624
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\int (d \tan (e+f x))^{3/2} \left (2 a^2+2 i a^2 \tan (e+f x)\right ) \, dx \\ & = \frac {4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\int \sqrt {d \tan (e+f x)} \left (-2 i a^2 d+2 a^2 d \tan (e+f x)\right ) \, dx \\ & = \frac {4 a^2 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\int \frac {-2 a^2 d^2-2 i a^2 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx \\ & = \frac {4 a^2 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f}+\frac {\left (8 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{-2 a^2 d^3+2 i a^2 d^2 x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f} \\ & = \frac {4 \sqrt [4]{-1} a^2 d^{3/2} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {4 a^2 d \sqrt {d \tan (e+f x)}}{f}+\frac {4 i a^2 (d \tan (e+f x))^{3/2}}{3 f}-\frac {2 a^2 (d \tan (e+f x))^{5/2}}{5 d f} \\ \end{align*}
Time = 0.99 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78 \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=\frac {2 a^2 d \left ((-15+15 i) \sqrt {2} \sqrt {d} \text {arctanh}\left (\frac {(1+i) \sqrt {d \tan (e+f x)}}{\sqrt {2} \sqrt {d}}\right )+\sqrt {d \tan (e+f x)} \left (30+10 i \tan (e+f x)-3 \tan ^2(e+f x)\right )\right )}{15 f} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (92 ) = 184\).
Time = 0.99 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.88
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (-\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 d^{2} \sqrt {d \tan \left (f x +e \right )}-2 d^{3} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f d}\) | \(326\) |
default | \(\frac {2 a^{2} \left (-\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 i d \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+2 d^{2} \sqrt {d \tan \left (f x +e \right )}-2 d^{3} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f d}\) | \(326\) |
parts | \(\frac {2 a^{2} d \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}+\frac {2 i a^{2} \left (\frac {2 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d^{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 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}-\frac {2 a^{2} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-d^{2} \sqrt {d \tan \left (f x +e \right )}+\frac {d^{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 )}{8}\right )}{f d}\) | \(482\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (91) = 182\).
Time = 0.25 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.29 \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=-\frac {15 \, \sqrt {-\frac {16 i \, a^{4} d^{3}}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-4 i \, a^{2} d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + \sqrt {-\frac {16 i \, a^{4} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2} d}\right ) - 15 \, \sqrt {-\frac {16 i \, a^{4} d^{3}}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-4 i \, a^{2} d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - \sqrt {-\frac {16 i \, a^{4} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a^{2} d}\right ) - 8 \, {\left (43 \, a^{2} d e^{\left (4 i \, f x + 4 i \, e\right )} + 54 \, a^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} + 23 \, a^{2} d\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{60 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
[In]
[Out]
\[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=- a^{2} \left (\int \left (- \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\right )\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\right )\, dx\right ) \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (91) = 182\).
Time = 0.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.89 \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=\frac {15 \, a^{2} d^{3} {\left (-\frac {\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (i - 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\left (i - 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - 12 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{2} + 40 i \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{2} d + 120 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{2}}{30 \, d f} \]
[In]
[Out]
none
Time = 0.75 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.39 \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=-\frac {2}{15} \, {\left (\frac {30 i \, \sqrt {2} a^{2} \sqrt {d} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{10} f^{4} \tan \left (f x + e\right )^{2} - 10 i \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{10} f^{4} \tan \left (f x + e\right ) - 30 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{10} f^{4}}{d^{10} f^{5}}\right )} d \]
[In]
[Out]
Time = 5.69 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2 \, dx=\frac {a^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,4{}\mathrm {i}}{3\,f}-\frac {2\,a^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}+\frac {4\,a^2\,d\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}-\frac {\sqrt {4{}\mathrm {i}}\,a^2\,{\left (-d\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {4{}\mathrm {i}}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {-d}}\right )\,2{}\mathrm {i}}{f} \]
[In]
[Out]