Integrand size = 28, antiderivative size = 287 \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))} \]
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Time = 0.35 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3631, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a f}+\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a f}-\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a f}-\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3631
Rubi steps \begin{align*} \text {integral}& = -\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int \sqrt {d \tan (e+f x)} \left (\frac {3 a d^2}{2}-\frac {5}{2} i a d^2 \tan (e+f x)\right ) \, dx}{2 a^2} \\ & = -\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int \frac {\frac {5}{2} i a d^3+\frac {3}{2} a d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {\text {Subst}\left (\int \frac {\frac {5}{2} i a d^4+\frac {3}{2} a d^3 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f} \\ & = -\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+-\frac {\left (\left (\frac {3}{4}-\frac {5 i}{4}\right ) d^3\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}+\frac {\left (\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^3\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f} \\ & = -\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {\left (\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2}\right ) \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} a f}+\frac {\left (\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2}\right ) \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} a f}+\frac {\left (\left (\frac {3}{8}+\frac {5 i}{8}\right ) d^3\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}+\frac {\left (\left (\frac {3}{8}+\frac {5 i}{8}\right ) d^3\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f} \\ & = \frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+-\frac {\left (\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2}\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 )}{\sqrt {2} a f}+\frac {\left (\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2}\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 )}{\sqrt {2} a f} \\ & = -\frac {\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {3}{4}+\frac {5 i}{4}\right ) d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {\left (\frac {3}{8}-\frac {5 i}{8}\right ) d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {5 i d^2 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {d (d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))} \\ \end{align*}
Time = 2.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.43 \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {d^2 \left (-(-1)^{3/4} \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-4 (-1)^{3/4} \sqrt {d} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )+\frac {(-5-4 i \tan (e+f x)) \sqrt {d \tan (e+f x)}}{-i+\tan (e+f x)}\right )}{2 a f} \]
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Time = 0.86 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.38
method | result | size |
derivativedivides | \(\frac {2 d^{2} \left (-i \sqrt {d \tan \left (f x +e \right )}-\frac {d \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{d \tan \left (f x +e \right )-i d}-\frac {4 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{\sqrt {-i d}}\right )}{4}-\frac {d \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 \sqrt {i d}}\right )}{f a}\) | \(109\) |
default | \(\frac {2 d^{2} \left (-i \sqrt {d \tan \left (f x +e \right )}-\frac {d \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{d \tan \left (f x +e \right )-i d}-\frac {4 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{\sqrt {-i d}}\right )}{4}-\frac {d \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 \sqrt {i d}}\right )}{f a}\) | \(109\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (207) = 414\).
Time = 0.25 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.87 \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {{\left (a \sqrt {\frac {i \, d^{5}}{4 \, a^{2} f^{2}}} f e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (i \, a f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, a f\right )} \sqrt {\frac {i \, d^{5}}{4 \, a^{2} f^{2}}} \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 )}}{d^{2}}\right ) - a \sqrt {\frac {i \, d^{5}}{4 \, a^{2} f^{2}}} f e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (-i \, a f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a f\right )} \sqrt {\frac {i \, d^{5}}{4 \, a^{2} f^{2}}} \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 )}}{d^{2}}\right ) - a \sqrt {-\frac {4 i \, d^{5}}{a^{2} f^{2}}} f e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {{\left (2 \, d^{3} + {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {4 i \, d^{5}}{a^{2} f^{2}}} \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 )}}{a f}\right ) + a \sqrt {-\frac {4 i \, d^{5}}{a^{2} f^{2}}} f e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {{\left (2 \, d^{3} - {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {4 i \, d^{5}}{a^{2} f^{2}}} \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 )}}{a f}\right ) + {\left (-9 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, d^{2}\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 )}}{4 \, a f} \]
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\[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=- \frac {i \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \]
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Exception generated. \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.54 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.67 \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {1}{2} \, d^{2} {\left (\frac {\sqrt {2} \sqrt {d} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} - \frac {4 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {4 i \, \sqrt {d \tan \left (f x + e\right )}}{a f} + \frac {\sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) - i \, d\right )} a f}\right )} \]
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Time = 7.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.56 \[ \int \frac {(d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\mathrm {atan}\left (\frac {a\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {d^5\,1{}\mathrm {i}}{a^2\,f^2}}\,1{}\mathrm {i}}{d^3}\right )\,\sqrt {-\frac {d^5\,1{}\mathrm {i}}{a^2\,f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {d^5\,1{}\mathrm {i}}{16\,a^2\,f^2}}\,4{}\mathrm {i}}{d^3}\right )\,\sqrt {\frac {d^5\,1{}\mathrm {i}}{16\,a^2\,f^2}}\,2{}\mathrm {i}-\frac {d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,2{}\mathrm {i}}{a\,f}+\frac {d^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,a\,f\,\left (-d\,\mathrm {tan}\left (e+f\,x\right )+d\,1{}\mathrm {i}\right )} \]
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