Integrand size = 43, antiderivative size = 275 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {7 (5 i A-13 B) c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a^2 f}-\frac {7 (5 i A-13 B) c^4 \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac {7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
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Time = 0.52 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3669, 79, 43, 52, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {7 c^{9/2} (-13 B+5 i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a^2 f}-\frac {7 c^4 (-13 B+5 i A) \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {7 c^3 (-13 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac {7 c^2 (-13 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {c (-13 B+5 i A) (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(-B+i A) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{7/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}-\frac {((5 A+13 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{7/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = -\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (7 (5 A+13 i B) c^2\right ) \text {Subst}\left (\int \frac {(c-i c x)^{5/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{16 a f} \\ & = -\frac {7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (7 (5 A+13 i B) c^3\right ) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{8 a f} \\ & = -\frac {7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac {7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (7 (5 A+13 i B) c^4\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{4 a f} \\ & = -\frac {7 (5 i A-13 B) c^4 \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac {7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (7 (5 A+13 i B) c^5\right ) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 a f} \\ & = -\frac {7 (5 i A-13 B) c^4 \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac {7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (7 (5 i A-13 B) c^4\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{a f} \\ & = \frac {7 (5 i A-13 B) c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a^2 f}-\frac {7 (5 i A-13 B) c^4 \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac {7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac {(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2} \\ \end{align*}
Time = 7.92 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {105 \sqrt {2} (-5 i A+13 B) c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right ) \sec ^2(e+f x) (\cos (2 (e+f x))+i \sin (2 (e+f x)))-2 c^4 \sqrt {c-i c \tan (e+f x)} \left (-325 i A+851 B+5 (113 A+289 i B) \tan (e+f x)+(170 i A-478 B) \tan ^2(e+f x)+10 (A+5 i B) \tan ^3(e+f x)+6 B \tan ^4(e+f x)\right )}{30 a^2 f (-i+\tan (e+f x))^2} \]
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Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {2 i c^{2} \left (-\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {5 i B c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {A c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-18 i \sqrt {c -i c \tan \left (f x +e \right )}\, B \,c^{2}-6 \sqrt {c -i c \tan \left (f x +e \right )}\, A \,c^{2}+8 c^{3} \left (\frac {4 \left (\frac {21 i B}{64}+\frac {13 A}{64}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+4 \left (-\frac {19}{32} i B c -\frac {11}{32} c A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{2}}+\frac {7 \left (\frac {13 i B}{4}+\frac {5 A}{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 \sqrt {c}}\right )\right )}{f \,a^{2}}\) | \(220\) |
default | \(\frac {2 i c^{2} \left (-\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {5 i B c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {A c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-18 i \sqrt {c -i c \tan \left (f x +e \right )}\, B \,c^{2}-6 \sqrt {c -i c \tan \left (f x +e \right )}\, A \,c^{2}+8 c^{3} \left (\frac {4 \left (\frac {21 i B}{64}+\frac {13 A}{64}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+4 \left (-\frac {19}{32} i B c -\frac {11}{32} c A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{2}}+\frac {7 \left (\frac {13 i B}{4}+\frac {5 A}{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 \sqrt {c}}\right )\right )}{f \,a^{2}}\) | \(220\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (216) = 432\).
Time = 0.28 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.87 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=-\frac {105 \, \sqrt {2} \sqrt {-\frac {{\left (25 \, A^{2} + 130 i \, A B - 169 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}} {\left (a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (-\frac {14 \, {\left ({\left (-5 i \, A + 13 \, B\right )} c^{5} + \sqrt {-\frac {{\left (25 \, A^{2} + 130 i \, A B - 169 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a^{2} f}\right ) - 105 \, \sqrt {2} \sqrt {-\frac {{\left (25 \, A^{2} + 130 i \, A B - 169 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}} {\left (a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (-\frac {14 \, {\left ({\left (-5 i \, A + 13 \, B\right )} c^{5} - \sqrt {-\frac {{\left (25 \, A^{2} + 130 i \, A B - 169 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a^{2} f}\right ) + 2 \, \sqrt {2} {\left (105 \, {\left (5 i \, A - 13 \, B\right )} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 245 \, {\left (5 i \, A - 13 \, B\right )} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 161 \, {\left (5 i \, A - 13 \, B\right )} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 15 \, {\left (5 i \, A - 13 \, B\right )} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 30 \, {\left (-i \, A + B\right )} c^{4}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{60 \, {\left (a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \]
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Timed out. \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=-\frac {i \, {\left (\frac {105 \, \sqrt {2} {\left (5 \, A + 13 i \, B\right )} c^{\frac {11}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{2}} - \frac {60 \, {\left ({\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (13 \, A + 21 i \, B\right )} c^{6} - 2 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (11 \, A + 19 i \, B\right )} c^{7}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{2} - 4 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} c + 4 \, a^{2} c^{2}} + \frac {8 \, {\left (3 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} B c^{3} + 5 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (A + 5 i \, B\right )} c^{4} + 90 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + 3 i \, B\right )} c^{5}\right )}}{a^{2}}\right )}}{60 \, c f} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Time = 8.95 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {38\,B\,c^6\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-21\,B\,c^5\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{4\,a^2\,c^2\,f+a^2\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,a^2\,c\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}-\frac {\frac {A\,c^6\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,22{}\mathrm {i}}{a^2\,f}-\frac {A\,c^5\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,13{}\mathrm {i}}{a^2\,f}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,c\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )+4\,c^2}-\frac {A\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,12{}\mathrm {i}}{a^2\,f}-\frac {A\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{3\,a^2\,f}+\frac {36\,B\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a^2\,f}+\frac {10\,B\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a^2\,f}+\frac {2\,B\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,a^2\,f}+\frac {\sqrt {2}\,A\,{\left (-c\right )}^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,35{}\mathrm {i}}{2\,a^2\,f}+\frac {\sqrt {2}\,B\,c^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,91{}\mathrm {i}}{2\,a^2\,f} \]
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