Integrand size = 43, antiderivative size = 209 \[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=\frac {(5 i A+7 B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{64 \sqrt {2} a^3 f}+\frac {(i A-B) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))} \]
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Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3669, 79, 44, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=\frac {\sqrt {c} (7 B+5 i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{64 \sqrt {2} a^3 f}+\frac {(-B+i A) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(7 B+5 i A) \sqrt {c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac {(7 B+5 i A) \sqrt {c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2} \]
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Rule 44
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}{(a+i a x)^4 \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {((5 A-7 i B) c) \text {Subst}\left (\int \frac {1}{(a+i a x)^3 \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{12 f} \\ & = \frac {(i A-B) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac {((5 A-7 i B) c) \text {Subst}\left (\int \frac {1}{(a+i a x)^2 \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{32 a f} \\ & = \frac {(i A-B) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac {((5 A-7 i B) c) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{128 a^2 f} \\ & = \frac {(i A-B) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac {(5 i A+7 B) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{64 a^2 f} \\ & = \frac {(5 i A+7 B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{64 \sqrt {2} a^3 f}+\frac {(i A-B) \sqrt {c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac {(5 i A+7 B) \sqrt {c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))} \\ \end{align*}
Time = 5.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {\sec ^3(e+f x) \left (3 \sqrt {2} (5 A-7 i B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right ) (\cos (3 (e+f x))+i \sin (3 (e+f x)))+2 \cos (e+f x) (26 A+2 i B+(41 A-19 i B) \cos (2 (e+f x))+5 (5 i A+7 B) \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}\right )}{384 a^3 f (-i+\tan (e+f x))^3} \]
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Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {2 i c^{3} \left (\frac {\frac {\left (-7 i B +5 A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{128 c^{2}}-\frac {\left (-7 i B +5 A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{24 c}+8 \left (\frac {11 A}{256}-\frac {9 i B}{256}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{3}}+\frac {\left (-7 i B +5 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{256 c^{\frac {5}{2}}}\right )}{f \,a^{3}}\) | \(147\) |
default | \(\frac {2 i c^{3} \left (\frac {\frac {\left (-7 i B +5 A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{128 c^{2}}-\frac {\left (-7 i B +5 A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{24 c}+8 \left (\frac {11 A}{256}-\frac {9 i B}{256}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{3}}+\frac {\left (-7 i B +5 A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{256 c^{\frac {5}{2}}}\right )}{f \,a^{3}}\) | \(147\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (162) = 324\).
Time = 0.26 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.85 \[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} + {\left (5 i \, A + 7 \, B\right )} c\right )} e^{\left (-i \, f x - i \, e\right )}}{32 \, a^{3} f}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} - {\left (5 i \, A + 7 \, B\right )} c\right )} e^{\left (-i \, f x - i \, e\right )}}{32 \, a^{3} f}\right ) - \sqrt {2} {\left (3 \, {\left (-11 i \, A - 9 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - {\left (59 i \, A + 25 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (-17 i \, A + 5 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, A + 8 \, B\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{384 \, a^{3} f} \]
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\[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=\frac {i \left (\int \frac {A \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \frac {B \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx\right )}{a^{3}} \]
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Time = 0.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {i \, {\left (\frac {3 \, \sqrt {2} {\left (5 \, A - 7 i \, B\right )} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{3}} + \frac {4 \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (5 \, A - 7 i \, B\right )} c^{2} - 16 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (5 \, A - 7 i \, B\right )} c^{3} + 12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (11 \, A - 9 i \, B\right )} c^{4}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} a^{3} - 6 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{3} c + 12 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} c^{2} - 8 \, a^{3} c^{3}}\right )}}{768 \, c f} \]
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\[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Time = 9.05 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx=\frac {\frac {7\,B\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{64}+\frac {9\,B\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{16}-\frac {7\,B\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{12}}{8\,a^3\,c^3\,f-a^3\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+6\,a^3\,c\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,a^3\,c^2\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}+\frac {\frac {A\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,11{}\mathrm {i}}{16\,a^3\,f}-\frac {A\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,5{}\mathrm {i}}{12\,a^3\,f}+\frac {A\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,5{}\mathrm {i}}{64\,a^3\,f}}{6\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,c^2\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+8\,c^3}+\frac {\sqrt {2}\,A\,\sqrt {-c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,5{}\mathrm {i}}{128\,a^3\,f}+\frac {7\,\sqrt {2}\,B\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {c}}\right )}{128\,a^3\,f} \]
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