Integrand size = 43, antiderivative size = 211 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {(i A+3 B) c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^3 f}+\frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}-\frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
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Time = 0.29 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3669, 79, 43, 44, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {c^{3/2} (3 B+i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^3 f}-\frac {c (3 B+i A) \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (1+i \tan (e+f x))}+\frac {c (3 B+i A) \sqrt {c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(-B+i A) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
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Rule 43
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) \sqrt {c-i c x}}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac {((A-3 i B) c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = \frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}+\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {\left ((A-3 i B) c^2\right ) \text {Subst}\left (\int \frac {1}{(a+i a x)^2 \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{16 a f} \\ & = \frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}-\frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {\left ((A-3 i B) c^2\right ) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{64 a^2 f} \\ & = \frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}-\frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac {((i A+3 B) c) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{32 a^2 f} \\ & = -\frac {(i A+3 B) c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{32 \sqrt {2} a^3 f}+\frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{8 a^3 f (1+i \tan (e+f x))^2}-\frac {(i A+3 B) c \sqrt {c-i c \tan (e+f x)}}{32 a^3 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{3/2}}{6 a^3 f (1+i \tan (e+f x))^3} \\ \end{align*}
Time = 6.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.82 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {c \sec ^3(e+f x) \left (3 \sqrt {2} (A-3 i B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right ) (\cos (3 (e+f x))+i \sin (3 (e+f x)))-2 \cos (e+f x) (2 (7 A-5 i B)+(11 A-i B) \cos (2 (e+f x))+(-5 i A+17 B) \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}\right )}{192 a^3 f (-i+\tan (e+f x))^3} \]
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Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {2 i c^{3} \left (\frac {-\frac {\left (-3 i B +A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{64 c}+8 \left (\frac {i B}{96}+\frac {A}{96}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+\frac {c \left (-3 i B +A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{16}}{\left (c +i c \tan \left (f x +e \right )\right )^{3}}-\frac {\left (-3 i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{128 c^{\frac {3}{2}}}\right )}{f \,a^{3}}\) | \(139\) |
default | \(\frac {2 i c^{3} \left (\frac {-\frac {\left (-3 i B +A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{64 c}+8 \left (\frac {i B}{96}+\frac {A}{96}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+\frac {c \left (-3 i B +A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{16}}{\left (c +i c \tan \left (f x +e \right )\right )^{3}}-\frac {\left (-3 i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{128 c^{\frac {3}{2}}}\right )}{f \,a^{3}}\) | \(139\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (164) = 328\).
Time = 0.26 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.87 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {{\left (A^{2} - 6 i \, A B - 9 \, B^{2}\right )} c^{3}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left ({\left (-i \, A - 3 \, B\right )} c^{2} + \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {{\left (A^{2} - 6 i \, A B - 9 \, B^{2}\right )} c^{3}}{a^{6} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{16 \, a^{3} f}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{3} f \sqrt {-\frac {{\left (A^{2} - 6 i \, A B - 9 \, B^{2}\right )} c^{3}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left ({\left (-i \, A - 3 \, B\right )} c^{2} - \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {{\left (A^{2} - 6 i \, A B - 9 \, B^{2}\right )} c^{3}}{a^{6} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{16 \, a^{3} f}\right ) - \sqrt {2} {\left (3 \, {\left (-i \, A - 3 \, B\right )} c e^{\left (6 i \, f x + 6 i \, e\right )} - {\left (17 i \, A + 19 \, B\right )} c e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, {\left (-11 i \, A - B\right )} c e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (-i \, A + B\right )} c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{192 \, a^{3} f} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {i \left (\int \frac {A c \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \frac {B c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \left (- \frac {i A c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\right )\, dx + \int \left (- \frac {i B c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\right )\, dx\right )}{a^{3}} \]
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Time = 0.45 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {i \, {\left (\frac {3 \, \sqrt {2} {\left (A - 3 i \, B\right )} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{3}} + \frac {4 \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - 3 i \, B\right )} c^{3} - 16 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (A + i \, B\right )} c^{4} - 12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A - 3 i \, B\right )} c^{5}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} a^{3} - 6 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{3} c + 12 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} c^{2} - 8 \, a^{3} c^{3}}\right )}}{384 \, c f} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Time = 9.10 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {\frac {A\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,a^3\,f}+\frac {A\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,1{}\mathrm {i}}{6\,a^3\,f}-\frac {A\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,1{}\mathrm {i}}{32\,a^3\,f}}{6\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,c^2\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+8\,c^3}-\frac {-\frac {3\,B\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{8}+\frac {B\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{6}+\frac {3\,B\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{32}}{8\,a^3\,c^3\,f-a^3\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3+6\,a^3\,c\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-12\,a^3\,c^2\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}+\frac {\sqrt {2}\,A\,{\left (-c\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,1{}\mathrm {i}}{64\,a^3\,f}-\frac {3\,\sqrt {2}\,B\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {c}}\right )}{64\,a^3\,f} \]
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