Integrand size = 43, antiderivative size = 141 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {(3 i A+B) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a \sqrt {c} f}-\frac {3 i A+B}{4 a f \sqrt {c-i c \tan (e+f x)}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3669, 79, 53, 65, 214} \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {(B+3 i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a \sqrt {c} f}+\frac {-B+i A}{2 a f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}-\frac {B+3 i A}{4 a f \sqrt {c-i c \tan (e+f x)}} \]
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Rule 53
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)^2 (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {i A-B}{2 a f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {((3 A-i B) c) \text {Subst}\left (\int \frac {1}{(a+i a x) (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = -\frac {3 i A+B}{4 a f \sqrt {c-i c \tan (e+f x)}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {(3 A-i B) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = -\frac {3 i A+B}{4 a f \sqrt {c-i c \tan (e+f x)}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}+\frac {(3 i A+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 )}{4 c f} \\ & = \frac {(3 i A+B) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a \sqrt {c} f}-\frac {3 i A+B}{4 a f \sqrt {c-i c \tan (e+f x)}}+\frac {i A-B}{2 a f (1+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \\ \end{align*}
Time = 3.91 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\frac {\sqrt {2} (3 i A+B) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}+\frac {-2 A+6 i B+(-6 i A-2 B) \tan (e+f x)}{(-i+\tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}}{8 a f} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {2 i c \left (-\frac {-i B +A}{4 c \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {\frac {\left (\frac {i B}{4}+\frac {A}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {3 A}{2}-\frac {i B}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}}{4 c}\right )}{f a}\) | \(121\) |
default | \(\frac {2 i c \left (-\frac {-i B +A}{4 c \sqrt {c -i c \tan \left (f x +e \right )}}+\frac {\frac {\left (\frac {i B}{4}+\frac {A}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {3 A}{2}-\frac {i B}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}}{4 c}\right )}{f a}\) | \(121\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (108) = 216\).
Time = 0.26 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.52 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {{\left (\sqrt {\frac {1}{2}} a c f \sqrt {-\frac {9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} + 3 i \, A + B\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a f}\right ) - \sqrt {\frac {1}{2}} a c f \sqrt {-\frac {9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {9 \, A^{2} - 6 i \, A B - B^{2}}{a^{2} c f^{2}}} - 3 i \, A - B\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a f}\right ) - \sqrt {2} {\left (2 \, {\left (i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - {\left (-i \, A - 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, A + B\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \]
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\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=- \frac {i \left (\int \frac {A}{\sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} - i \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \frac {B \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} - i \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx\right )}{a} \]
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Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {i \, {\left (\frac {\sqrt {2} {\left (3 \, A - i \, B\right )} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a} + \frac {4 \, {\left ({\left (-i \, c \tan \left (f x + e\right ) + c\right )} {\left (3 \, A - i \, B\right )} c - 4 \, {\left (A - i \, B\right )} c^{2}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a - 2 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a c}\right )}}{16 \, c f} \]
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\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )} \sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]
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Time = 9.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.50 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {B\,c-\frac {B\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}{4}}{a\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}-2\,a\,c\,f\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}+\frac {\frac {A\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{4\,a\,f}-\frac {A\,c\,1{}\mathrm {i}}{a\,f}}{2\,c\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}-\frac {\sqrt {2}\,A\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,3{}\mathrm {i}}{8\,a\,\sqrt {-c}\,f}+\frac {\sqrt {2}\,B\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {c}}\right )}{8\,a\,\sqrt {c}\,f} \]
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