Integrand size = 43, antiderivative size = 180 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {\sqrt {2} (3 i A-7 B) c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {(3 i A-7 B) c^2 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(3 i A-7 B) c (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))} \]
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Time = 0.29 (sec) , antiderivative size = 180, 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, 52, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {\sqrt {2} c^{5/2} (-7 B+3 i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {c^2 (-7 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {c (-7 B+3 i A) (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))} \]
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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)^{3/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))}-\frac {((3 A+7 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = \frac {(3 i A-7 B) c (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left ((3 A+7 i B) c^2\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {(3 i A-7 B) c^2 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(3 i A-7 B) c (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left ((3 A+7 i B) c^3\right ) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(3 i A-7 B) c^2 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(3 i A-7 B) c (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left (2 (3 i A-7 B) c^2\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 )}{f} \\ & = -\frac {\sqrt {2} (3 i A-7 B) c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {(3 i A-7 B) c^2 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(3 i A-7 B) c (c-i c \tan (e+f x))^{3/2}}{6 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{2 a f (1+i \tan (e+f x))} \\ \end{align*}
Time = 6.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {c^2 \left (3 \sqrt {2} (-3 i A+7 B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )+\frac {2 \sqrt {c-i c \tan (e+f x)} \left (6 A+13 i B+3 i (A+3 i B) \tan (e+f x)+i B \tan ^2(e+f x)\right )}{-i+\tan (e+f x)}\right )}{3 a f} \]
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Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 i c \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+3 i \sqrt {c -i c \tan \left (f x +e \right )}\, B c +\sqrt {c -i c \tan \left (f x +e \right )}\, c A -4 c^{2} \left (\frac {\left (-\frac {i B}{8}-\frac {A}{8}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {7 i B}{2}+\frac {3 A}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )\right )}{f a}\) | \(150\) |
default | \(\frac {2 i c \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+3 i \sqrt {c -i c \tan \left (f x +e \right )}\, B c +\sqrt {c -i c \tan \left (f x +e \right )}\, c A -4 c^{2} \left (\frac {\left (-\frac {i B}{8}-\frac {A}{8}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {7 i B}{2}+\frac {3 A}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )\right )}{f a}\) | \(150\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (140) = 280\).
Time = 0.27 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.26 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {3 \, \sqrt {2} {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 42 i \, A B - 49 \, B^{2}\right )} c^{5}}{a^{2} f^{2}}} \log \left (-\frac {4 \, {\left ({\left (3 i \, A - 7 \, B\right )} c^{3} + {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 42 i \, A B - 49 \, B^{2}\right )} c^{5}}{a^{2} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) - 3 \, \sqrt {2} {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 42 i \, A B - 49 \, B^{2}\right )} c^{5}}{a^{2} f^{2}}} \log \left (-\frac {4 \, {\left ({\left (3 i \, A - 7 \, B\right )} c^{3} - {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 42 i \, A B - 49 \, B^{2}\right )} c^{5}}{a^{2} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) + 2 \, \sqrt {2} {\left (3 \, {\left (-3 i \, A + 7 \, B\right )} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, {\left (-3 i \, A + 7 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 \, {\left (-i \, A + B\right )} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )}} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=- \frac {i \left (\int \frac {A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \frac {B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \left (- \frac {2 i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \left (- \frac {2 i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx\right )}{a} \]
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Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {i \, {\left (\frac {3 \, \sqrt {2} {\left (3 \, A + 7 i \, B\right )} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a} - \frac {12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + i \, B\right )} c^{4}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a - 2 \, a c} + \frac {4 \, {\left (i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} B c^{2} + 3 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + 3 i \, B\right )} c^{3}\right )}}{a}\right )}}{6 \, c f} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]
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Time = 9.39 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {2\,B\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-2\,a\,c\,f}+\frac {A\,c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,f}-\frac {6\,B\,c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a\,f}-\frac {2\,B\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a\,f}-\frac {\sqrt {2}\,A\,{\left (-c\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,3{}\mathrm {i}}{a\,f}-\frac {\sqrt {2}\,B\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,7{}\mathrm {i}}{a\,f}+\frac {A\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,f\,\left (c+c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \]
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