Integrand size = 43, antiderivative size = 220 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=-\frac {2 \sqrt {2} (5 i A-9 B) c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {2 (5 i A-9 B) c^3 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac {(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))} \]
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Time = 0.33 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3669, 79, 52, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=-\frac {2 \sqrt {2} c^{7/2} (-9 B+5 i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {2 c^3 (-9 B+5 i A) \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {c^2 (-9 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac {c (-9 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))} \]
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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)^{5/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac {((5 A+9 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{5/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = \frac {(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left ((5 A+9 i B) c^2\right ) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac {(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left ((5 A+9 i B) c^3\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {2 (5 i A-9 B) c^3 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac {(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left (2 (5 A+9 i B) c^4\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 {2 (5 i A-9 B) c^3 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac {(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac {\left (4 (5 i A-9 B) c^3\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 {2 \sqrt {2} (5 i A-9 B) c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {2 (5 i A-9 B) c^3 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac {(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))} \\ \end{align*}
Time = 7.75 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=\frac {2 c^3 \left (15 \sqrt {2} (-5 i A+9 B) \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )+\frac {\sqrt {c-i c \tan (e+f x)} \left (95 A+168 i B+(60 i A-117 B) \tan (e+f x)+(5 A+18 i B) \tan ^2(e+f x)+3 B \tan ^3(e+f x)\right )}{-i+\tan (e+f x)}\right )}{15 a f} \]
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Time = 0.23 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {2 i c \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+i B c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+\frac {A c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+8 i \sqrt {c -i c \tan \left (f x +e \right )}\, B \,c^{2}+4 \sqrt {c -i c \tan \left (f x +e \right )}\, A \,c^{2}-4 c^{3} \left (\frac {\left (-\frac {i B}{4}-\frac {A}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {9 i B}{2}+\frac {5 A}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\right )\right )}{f a}\) | \(192\) |
default | \(\frac {2 i c \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+i B c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+\frac {A c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+8 i \sqrt {c -i c \tan \left (f x +e \right )}\, B \,c^{2}+4 \sqrt {c -i c \tan \left (f x +e \right )}\, A \,c^{2}-4 c^{3} \left (\frac {\left (-\frac {i B}{4}-\frac {A}{4}\right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}}+\frac {\left (\frac {9 i B}{2}+\frac {5 A}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\right )\right )}{f a}\) | \(192\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (173) = 346\).
Time = 0.30 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=-\frac {15 \, \sqrt {2} \sqrt {-\frac {{\left (25 \, A^{2} + 90 i \, A B - 81 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (-\frac {8 \, {\left ({\left (5 i \, A - 9 \, B\right )} c^{4} + \sqrt {-\frac {{\left (25 \, A^{2} + 90 i \, A B - 81 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) - 15 \, \sqrt {2} \sqrt {-\frac {{\left (25 \, A^{2} + 90 i \, A B - 81 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (-\frac {8 \, {\left ({\left (5 i \, A - 9 \, B\right )} c^{4} - \sqrt {-\frac {{\left (25 \, A^{2} + 90 i \, A B - 81 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) + 2 \, \sqrt {2} {\left (15 \, {\left (-5 i \, A + 9 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 35 \, {\left (-5 i \, A + 9 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 23 \, {\left (-5 i \, A + 9 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 15 \, {\left (-i \, A + B\right )} c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, {\left (a f e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )}} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=- \frac {i \left (\int \frac {A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {3 A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \frac {B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {3 B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \left (- \frac {3 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \frac {i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {3 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \frac {i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx\right )}{a} \]
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Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=\frac {i \, {\left (\frac {15 \, \sqrt {2} {\left (5 \, A + 9 i \, B\right )} c^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a} - \frac {60 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + i \, B\right )} c^{5}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a - 2 \, a c} + \frac {2 \, {\left (3 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} B c^{2} + 5 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (A + 3 i \, B\right )} c^{3} + 60 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + 2 i \, B\right )} c^{4}\right )}}{a}\right )}}{15 \, c f} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]
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Time = 1.37 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx=\frac {4\,B\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-2\,a\,c\,f}+\frac {A\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,8{}\mathrm {i}}{a\,f}+\frac {A\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{3\,a\,f}-\frac {16\,B\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a\,f}-\frac {2\,B\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{a\,f}-\frac {2\,B\,c\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,a\,f}+\frac {\sqrt {2}\,A\,{\left (-c\right )}^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,10{}\mathrm {i}}{a\,f}-\frac {\sqrt {2}\,B\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,18{}\mathrm {i}}{a\,f}+\frac {A\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,4{}\mathrm {i}}{a\,f\,\left (c+c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \]
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