Integrand size = 43, antiderivative size = 238 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {5 (3 i A-11 B) c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a^2 f}-\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
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Time = 0.48 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3669, 79, 43, 52, 65, 214} \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {5 c^{7/2} (-11 B+3 i A) \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a^2 f}-\frac {5 c^3 (-11 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 c^2 (-11 B+3 i A) (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {c (-11 B+3 i A) (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
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Rule 43
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)^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}-\frac {((3 A+11 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{5/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = -\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 A+11 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 )}{16 a f} \\ & = -\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 A+11 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 )}{8 a f} \\ & = -\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 A+11 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 )}{4 a f} \\ & = -\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (5 (3 i A-11 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 )}{2 a f} \\ & = \frac {5 (3 i A-11 B) c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a^2 f}-\frac {5 (3 i A-11 B) c^3 \sqrt {c-i c \tan (e+f x)}}{4 a^2 f}-\frac {5 (3 i A-11 B) c^2 (c-i c \tan (e+f x))^{3/2}}{24 a^2 f}-\frac {(3 i A-11 B) c (c-i c \tan (e+f x))^{5/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{4 a^2 f (1+i \tan (e+f x))^2} \\ \end{align*}
Time = 6.89 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.76 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {15 \sqrt {2} (-3 i A+11 B) c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right ) \sec ^2(e+f x) (\cos (2 (e+f x))+i \sin (2 (e+f x)))+2 c^3 \sqrt {c-i c \tan (e+f x)} \left (27 i A-103 B-(51 A+175 i B) \tan (e+f x)+4 (-3 i A+14 B) \tan ^2(e+f x)-4 i B \tan ^3(e+f x)\right )}{12 a^2 f (-i+\tan (e+f x))^2} \]
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Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {2 i c^{2} \left (-\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-5 i \sqrt {c -i c \tan \left (f x +e \right )}\, B c -\sqrt {c -i c \tan \left (f x +e \right )}\, c A +2 c^{2} \left (\frac {4 \left (\frac {17 i B}{32}+\frac {9 A}{32}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+4 \left (-\frac {15}{16} i B c -\frac {7}{16} c A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{2}}+\frac {5 \left (\frac {11 i B}{4}+\frac {3 A}{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )\right )}{f \,a^{2}}\) | \(179\) |
default | \(\frac {2 i c^{2} \left (-\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-5 i \sqrt {c -i c \tan \left (f x +e \right )}\, B c -\sqrt {c -i c \tan \left (f x +e \right )}\, c A +2 c^{2} \left (\frac {4 \left (\frac {17 i B}{32}+\frac {9 A}{32}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+4 \left (-\frac {15}{16} i B c -\frac {7}{16} c A \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{2}}+\frac {5 \left (\frac {11 i B}{4}+\frac {3 A}{4}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )\right )}{f \,a^{2}}\) | \(179\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (185) = 370\).
Time = 0.27 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.92 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=-\frac {15 \, \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 66 i \, A B - 121 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \log \left (-\frac {5 \, {\left ({\left (-3 i \, A + 11 \, B\right )} c^{4} + \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 66 i \, A B - 121 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a^{2} f}\right ) - 15 \, \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 66 i \, A B - 121 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \log \left (-\frac {5 \, {\left ({\left (-3 i \, A + 11 \, B\right )} c^{4} - \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {-\frac {{\left (9 \, A^{2} + 66 i \, A B - 121 \, B^{2}\right )} c^{7}}{a^{4} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a^{2} f}\right ) + \sqrt {2} {\left (15 \, {\left (3 i \, A - 11 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 20 \, {\left (3 i \, A - 11 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (3 i \, A - 11 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 \, {\left (-i \, A + B\right )} c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )}} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=- \frac {\int \frac {A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \left (- \frac {3 A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \frac {B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \left (- \frac {3 B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \left (- \frac {3 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \frac {i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \left (- \frac {3 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \frac {i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx}{a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=-\frac {i \, {\left (\frac {15 \, \sqrt {2} {\left (3 \, A + 11 i \, B\right )} c^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{2}} - \frac {12 \, {\left ({\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (9 \, A + 17 i \, B\right )} c^{5} - 2 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (7 \, A + 15 i \, B\right )} c^{6}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{2} - 4 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} c + 4 \, a^{2} c^{2}} + \frac {16 \, {\left (i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} B c^{3} + 3 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A + 5 i \, B\right )} c^{4}\right )}}{a^{2}}\right )}}{24 \, c f} \]
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\[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Time = 8.91 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^2} \, dx=\frac {15\,B\,c^5\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-\frac {17\,B\,c^4\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{2}}{4\,a^2\,c^2\,f+a^2\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,a^2\,c\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}-\frac {\frac {A\,c^5\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,7{}\mathrm {i}}{a^2\,f}-\frac {A\,c^4\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,9{}\mathrm {i}}{2\,a^2\,f}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,c\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )+4\,c^2}-\frac {A\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a^2\,f}+\frac {10\,B\,c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a^2\,f}+\frac {2\,B\,c^2\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a^2\,f}-\frac {\sqrt {2}\,A\,{\left (-c\right )}^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,15{}\mathrm {i}}{4\,a^2\,f}+\frac {\sqrt {2}\,B\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,55{}\mathrm {i}}{4\,a^2\,f} \]
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