Integrand size = 45, antiderivative size = 208 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{1287 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{9009 c^3 f (c-i c \tan (e+f x))^{7/2}} \]
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Time = 0.48 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47, 37} \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {2 (-10 B+3 i A) (a+i a \tan (e+f x))^{7/2}}{9009 c^3 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (-10 B+3 i A) (a+i a \tan (e+f x))^{7/2}}{1287 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {(-10 B+3 i A) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}} \]
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Rule 37
Rule 47
Rule 79
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{5/2} (A+B x)}{(c-i c x)^{15/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}+\frac {(a (3 A+10 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{13/2}} \, dx,x,\tan (e+f x)\right )}{13 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}+\frac {(2 a (3 A+10 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{143 c f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{1287 c^2 f (c-i c \tan (e+f x))^{9/2}}+\frac {(2 a (3 A+10 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{1287 c^2 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{1287 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{9009 c^3 f (c-i c \tan (e+f x))^{7/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(495\) vs. \(2(208)=416\).
Time = 19.12 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.38 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\frac {\cos ^4(e+f x) \left ((-i A+B) \cos (6 f x) \left (\frac {\cos (3 e)}{112 c^7}+\frac {i \sin (3 e)}{112 c^7}\right )+(-15 i A+8 B) \cos (8 f x) \left (\frac {\cos (5 e)}{504 c^7}+\frac {i \sin (5 e)}{504 c^7}\right )+(-30 i A+B) \cos (10 f x) \left (\frac {\cos (7 e)}{792 c^7}+\frac {i \sin (7 e)}{792 c^7}\right )+(25 A-12 i B) \cos (12 f x) \left (-\frac {i \cos (9 e)}{1144 c^7}+\frac {\sin (9 e)}{1144 c^7}\right )+(A-i B) \cos (14 f x) \left (-\frac {i \cos (11 e)}{208 c^7}+\frac {\sin (11 e)}{208 c^7}\right )+(A+i B) \left (\frac {\cos (3 e)}{112 c^7}+\frac {i \sin (3 e)}{112 c^7}\right ) \sin (6 f x)+(15 A+8 i B) \left (\frac {\cos (5 e)}{504 c^7}+\frac {i \sin (5 e)}{504 c^7}\right ) \sin (8 f x)+(30 A+i B) \left (\frac {\cos (7 e)}{792 c^7}+\frac {i \sin (7 e)}{792 c^7}\right ) \sin (10 f x)+(25 A-12 i B) \left (\frac {\cos (9 e)}{1144 c^7}+\frac {i \sin (9 e)}{1144 c^7}\right ) \sin (12 f x)+(A-i B) \left (\frac {\cos (11 e)}{208 c^7}+\frac {i \sin (11 e)}{208 c^7}\right ) \sin (14 f x)\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
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Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (693 i A \,{\mathrm e}^{12 i \left (f x +e \right )}+693 B \,{\mathrm e}^{12 i \left (f x +e \right )}+2457 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+819 B \,{\mathrm e}^{10 i \left (f x +e \right )}+3003 i A \,{\mathrm e}^{8 i \left (f x +e \right )}-1001 B \,{\mathrm e}^{8 i \left (f x +e \right )}+1287 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-1287 B \,{\mathrm e}^{6 i \left (f x +e \right )}\right )}{72072 c^{6} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(156\) |
derivativedivides | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{5}-160 i B \tan \left (f x +e \right )^{4}-20 B \tan \left (f x +e \right )^{5}-177 i A \tan \left (f x +e \right )^{3}-48 A \tan \left (f x +e \right )^{4}-1643 i B \tan \left (f x +e \right )^{2}+590 B \tan \left (f x +e \right )^{3}-1569 i A \tan \left (f x +e \right )+408 A \tan \left (f x +e \right )^{2}-97 i B -776 B \tan \left (f x +e \right )-930 A \right )}{9009 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(184\) |
default | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{5}-160 i B \tan \left (f x +e \right )^{4}-20 B \tan \left (f x +e \right )^{5}-177 i A \tan \left (f x +e \right )^{3}-48 A \tan \left (f x +e \right )^{4}-1643 i B \tan \left (f x +e \right )^{2}+590 B \tan \left (f x +e \right )^{3}-1569 i A \tan \left (f x +e \right )+408 A \tan \left (f x +e \right )^{2}-97 i B -776 B \tan \left (f x +e \right )-930 A \right )}{9009 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(184\) |
parts | \(\frac {i A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (2 i \tan \left (f x +e \right )^{5}-59 i \tan \left (f x +e \right )^{3}-16 \tan \left (f x +e \right )^{4}-523 i \tan \left (f x +e \right )+136 \tan \left (f x +e \right )^{2}-310\right )}{3003 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (160 i \tan \left (f x +e \right )^{4}+20 \tan \left (f x +e \right )^{5}+1643 i \tan \left (f x +e \right )^{2}-590 \tan \left (f x +e \right )^{3}+97 i+776 \tan \left (f x +e \right )\right )}{9009 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(238\) |
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Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {{\left (693 \, {\left (i \, A + B\right )} a^{3} e^{\left (15 i \, f x + 15 i \, e\right )} + 126 \, {\left (25 i \, A + 12 \, B\right )} a^{3} e^{\left (13 i \, f x + 13 i \, e\right )} + 182 \, {\left (30 i \, A - B\right )} a^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + 286 \, {\left (15 i \, A - 8 \, B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 1287 \, {\left (i \, A - B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{72072 \, c^{7} f} \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\text {Timed out} \]
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Time = 0.67 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\frac {{\left (693 \, {\left (-i \, A - B\right )} a^{3} \cos \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 819 \, {\left (-3 i \, A - B\right )} a^{3} \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1001 \, {\left (-3 i \, A + B\right )} a^{3} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1287 \, {\left (-i \, A + B\right )} a^{3} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 693 \, {\left (A - i \, B\right )} a^{3} \sin \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 819 \, {\left (3 \, A - i \, B\right )} a^{3} \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1001 \, {\left (3 \, A + i \, B\right )} a^{3} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1287 \, {\left (A + i \, B\right )} a^{3} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{72072 \, c^{\frac {13}{2}} f} \]
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\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]
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Time = 13.01 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.80 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {\sqrt {a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^3\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (3\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{72\,c^6\,f}+\frac {a^3\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\left (3\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{88\,c^6\,f}+\frac {a^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{56\,c^6\,f}+\frac {a^3\,{\mathrm {e}}^{e\,12{}\mathrm {i}+f\,x\,12{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{104\,c^6\,f}\right )}{\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}} \]
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