Integrand size = 45, antiderivative size = 208 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{3465 c^3 f (c-i c \tan (e+f x))^{5/2}} \]
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Time = 0.34 (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} \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {2 (-8 B+3 i A) (a+i a \tan (e+f x))^{5/2}}{3465 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (-8 B+3 i A) (a+i a \tan (e+f x))^{5/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {(-8 B+3 i A) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}} \]
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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)^{3/2} (A+B x)}{(c-i c x)^{13/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}+\frac {(a (3 A+8 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{11 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}+\frac {(2 a (3 A+8 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{99 c f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}+\frac {(2 a (3 A+8 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{693 c^2 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{3465 c^3 f (c-i c \tan (e+f x))^{5/2}} \\ \end{align*}
Time = 16.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.75 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\frac {a^2 \cos (e+f x) (55 (-24 i A+B) \cos (e+f x)+63 (-8 i A+3 B) \cos (3 (e+f x))-(3 A+8 i B) (55 \sin (e+f x)+63 \sin (3 (e+f x)))) (\cos (8 e+10 f x)+i \sin (8 e+10 f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{13860 c^6 f (\cos (f x)+i \sin (f x))^2} \]
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Time = 0.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (315 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+315 B \,{\mathrm e}^{10 i \left (f x +e \right )}+1155 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+385 B \,{\mathrm e}^{8 i \left (f x +e \right )}+1485 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-495 B \,{\mathrm e}^{6 i \left (f x +e \right )}+693 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-693 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{27720 c^{5} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(156\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{4}-112 i B \tan \left (f x +e \right )^{3}-16 B \tan \left (f x +e \right )^{4}-135 i A \tan \left (f x +e \right )^{2}-42 A \tan \left (f x +e \right )^{3}-427 i \tan \left (f x +e \right ) B +360 B \tan \left (f x +e \right )^{2}-456 i A +273 A \tan \left (f x +e \right )+61 B \right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\) | \(161\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{4}-112 i B \tan \left (f x +e \right )^{3}-16 B \tan \left (f x +e \right )^{4}-135 i A \tan \left (f x +e \right )^{2}-42 A \tan \left (f x +e \right )^{3}-427 i \tan \left (f x +e \right ) B +360 B \tan \left (f x +e \right )^{2}-456 i A +273 A \tan \left (f x +e \right )+61 B \right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\) | \(161\) |
parts | \(-\frac {i A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (2 i \tan \left (f x +e \right )^{4}-45 i \tan \left (f x +e \right )^{2}-14 \tan \left (f x +e \right )^{3}-152 i+91 \tan \left (f x +e \right )\right )}{1155 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}+\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-61+427 i \tan \left (f x +e \right )-360 \tan \left (f x +e \right )^{2}+112 i \tan \left (f x +e \right )^{3}+16 \tan \left (f x +e \right )^{4}\right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\) | \(217\) |
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Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {{\left (315 \, {\left (i \, A + B\right )} a^{2} e^{\left (13 i \, f x + 13 i \, e\right )} + 70 \, {\left (21 i \, A + 10 \, B\right )} a^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + 110 \, {\left (24 i \, A - B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 198 \, {\left (11 i \, A - 6 \, B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 693 \, {\left (i \, A - B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{27720 \, c^{6} f} \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\text {Timed out} \]
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Time = 0.61 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\frac {{\left (315 \, {\left (-i \, A - B\right )} a^{2} \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 385 \, {\left (-3 i \, A - B\right )} a^{2} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 495 \, {\left (-3 i \, A + B\right )} a^{2} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 693 \, {\left (-i \, A + B\right )} a^{2} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 315 \, {\left (A - i \, B\right )} a^{2} \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 385 \, {\left (3 \, A - i \, B\right )} a^{2} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 495 \, {\left (3 \, A + i \, B\right )} a^{2} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 693 \, {\left (A + i \, B\right )} a^{2} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{27720 \, c^{\frac {11}{2}} f} \]
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\[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
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Time = 12.41 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.40 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,693{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,1485{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,1155{}\mathrm {i}+A\,\cos \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}-693\,B\,\cos \left (4\,e+4\,f\,x\right )-495\,B\,\cos \left (6\,e+6\,f\,x\right )+385\,B\,\cos \left (8\,e+8\,f\,x\right )+315\,B\,\cos \left (10\,e+10\,f\,x\right )-693\,A\,\sin \left (4\,e+4\,f\,x\right )-1485\,A\,\sin \left (6\,e+6\,f\,x\right )-1155\,A\,\sin \left (8\,e+8\,f\,x\right )-315\,A\,\sin \left (10\,e+10\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,693{}\mathrm {i}-B\,\sin \left (6\,e+6\,f\,x\right )\,495{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,385{}\mathrm {i}+B\,\sin \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}\right )}{27720\,c^5\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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