Optimal. Leaf size=261 \[ -\frac {2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ -\frac {2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x} (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))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}+\frac {(a (4 A+7 i B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x}}{(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))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}+\frac {(a (4 A+7 i B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{33 c f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}+\frac {(2 a (4 A+7 i B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{231 c^2 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {(2 a (4 A+7 i B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{1155 c^3 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 13.83, size = 179, normalized size = 0.69 \[ -\frac {i a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} (\cos (7 e+8 f x)+i \sin (7 e+8 f x)) (308 (7 A+i B) \cos (2 (e+f x))+105 (7 A+4 i B) \cos (4 (e+f x))-616 i A \sin (2 (e+f x))-420 i A \sin (4 (e+f x))+1485 A+1078 B \sin (2 (e+f x))+735 B \sin (4 (e+f x)))}{27720 c^6 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 151, normalized size = 0.58 \[ \frac {{\left ({\left (-315 i \, A - 315 \, B\right )} a e^{\left (13 i \, f x + 13 i \, e\right )} + {\left (-1855 i \, A - 1085 \, B\right )} a e^{\left (11 i \, f x + 11 i \, e\right )} + {\left (-4510 i \, A - 770 \, B\right )} a e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (-5742 i \, A + 1386 \, B\right )} a e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (-3927 i \, A + 2541 \, B\right )} a e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (-1155 i \, A + 1155 \, B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{55440 \, c^{6} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 158, normalized size = 0.61 \[ -\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (14 i B \left (\tan ^{4}\left (f x +e \right )\right )+56 i A \left (\tan ^{3}\left (f x +e \right )\right )+8 A \left (\tan ^{4}\left (f x +e \right )\right )-315 i B \left (\tan ^{2}\left (f x +e \right )\right )-98 B \left (\tan ^{3}\left (f x +e \right )\right )-364 i A \tan \left (f x +e \right )-180 A \left (\tan ^{2}\left (f x +e \right )\right )+91 i B +637 B \tan \left (f x +e \right )+547 A \right )}{3465 f \,c^{6} \left (\tan \left (f x +e \right )+i\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 312, normalized size = 1.20 \[ -\frac {{\left ({\left (315 i \, A + 315 \, B\right )} a \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + {\left (1540 i \, A + 770 \, B\right )} a \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 2970 i \, A a \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + {\left (2772 i \, A - 1386 \, B\right )} a \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + {\left (1155 i \, A - 1155 \, B\right )} a \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 315 \, {\left (A - i \, B\right )} a \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 770 \, {\left (2 \, A - i \, B\right )} a \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 2970 \, A a \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 1386 \, {\left (2 \, A + i \, B\right )} a \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 1155 \, {\left (A + i \, B\right )} a \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{55440 \, c^{\frac {11}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.51, size = 315, normalized size = 1.21 \[ -\frac {a\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (2\,e+2\,f\,x\right )\,1155{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,2772{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,2970{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,1540{}\mathrm {i}+A\,\cos \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}-1155\,B\,\cos \left (2\,e+2\,f\,x\right )-1386\,B\,\cos \left (4\,e+4\,f\,x\right )+770\,B\,\cos \left (8\,e+8\,f\,x\right )+315\,B\,\cos \left (10\,e+10\,f\,x\right )-1155\,A\,\sin \left (2\,e+2\,f\,x\right )-2772\,A\,\sin \left (4\,e+4\,f\,x\right )-2970\,A\,\sin \left (6\,e+6\,f\,x\right )-1540\,A\,\sin \left (8\,e+8\,f\,x\right )-315\,A\,\sin \left (10\,e+10\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,1155{}\mathrm {i}-B\,\sin \left (4\,e+4\,f\,x\right )\,1386{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,770{}\mathrm {i}+B\,\sin \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}\right )}{55440\,c^5\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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