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 = 0.320727, antiderivative size = 261, normalized size of antiderivative = 1., 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 3588
Rule 78
Rule 45
Rule 37
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*}
Mathematica [A] time = 12.7293, 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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Maple [A] time = 0.114, size = 158, normalized size = 0.6 \begin{align*} -{\frac{a \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 14\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{4}+56\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{3}+8\,A \left ( \tan \left ( fx+e \right ) \right ) ^{4}-315\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{2}-98\,B \left ( \tan \left ( fx+e \right ) \right ) ^{3}-364\,iA\tan \left ( fx+e \right ) -180\,A \left ( \tan \left ( fx+e \right ) \right ) ^{2}+91\,iB+637\,B\tan \left ( fx+e \right ) +547\,A \right ) }{3465\,f{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{7}}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.32133, size = 421, normalized size = 1.61 \begin{align*} \frac{{\left (315 \,{\left (-i \, A - 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 ) + 770 \,{\left (-2 i \, A - 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 ) + 1386 \,{\left (-2 i \, A + 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 ) + 1155 \,{\left (-i \, A + 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 ) +{\left (315 \, A - 315 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 ) +{\left (1540 \, A - 770 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 ) +{\left (2772 \, A + 1386 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 ) +{\left (1155 \, A + 1155 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40657, size = 502, normalized size = 1.92 \begin{align*} \frac{{\left ({\left (-315 i \, A - 315 \, B\right )} a e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-1855 i \, A - 1085 \, B\right )} a e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-4510 i \, A - 770 \, B\right )} a e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-5742 i \, A + 1386 \, B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-3927 i \, A + 2541 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-1155 i \, A + 1155 \, B\right )} a e^{\left (2 i \, f x + 2 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}} e^{\left (i \, f x + i \, e\right )}}{55440 \, c^{6} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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