Optimal. Leaf size=208 \[ -\frac{2 (-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{315 c^3 f (c-i c \tan (e+f x))^{3/2}}-\frac{2 (-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{105 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac{(-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{21 c f (c-i c \tan (e+f x))^{7/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]
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Rubi [A] time = 0.282241, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ -\frac{2 (-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{315 c^3 f (c-i c \tan (e+f x))^{3/2}}-\frac{2 (-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{105 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac{(-2 B+i A) (a+i a \tan (e+f x))^{3/2}}{21 c f (c-i c \tan (e+f x))^{7/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]
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))^{9/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{\sqrt{a+i a x} (A+B x)}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}}+\frac{(a (A+2 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 )}{3 f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac{(i A-2 B) (a+i a \tan (e+f x))^{3/2}}{21 c f (c-i c \tan (e+f x))^{7/2}}+\frac{(2 a (A+2 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 )}{21 c f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac{(i A-2 B) (a+i a \tan (e+f x))^{3/2}}{21 c f (c-i c \tan (e+f x))^{7/2}}-\frac{2 (i A-2 B) (a+i a \tan (e+f x))^{3/2}}{105 c^2 f (c-i c \tan (e+f x))^{5/2}}+\frac{(2 a (A+2 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 )}{105 c^2 f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{3/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac{(i A-2 B) (a+i a \tan (e+f x))^{3/2}}{21 c f (c-i c \tan (e+f x))^{7/2}}-\frac{2 (i A-2 B) (a+i a \tan (e+f x))^{3/2}}{105 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 (i A-2 B) (a+i a \tan (e+f x))^{3/2}}{315 c^3 f (c-i c \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 8.81579, size = 148, normalized size = 0.71 \[ \frac{a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)} (\cos (6 e+7 f x)+i \sin (6 e+7 f x)) (-(A+2 i B) (27 \sin (e+f x)+35 \sin (3 (e+f x)))+9 (B-18 i A) \cos (e+f x)+35 (B-2 i A) \cos (3 (e+f x)))}{1260 c^5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.117, size = 136, normalized size = 0.7 \begin{align*}{\frac{{\frac{i}{315}}a \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 2\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{3}-24\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{2}-4\,B \left ( \tan \left ( fx+e \right ) \right ) ^{3}-33\,iA\tan \left ( fx+e \right ) -12\,A \left ( \tan \left ( fx+e \right ) \right ) ^{2}+11\,iB+66\,B\tan \left ( fx+e \right ) +58\,A \right ) }{f{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}\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 = 2.39516, size = 351, normalized size = 1.69 \begin{align*} \frac{{\left (35 \,{\left (-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 ) + 45 \,{\left (-3 i \, A - B\right )} 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 ) + 63 \,{\left (-3 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 ) + 105 \,{\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 (35 \, A - 35 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 ) +{\left (135 \, A - 45 i \, B\right )} 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 (189 \, A + 63 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 (105 \, A + 105 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}}{2520 \, c^{\frac{9}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33321, size = 423, normalized size = 2.03 \begin{align*} \frac{{\left ({\left (-35 i \, A - 35 \, B\right )} a e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-170 i \, A - 80 \, B\right )} a e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-324 i \, A + 18 \, B\right )} a e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-294 i \, A + 168 \, B\right )} a e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-105 i \, A + 105 \, 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 )}}{2520 \, c^{5} 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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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