Optimal. Leaf size=155 \[ -\frac{(-9 B+2 i A) (a+i a \tan (e+f x))^{7/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac{(-9 B+2 i A) (a+i a \tan (e+f x))^{7/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{11 f (c-i c \tan (e+f x))^{11/2}} \]
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Rubi [A] time = 0.263789, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ -\frac{(-9 B+2 i A) (a+i a \tan (e+f x))^{7/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac{(-9 B+2 i A) (a+i a \tan (e+f x))^{7/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{11 f (c-i c \tan (e+f x))^{11/2}} \]
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))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(a+i a x)^{5/2} (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))^{7/2}}{11 f (c-i c \tan (e+f x))^{11/2}}+\frac{(a (2 A+9 i B)) \operatorname{Subst}\left (\int \frac{(a+i a x)^{5/2}}{(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))^{7/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac{(2 i A-9 B) (a+i a \tan (e+f x))^{7/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}+\frac{(a (2 A+9 i B)) \operatorname{Subst}\left (\int \frac{(a+i a x)^{5/2}}{(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))^{7/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac{(2 i A-9 B) (a+i a \tan (e+f x))^{7/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac{(2 i A-9 B) (a+i a \tan (e+f x))^{7/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}\\ \end{align*}
Mathematica [B] time = 15.7224, size = 417, normalized size = 2.69 \[ \frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left ((B-i A) \cos (6 f x) \left (\frac{\cos (3 e)}{56 c^6}+\frac{i \sin (3 e)}{56 c^6}\right )+(A+i B) \sin (6 f x) \left (\frac{\cos (3 e)}{56 c^6}+\frac{i \sin (3 e)}{56 c^6}\right )+(9 B-23 i A) \cos (8 f x) \left (\frac{\cos (5 e)}{504 c^6}+\frac{i \sin (5 e)}{504 c^6}\right )+(31 A-9 i B) \cos (10 f x) \left (\frac{\sin (7 e)}{792 c^6}-\frac{i \cos (7 e)}{792 c^6}\right )+(A-i B) \cos (12 f x) \left (\frac{\sin (9 e)}{88 c^6}-\frac{i \cos (9 e)}{88 c^6}\right )+(23 A+9 i B) \sin (8 f x) \left (\frac{\cos (5 e)}{504 c^6}+\frac{i \sin (5 e)}{504 c^6}\right )+(31 A-9 i B) \sin (10 f x) \left (\frac{\cos (7 e)}{792 c^6}+\frac{i \sin (7 e)}{792 c^6}\right )+(A-i B) \sin (12 f x) \left (\frac{\cos (9 e)}{88 c^6}+\frac{i \sin (9 e)}{88 c^6}\right )\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.11, size = 161, normalized size = 1. \begin{align*}{\frac{{\frac{i}{693}}{a}^{3} \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 2\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{4}-63\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{3}-9\,B \left ( \tan \left ( fx+e \right ) \right ) ^{4}-45\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{2}-14\,A \left ( \tan \left ( fx+e \right ) \right ) ^{3}+63\,iB\tan \left ( fx+e \right ) -144\,B \left ( \tan \left ( fx+e \right ) \right ) ^{2}+79\,iA-140\,A\tan \left ( fx+e \right ) -9\,B \right ) }{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 = 3.05426, size = 270, normalized size = 1.74 \begin{align*} \frac{{\left (63 \,{\left (-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 ) - 154 i \, A 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 ) + 99 \,{\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 ) +{\left (63 \, A - 63 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 ) + 154 \, A 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 ) +{\left (99 \, A + 99 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}}{2772 \, c^{\frac{11}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41822, size = 375, normalized size = 2.42 \begin{align*} \frac{{\left ({\left (-63 i \, A - 63 \, B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-217 i \, A - 63 \, B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-253 i \, A + 99 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-99 i \, A + 99 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 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 )}}{2772 \, 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{7}{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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