Optimal. Leaf size=261 \[ -\frac{2 (-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{45045 c^4 f (c-i c \tan (e+f x))^{7/2}}-\frac{2 (-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{6435 c^3 f (c-i c \tan (e+f x))^{9/2}}-\frac{(-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}-\frac{(-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}} \]
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Rubi [A] time = 0.315509, 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} \[ -\frac{2 (-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{45045 c^4 f (c-i c \tan (e+f x))^{7/2}}-\frac{2 (-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{6435 c^3 f (c-i c \tan (e+f x))^{9/2}}-\frac{(-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}-\frac{(-11 B+4 i A) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac{(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}} \]
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))^{15/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(a+i a x)^{5/2} (A+B x)}{(c-i c x)^{17/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}+\frac{(a (4 A+11 i B)) \operatorname{Subst}\left (\int \frac{(a+i a x)^{5/2}}{(c-i c x)^{15/2}} \, dx,x,\tan (e+f x)\right )}{15 f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}+\frac{(a (4 A+11 i B)) \operatorname{Subst}\left (\int \frac{(a+i a x)^{5/2}}{(c-i c x)^{13/2}} \, dx,x,\tan (e+f x)\right )}{65 c f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}+\frac{(2 a (4 A+11 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 )}{715 c^2 f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}-\frac{2 (4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{6435 c^3 f (c-i c \tan (e+f x))^{9/2}}+\frac{(2 a (4 A+11 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 )}{6435 c^3 f}\\ &=-\frac{(i A+B) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac{(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}-\frac{2 (4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{6435 c^3 f (c-i c \tan (e+f x))^{9/2}}-\frac{2 (4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{45045 c^4 f (c-i c \tan (e+f x))^{7/2}}\\ \end{align*}
Mathematica [B] time = 17.3174, size = 577, normalized size = 2.21 \[ \frac{\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt{\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left ((B-i A) \cos (6 f x) \left (\frac{\cos (3 e)}{224 c^8}+\frac{i \sin (3 e)}{224 c^8}\right )+(A+i B) \sin (6 f x) \left (\frac{\cos (3 e)}{224 c^8}+\frac{i \sin (3 e)}{224 c^8}\right )+(23 B-37 i A) \cos (8 f x) \left (\frac{\cos (5 e)}{2016 c^8}+\frac{i \sin (5 e)}{2016 c^8}\right )+(11 B-49 i A) \cos (10 f x) \left (\frac{\cos (7 e)}{1584 c^8}+\frac{i \sin (7 e)}{1584 c^8}\right )+(61 A-11 i B) \cos (12 f x) \left (\frac{\sin (9 e)}{2288 c^8}-\frac{i \cos (9 e)}{2288 c^8}\right )+(73 A-43 i B) \cos (14 f x) \left (\frac{\sin (11 e)}{6240 c^8}-\frac{i \cos (11 e)}{6240 c^8}\right )+(A-i B) \cos (16 f x) \left (\frac{\sin (13 e)}{480 c^8}-\frac{i \cos (13 e)}{480 c^8}\right )+(37 A+23 i B) \sin (8 f x) \left (\frac{\cos (5 e)}{2016 c^8}+\frac{i \sin (5 e)}{2016 c^8}\right )+(49 A+11 i B) \sin (10 f x) \left (\frac{\cos (7 e)}{1584 c^8}+\frac{i \sin (7 e)}{1584 c^8}\right )+(61 A-11 i B) \sin (12 f x) \left (\frac{\cos (9 e)}{2288 c^8}+\frac{i \sin (9 e)}{2288 c^8}\right )+(73 A-43 i B) \sin (14 f x) \left (\frac{\cos (11 e)}{6240 c^8}+\frac{i \sin (11 e)}{6240 c^8}\right )+(A-i B) \sin (16 f x) \left (\frac{\cos (13 e)}{480 c^8}+\frac{i \sin (13 e)}{480 c^8}\right )\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 206, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3} \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 22\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{6}+72\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{5}+8\,A \left ( \tan \left ( fx+e \right ) \right ) ^{6}-825\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{4}-198\,B \left ( \tan \left ( fx+e \right ) \right ) ^{5}-780\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{3}-300\,A \left ( \tan \left ( fx+e \right ) \right ) ^{4}-7260\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{2}+2145\,B \left ( \tan \left ( fx+e \right ) \right ) ^{3}-6858\,iA\tan \left ( fx+e \right ) +1455\,A \left ( \tan \left ( fx+e \right ) \right ) ^{2}-407\,iB-3663\,B\tan \left ( fx+e \right ) -4243\,A \right ) }{45045\,f{c}^{8} \left ( \tan \left ( fx+e \right ) +i \right ) ^{9}}\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 = 3.28053, size = 448, normalized size = 1.72 \begin{align*} \frac{{\left (3003 \,{\left (-i \, A - B\right )} a^{3} \cos \left (\frac{15}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 6930 \,{\left (-2 i \, A - B\right )} a^{3} \cos \left (\frac{13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 24570 i \, A 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 ) + 10010 \,{\left (-2 i \, A + B\right )} 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 ) + 6435 \,{\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 (3003 \, A - 3003 i \, B\right )} a^{3} \sin \left (\frac{15}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) +{\left (13860 \, A - 6930 i \, B\right )} a^{3} \sin \left (\frac{13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 24570 \, A 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 ) +{\left (20020 \, A + 10010 i \, B\right )} 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 (6435 \, A + 6435 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}}{720720 \, c^{\frac{15}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41013, size = 537, normalized size = 2.06 \begin{align*} \frac{{\left ({\left (-3003 i \, A - 3003 \, B\right )} a^{3} e^{\left (16 i \, f x + 16 i \, e\right )} +{\left (-16863 i \, A - 9933 \, B\right )} a^{3} e^{\left (14 i \, f x + 14 i \, e\right )} +{\left (-38430 i \, A - 6930 \, B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-44590 i \, A + 10010 \, B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-26455 i \, A + 16445 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-6435 i \, A + 6435 \, 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 )}}{720720 \, c^{8} 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{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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