Optimal. Leaf size=220 \[ \frac{2 c^3 (-9 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{c^2 (-9 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{3 a f}-\frac{2 \sqrt{2} c^{7/2} (-9 B+5 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{a f}+\frac{c (-9 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))} \]
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Rubi [A] time = 0.275552, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116, Rules used = {3588, 78, 50, 63, 208} \[ \frac{2 c^3 (-9 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{c^2 (-9 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{3 a f}-\frac{2 \sqrt{2} c^{7/2} (-9 B+5 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{a f}+\frac{c (-9 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(-B+i A) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)^{5/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac{((5 A+9 i B) c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{5/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=\frac{(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac{\left ((5 A+9 i B) c^2\right ) \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac{(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac{\left ((5 A+9 i B) c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 (5 i A-9 B) c^3 \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac{(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac{\left (2 (5 A+9 i B) c^4\right ) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 (5 i A-9 B) c^3 \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac{(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}-\frac{\left (4 (5 i A-9 B) c^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{f}\\ &=-\frac{2 \sqrt{2} (5 i A-9 B) c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{a f}+\frac{2 (5 i A-9 B) c^3 \sqrt{c-i c \tan (e+f x)}}{a f}+\frac{(5 i A-9 B) c^2 (c-i c \tan (e+f x))^{3/2}}{3 a f}+\frac{(5 i A-9 B) c (c-i c \tan (e+f x))^{5/2}}{10 a f}+\frac{(i A-B) (c-i c \tan (e+f x))^{7/2}}{2 a f (1+i \tan (e+f x))}\\ \end{align*}
Mathematica [F] time = 180.005, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [A] time = 0.097, size = 192, normalized size = 0.9 \begin{align*}{\frac{2\,ic}{af} \left ({\frac{i}{5}}B \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}+iB \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}c+{\frac{Ac}{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+8\,iB{c}^{2}\sqrt{c-ic\tan \left ( fx+e \right ) }+4\,A{c}^{2}\sqrt{c-ic\tan \left ( fx+e \right ) }+4\,{c}^{3} \left ({\frac{ \left ( -A/2-i/2B \right ) \sqrt{c-ic\tan \left ( fx+e \right ) }}{-c-ic\tan \left ( fx+e \right ) }}-1/4\,{\frac{ \left ( 5\,A+9\,iB \right ) \sqrt{2}}{\sqrt{c}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c-ic\tan \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.28195, size = 1261, normalized size = 5.73 \begin{align*} \frac{15 \, \sqrt{-\frac{{\left (800 \, A^{2} + 2880 i \, A B - 2592 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}}{\left (a f e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac{{\left ({\left (-40 i \, A + 72 \, B\right )} c^{4} + \sqrt{2} \sqrt{-\frac{{\left (800 \, A^{2} + 2880 i \, A B - 2592 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}}{\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) - 15 \, \sqrt{-\frac{{\left (800 \, A^{2} + 2880 i \, A B - 2592 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}}{\left (a f e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac{{\left ({\left (-40 i \, A + 72 \, B\right )} c^{4} - \sqrt{2} \sqrt{-\frac{{\left (800 \, A^{2} + 2880 i \, A B - 2592 \, B^{2}\right )} c^{7}}{a^{2} f^{2}}}{\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) + \sqrt{2}{\left ({\left (600 i \, A - 1080 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (1400 i \, A - 2520 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (920 i \, A - 1656 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (120 i \, A - 120 \, B\right )} c^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{60 \,{\left (a f e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}{i \, a \tan \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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