Optimal. Leaf size=209 \[ \frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(7 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac{(7 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{\sqrt{c} (7 B+5 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{64 \sqrt{2} a^3 f} \]
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Rubi [A] time = 0.239506, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116, Rules used = {3588, 78, 51, 63, 208} \[ \frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(7 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac{(7 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{\sqrt{c} (7 B+5 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{64 \sqrt{2} a^3 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^4 \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{((5 A-7 i B) c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^3 \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{12 f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{((5 A-7 i B) c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^2 \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{32 a f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac{((5 A-7 i B) c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{128 a^2 f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}+\frac{(5 i A+7 B) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{64 a^2 f}\\ &=\frac{(5 i A+7 B) \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{64 \sqrt{2} a^3 f}+\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{48 a^3 f (1+i \tan (e+f x))^2}+\frac{(5 i A+7 B) \sqrt{c-i c \tan (e+f x)}}{64 a^3 f (1+i \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 3.70111, size = 225, normalized size = 1.08 \[ \frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \left (\frac{2}{3} \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt{c-i c \tan (e+f x)} (5 (7 B+5 i A) \sin (2 (e+f x))+(41 A-19 i B) \cos (2 (e+f x))+26 A+2 i B)+\sqrt{2} \sqrt{c} (7 B+5 i A) (\cos (3 e)+i \sin (3 e)) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )\right )}{128 f (a+i a \tan (e+f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.146, size = 148, normalized size = 0.7 \begin{align*}{\frac{2\,i{c}^{3}}{f{a}^{3}} \left ({\frac{1}{ \left ( -c-ic\tan \left ( fx+e \right ) \right ) ^{3}} \left ( -{\frac{5\,A-7\,iB}{128\,{c}^{2}} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A-7\,iB}{24\,c} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+ \left ( -{\frac{11\,A}{32}}+{\frac{9\,i}{32}}B \right ) \sqrt{c-ic\tan \left ( fx+e \right ) } \right ) }+{\frac{ \left ( 5\,A-7\,iB \right ) \sqrt{2}}{256}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}} \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 = 1.49838, size = 1035, normalized size = 4.95 \begin{align*} \frac{{\left (3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} +{\left (5 i \, A + 7 \, B\right )} c\right )} e^{\left (-i \, f x - i \, e\right )}}{32 \, a^{3} f}\right ) - 3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{{\left (25 \, A^{2} - 70 i \, A B - 49 \, B^{2}\right )} c}{a^{6} f^{2}}} -{\left (5 i \, A + 7 \, B\right )} c\right )} e^{\left (-i \, f x - i \, e\right )}}{32 \, a^{3} f}\right ) + \sqrt{2}{\left ({\left (33 i \, A + 27 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (59 i \, A + 25 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (34 i \, A - 10 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, A - 8 \, B\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{384 \, a^{3} f} \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 )} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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