Optimal. Leaf size=213 \[ -\frac{c^2 (11 B+i A) \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (1+i \tan (e+f x))}+\frac{c^{5/2} (11 B+i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^3 f}+\frac{c (11 B+i A) (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
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Rubi [A] time = 0.248603, antiderivative size = 213, 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, 47, 63, 208} \[ -\frac{c^2 (11 B+i A) \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (1+i \tan (e+f x))}+\frac{c^{5/2} (11 B+i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^3 f}+\frac{c (11 B+i A) (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(-B+i A) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)^{3/2}}{(a+i a x)^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{((A-11 i B) c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{12 f}\\ &=\frac{(i A+11 B) c (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3}-\frac{\left ((A-11 i B) c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{16 a f}\\ &=-\frac{(i A+11 B) c^2 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (1+i \tan (e+f x))}+\frac{(i A+11 B) c (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{\left ((A-11 i B) c^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{32 a^2 f}\\ &=-\frac{(i A+11 B) c^2 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (1+i \tan (e+f x))}+\frac{(i A+11 B) c (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3}+\frac{\left ((i A+11 B) c^2\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 )}{16 a^2 f}\\ &=\frac{(i A+11 B) c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a^3 f}-\frac{(i A+11 B) c^2 \sqrt{c-i c \tan (e+f x)}}{16 a^3 f (1+i \tan (e+f x))}+\frac{(i A+11 B) c (c-i c \tan (e+f x))^{3/2}}{24 a^3 f (1+i \tan (e+f x))^2}+\frac{(i A-B) (c-i c \tan (e+f x))^{5/2}}{6 a^3 f (1+i \tan (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 7.38131, size = 227, normalized size = 1.07 \[ \frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^3 (A+B \tan (e+f x)) \left (\frac{2}{3} c^2 \cos (e+f x) (\cos (3 f x)-i \sin (3 f x)) \sqrt{c-i c \tan (e+f x)} ((11 A-25 i B) \sin (2 (e+f x))+(-41 B+5 i A) \cos (2 (e+f x))+2 i A+22 B)+\sqrt{2} c^{5/2} (11 B+i A) (\cos (3 e)+i \sin (3 e)) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )\right )}{32 f (a+i a \tan (e+f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.11, size = 146, 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 ( \left ( -{\frac{21\,i}{32}}B-{\frac{A}{32}} \right ) \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}+ \left ({\frac{11\,i}{6}}Bc-{\frac{Ac}{6}} \right ) \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}+ \left ( -{\frac{11\,i}{8}}B{c}^{2}+{\frac{A{c}^{2}}{8}} \right ) \sqrt{c-ic\tan \left ( fx+e \right ) } \right ) }+{\frac{ \left ( -11\,iB+A \right ) \sqrt{2}}{64}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.47521, size = 1054, normalized size = 4.95 \begin{align*} \frac{{\left (3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{{\left (A^{2} - 22 i \, A B - 121 \, B^{2}\right )} c^{5}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left ({\left (i \, A + 11 \, B\right )} c^{3} + \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 (A^{2} - 22 i \, A B - 121 \, B^{2}\right )} c^{5}}{a^{6} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a^{3} f}\right ) - 3 \, \sqrt{\frac{1}{2}} a^{3} f \sqrt{-\frac{{\left (A^{2} - 22 i \, A B - 121 \, B^{2}\right )} c^{5}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac{{\left ({\left (i \, A + 11 \, B\right )} c^{3} - \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 (A^{2} - 22 i \, A B - 121 \, B^{2}\right )} c^{5}}{a^{6} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a^{3} f}\right ) + \sqrt{2}{\left ({\left (-3 i \, A - 33 \, B\right )} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-i \, A - 11 \, B\right )} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (10 i \, A + 14 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (8 i \, A - 8 \, B\right )} c^{2}\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 )}}{96 \, 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 )}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{{\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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