Optimal. Leaf size=223 \[ -\frac{-3 B+7 i A}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{(-3 B+7 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a c^{5/2} f}-\frac{-3 B+7 i A}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{-3 B+7 i A}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.288617, antiderivative size = 223, 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, 51, 63, 208} \[ -\frac{-3 B+7 i A}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{(-3 B+7 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a c^{5/2} f}-\frac{-3 B+7 i A}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{-3 B+7 i A}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \]
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)}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^2 (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac{((7 A+3 i B) c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac{7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}+\frac{(7 A+3 i B) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac{7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac{7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}+\frac{(7 A+3 i B) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{16 c f}\\ &=-\frac{7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac{7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{7 i A-3 B}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{(7 A+3 i B) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{32 c^2 f}\\ &=-\frac{7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac{7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{7 i A-3 B}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{(7 i A-3 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 )}{16 c^3 f}\\ &=\frac{(7 i A-3 B) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{16 \sqrt{2} a c^{5/2} f}-\frac{7 i A-3 B}{20 a f (c-i c \tan (e+f x))^{5/2}}+\frac{i A-B}{2 a f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}-\frac{7 i A-3 B}{24 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{7 i A-3 B}{16 a c^2 f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 7.55901, size = 213, normalized size = 0.96 \[ -\frac{e^{-2 i (e+f x)} \sqrt{\frac{c}{1+e^{2 i (e+f x)}}} \left (\left (1+e^{2 i (e+f x)}\right ) \left (i A \left (116 e^{2 i (e+f x)}+32 e^{4 i (e+f x)}+6 e^{6 i (e+f x)}-15\right )+3 B \left (-8 e^{2 i (e+f x)}+4 e^{4 i (e+f x)}+2 e^{6 i (e+f x)}+5\right )\right )+15 (3 B-7 i A) e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (e+f x)}}\right )\right )}{240 \sqrt{2} a c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 168, normalized size = 0.8 \begin{align*}{\frac{2\,ic}{af} \left ( -{\frac{1}{16\,{c}^{3}} \left ({\frac{1}{-c-ic\tan \left ( fx+e \right ) } \left ({\frac{i}{2}}B+{\frac{A}{2}} \right ) \sqrt{c-ic\tan \left ( fx+e \right ) }}-{\frac{ \left ( 7\,A+3\,iB \right ) \sqrt{2}}{4}{\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 ) }-{\frac{A}{12\,{c}^{2}} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{3\,A+iB}{16\,{c}^{3}}{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}}-{\frac{A-iB}{20\,c} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\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 = 1.36938, size = 1112, normalized size = 4.99 \begin{align*} \frac{{\left (15 \, \sqrt{\frac{1}{2}} a c^{3} f \sqrt{-\frac{49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a c^{2} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} + 7 i \, A - 3 \, B\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a c^{2} f}\right ) - 15 \, \sqrt{\frac{1}{2}} a c^{3} f \sqrt{-\frac{49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (a c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a c^{2} f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{-\frac{49 \, A^{2} + 42 i \, A B - 9 \, B^{2}}{a^{2} c^{5} f^{2}}} - 7 i \, A + 3 \, B\right )} e^{\left (-i \, f x - i \, e\right )}}{8 \, a c^{2} f}\right ) + \sqrt{2}{\left ({\left (-6 i \, A - 6 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-38 i \, A - 18 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-148 i \, A + 12 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-101 i \, A + 9 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 15 i \, A - 15 \, B\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{480 \, a c^{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{B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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