Optimal. Leaf size=275 \[ -\frac{7 c^4 (-13 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{7 c^3 (-13 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 c^2 (-13 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}+\frac{7 c^{9/2} (-13 B+5 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a^2 f}-\frac{c (-13 B+5 i A) (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
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Rubi [A] time = 0.301172, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3588, 78, 47, 50, 63, 208} \[ -\frac{7 c^4 (-13 B+5 i A) \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{7 c^3 (-13 B+5 i A) (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 c^2 (-13 B+5 i A) (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}+\frac{7 c^{9/2} (-13 B+5 i A) \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a^2 f}-\frac{c (-13 B+5 i A) (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(-B+i A) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 47
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))^{9/2}}{(a+i a \tan (e+f x))^2} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(A+B x) (c-i c x)^{7/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}-\frac{((5 A+13 i B) c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{7/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac{(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{\left (7 (5 A+13 i B) c^2\right ) \operatorname{Subst}\left (\int \frac{(c-i c x)^{5/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{16 a f}\\ &=-\frac{7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac{(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{\left (7 (5 A+13 i B) c^3\right ) \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{8 a f}\\ &=-\frac{7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac{(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{\left (7 (5 A+13 i B) c^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{4 a f}\\ &=-\frac{7 (5 i A-13 B) c^4 \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac{(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{\left (7 (5 A+13 i B) c^5\right ) \operatorname{Subst}\left (\int \frac{1}{(a+i a x) \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 a f}\\ &=-\frac{7 (5 i A-13 B) c^4 \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac{(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac{\left (7 (5 i A-13 B) c^4\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 )}{a f}\\ &=\frac{7 (5 i A-13 B) c^{9/2} \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a^2 f}-\frac{7 (5 i A-13 B) c^4 \sqrt{c-i c \tan (e+f x)}}{2 a^2 f}-\frac{7 (5 i A-13 B) c^3 (c-i c \tan (e+f x))^{3/2}}{12 a^2 f}-\frac{7 (5 i A-13 B) c^2 (c-i c \tan (e+f x))^{5/2}}{40 a^2 f}-\frac{(5 i A-13 B) c (c-i c \tan (e+f x))^{7/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac{(i A-B) (c-i c \tan (e+f x))^{9/2}}{4 a^2 f (1+i \tan (e+f x))^2}\\ \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 = 0.104, size = 221, normalized size = 0.8 \begin{align*}{\frac{-2\,i{c}^{2}}{f{a}^{2}} \left ({\frac{i}{5}}B \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}+{\frac{5\,i}{3}}B \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}}}}+18\,iB{c}^{2}\sqrt{c-ic\tan \left ( fx+e \right ) }+6\,A{c}^{2}\sqrt{c-ic\tan \left ( fx+e \right ) }+8\,{c}^{3} \left ({\frac{1}{ \left ( -c-ic\tan \left ( fx+e \right ) \right ) ^{2}} \left ( \left ( -{\frac{21\,i}{16}}B-{\frac{13\,A}{16}} \right ) \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{3/2}+ \left ({\frac{19\,i}{8}}Bc+{\frac{11\,Ac}{8}} \right ) \sqrt{c-ic\tan \left ( fx+e \right ) } \right ) }-{\frac{ \left ( 91\,iB+35\,A \right ) \sqrt{2}}{32\,\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 = 1.68899, size = 1382, normalized size = 5.03 \begin{align*} \frac{15 \, \sqrt{-\frac{{\left (2450 \, A^{2} + 12740 i \, A B - 16562 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}}{\left (a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (\frac{{\left ({\left (70 i \, A - 182 \, B\right )} c^{5} + \sqrt{2} \sqrt{-\frac{{\left (2450 \, A^{2} + 12740 i \, A B - 16562 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}}{\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} 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^{2} f}\right ) - 15 \, \sqrt{-\frac{{\left (2450 \, A^{2} + 12740 i \, A B - 16562 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}}{\left (a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \log \left (\frac{{\left ({\left (70 i \, A - 182 \, B\right )} c^{5} - \sqrt{2} \sqrt{-\frac{{\left (2450 \, A^{2} + 12740 i \, A B - 16562 \, B^{2}\right )} c^{9}}{a^{4} f^{2}}}{\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} 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^{2} f}\right ) + \sqrt{2}{\left ({\left (-1050 i \, A + 2730 \, B\right )} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-2450 i \, A + 6370 \, B\right )} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-1610 i \, A + 4186 \, B\right )} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-150 i \, A + 390 \, B\right )} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (60 i \, A - 60 \, B\right )} c^{4}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{60 \,{\left (a^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, a^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} + a^{2} f e^{\left (4 i \, f x + 4 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{9}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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