Optimal. Leaf size=193 \[ \frac{8 \tan (e+f x)}{21 a^3 c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{21 a^2 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.169957, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {3523, 45, 40, 39} \[ \frac{8 \tan (e+f x)}{21 a^3 c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{21 a^2 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 45
Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{9/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}+\frac{(5 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{7/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{5/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{7 a f}\\ &=\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{4 \tan (e+f x)}{21 a^2 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{21 a^2 f}\\ &=\frac{i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}+\frac{i}{7 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}+\frac{4 \tan (e+f x)}{21 a^2 f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{8 \tan (e+f x)}{21 a^3 c f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.4826, size = 151, normalized size = 0.78 \[ \frac{\sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-70 i \sin (e+f x)+63 i \sin (3 (e+f x))+5 i \sin (5 (e+f x))-140 \cos (e+f x)+42 \cos (3 (e+f x))+2 \cos (5 (e+f x)))}{336 a^3 c^2 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 141, normalized size = 0.7 \begin{align*}{\frac{16\,i \left ( \tan \left ( fx+e \right ) \right ) ^{6}-8\, \left ( \tan \left ( fx+e \right ) \right ) ^{7}+40\,i \left ( \tan \left ( fx+e \right ) \right ) ^{4}-12\, \left ( \tan \left ( fx+e \right ) \right ) ^{5}+30\,i \left ( \tan \left ( fx+e \right ) \right ) ^{2}+5\, \left ( \tan \left ( fx+e \right ) \right ) ^{3}+6\,i+9\,\tan \left ( fx+e \right ) }{21\,f{a}^{4}{c}^{2} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47561, size = 458, normalized size = 2.37 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-7 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 112 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 192 i \, e^{\left (9 i \, f x + 9 i \, e\right )} + 105 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 192 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 280 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 91 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 24 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i\right )} e^{\left (-7 i \, f x - 7 i \, e\right )}}{672 \, a^{4} c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{7}{2}}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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