Optimal. Leaf size=90 \[ \frac{i (c-i c \tan (e+f x))^{5/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{7 f (a+i a \tan (e+f x))^{7/2}} \]
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Rubi [A] time = 0.125901, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 45, 37} \[ \frac{i (c-i c \tan (e+f x))^{5/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{7 f (a+i a \tan (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{7/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{(a+i a x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i (c-i c \tan (e+f x))^{5/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{c \operatorname{Subst}\left (\int \frac{(c-i c x)^{3/2}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=\frac{i (c-i c \tan (e+f x))^{5/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{i (c-i c \tan (e+f x))^{5/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 5.82262, size = 100, normalized size = 1.11 \[ -\frac{i c^2 (\tan (e+f x)-6 i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))-i \sin (2 (e+f x)))}{35 a^3 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 87, normalized size = 1. \begin{align*}{\frac{{c}^{2} \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 5\,i\tan \left ( fx+e \right ) - \left ( \tan \left ( fx+e \right ) \right ) ^{2}-6 \right ) }{35\,f{a}^{4} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{5}}\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }\sqrt{a \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 [A] time = 2.16719, size = 127, normalized size = 1.41 \begin{align*} \frac{{\left (5 i \, c^{2} \cos \left (7 \, f x + 7 \, e\right ) + 7 i \, c^{2} \cos \left (\frac{5}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 5 \, c^{2} \sin \left (7 \, f x + 7 \, e\right ) + 7 \, c^{2} \sin \left (\frac{5}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right )\right )} \sqrt{c}}{70 \, a^{\frac{7}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59916, size = 320, normalized size = 3.56 \begin{align*} \frac{{\left (-12 i \, c^{2} e^{\left (9 i \, f x + 9 i \, e\right )} - 12 i \, c^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 7 i \, c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 5 i \, c^{2}\right )} \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}} e^{\left (-7 i \, f x - 7 i \, e\right )}}{70 \, a^{4} 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{{\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 )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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