Optimal. Leaf size=182 \[ \frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^3 f \sqrt{a+i a \tan (e+f x)}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}} \]
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Rubi [A] time = 0.155375, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 45, 37} \[ \frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^3 f \sqrt{a+i a \tan (e+f x)}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{7/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{9/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{7/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{(6 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{5/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{35 a f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{35 a^2 f}\\ &=\frac{i \sqrt{c-i c \tan (e+f x)}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{3 i \sqrt{c-i c \tan (e+f x)}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i \sqrt{c-i c \tan (e+f x)}}{35 a^3 f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.75959, size = 105, normalized size = 0.58 \[ -\frac{\sec ^3(e+f x) \sqrt{c-i c \tan (e+f x)} (7 i \sin (e+f x)+15 i \sin (3 (e+f x))+28 \cos (e+f x)+20 \cos (3 (e+f x)))}{140 a^3 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 95, normalized size = 0.5 \begin{align*}{\frac{10\,i \left ( \tan \left ( fx+e \right ) \right ) ^{3}-2\, \left ( \tan \left ( fx+e \right ) \right ) ^{4}-25\,i\tan \left ( fx+e \right ) +21\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}-12}{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 [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 = 1.39001, size = 369, normalized size = 2.03 \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 (-96 i \, e^{\left (9 i \, f x + 9 i \, e\right )} + 35 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 96 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 70 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 56 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 26 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 5 i\right )} e^{\left (-7 i \, f x - 7 i \, e\right )}}{280 \, 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{\sqrt{-i \, c \tan \left (f x + e\right ) + c}}{{\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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