Optimal. Leaf size=157 \[ \frac{(3 B+2 i A) \sqrt{c-i c \tan (e+f x)}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(3 B+2 i A) \sqrt{c-i c \tan (e+f x)}}{15 a f (a+i a \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.244008, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ \frac{(3 B+2 i A) \sqrt{c-i c \tan (e+f x)}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}+\frac{(-B+i A) \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(3 B+2 i A) \sqrt{c-i c \tan (e+f x)}}{15 a f (a+i a \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{7/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{((2 A-3 i B) 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 )}{5 f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(2 i A+3 B) \sqrt{c-i c \tan (e+f x)}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{((2 A-3 i B) 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 )}{15 a f}\\ &=\frac{(i A-B) \sqrt{c-i c \tan (e+f x)}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{(2 i A+3 B) \sqrt{c-i c \tan (e+f x)}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 i A+3 B) \sqrt{c-i c \tan (e+f x)}}{15 a^2 f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.66213, size = 106, normalized size = 0.68 \[ \frac{\sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} ((6 A-9 i B) \sin (2 (e+f x))+(-6 B-9 i A) \cos (2 (e+f x))-5 i A)}{30 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 127, normalized size = 0.8 \begin{align*}{\frac{-{\frac{i}{15}} \left ( 2\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{3}-12\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{2}+3\,B \left ( \tan \left ( fx+e \right ) \right ) ^{3}-13\,iA\tan \left ( fx+e \right ) +8\,A \left ( \tan \left ( fx+e \right ) \right ) ^{2}+3\,iB-12\,B\tan \left ( fx+e \right ) -7\,A \right ) }{f{a}^{3} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{4}}\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 = 1.4206, size = 416, normalized size = 2.65 \begin{align*} \frac{{\left ({\left (-28 i \, A - 12 \, B\right )} e^{\left (7 i \, f x + 7 i \, e\right )} +{\left (15 i \, A + 15 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-28 i \, A - 12 \, B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} +{\left (25 i \, A + 15 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (13 i \, A - 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, A - 3 \, B\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 (-5 i \, f x - 5 i \, e\right )}}{60 \, a^{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{{\left (B \tan \left (f x + e\right ) + A\right )} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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