Optimal. Leaf size=213 \[ -\frac{2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{15 a c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{(-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.275042, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ -\frac{2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{15 a c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 (-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{(-2 B+3 i A) \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac{-B+i A}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \]
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{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{3/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}+\frac{((3 A+2 i B) c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac{(3 i A-2 B) \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac{(2 (3 A+2 i B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac{(3 i A-2 B) \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac{2 (3 i A-2 B) \sqrt{a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}+\frac{(2 (3 A+2 i B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 c f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac{(3 i A-2 B) \sqrt{a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac{2 (3 i A-2 B) \sqrt{a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 (3 i A-2 B) \sqrt{a+i a \tan (e+f x)}}{15 a c^2 f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 10.8351, size = 128, normalized size = 0.6 \[ \frac{\sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x))) ((3 A+2 i B) (3 \sin (3 (e+f x))-5 \sin (e+f x))+5 (B-6 i A) \cos (e+f x)+(-9 B+6 i A) \cos (3 (e+f x)))}{60 c^3 f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.188, size = 184, normalized size = 0.9 \begin{align*}{\frac{4\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{5}+12\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{4}+6\,A \left ( \tan \left ( fx+e \right ) \right ) ^{5}+2\,iB \left ( \tan \left ( fx+e \right ) \right ) ^{3}-8\,B \left ( \tan \left ( fx+e \right ) \right ) ^{4}+18\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{2}+3\,A \left ( \tan \left ( fx+e \right ) \right ) ^{3}-2\,iB\tan \left ( fx+e \right ) -7\,B \left ( \tan \left ( fx+e \right ) \right ) ^{2}+6\,iA-3\,A\tan \left ( fx+e \right ) +B}{15\,af{c}^{3} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}\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 = 1.41208, size = 447, normalized size = 2.1 \begin{align*} \frac{{\left ({\left (-3 i \, A - 3 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-18 i \, A - 8 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-60 i \, A + 10 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (48 i \, A + 8 \, B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} - 30 i \, A e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (48 i \, A + 8 \, B\right )} e^{\left (i \, f x + i \, e\right )} + 15 i \, A - 15 \, 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 (-i \, f x - i \, e\right )}}{120 \, a c^{3} 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{B \tan \left (f x + e\right ) + A}{\sqrt{i \, a \tan \left (f x + e\right ) + a}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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