Optimal. Leaf size=212 \[ \frac{2 (2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{15 a^2 c f \sqrt{a+i a \tan (e+f x)}}-\frac{B+i A}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{2 (2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.278414, antiderivative size = 212, 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} \[ \frac{2 (2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{15 a^2 c f \sqrt{a+i a \tan (e+f x)}}-\frac{B+i A}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{2 (2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 B+3 i A) \sqrt{c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}} \]
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)}{(a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{7/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A+B}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(a (3 A-2 i B)) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{7/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A+B}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i A+2 B) \sqrt{c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}+\frac{(2 (3 A-2 i B)) \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}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i A+2 B) \sqrt{c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}+\frac{2 (3 i A+2 B) \sqrt{c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 (3 A-2 i B)) \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}{f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i A+2 B) \sqrt{c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}+\frac{2 (3 i A+2 B) \sqrt{c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac{2 (3 i A+2 B) \sqrt{c-i c \tan (e+f x)}}{15 a^2 c f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.33682, size = 132, normalized size = 0.62 \[ -\frac{\sec (e+f x) \sqrt{c-i c \tan (e+f x)} (-i (3 A-2 i B) (5 \sin (e+f x)-3 \sin (3 (e+f x)))+(-30 A+5 i B) \cos (e+f x)+(6 A-9 i B) \cos (3 (e+f x)))}{60 a^2 c f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.183, size = 186, 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\,f{a}^{3}c \left ( -\tan \left ( fx+e \right ) +i \right ) ^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}\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 = 1.35331, size = 458, normalized size = 2.16 \begin{align*} \frac{{\left ({\left (-15 i \, A - 15 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-48 i \, A + 8 \, B\right )} e^{\left (7 i \, f x + 7 i \, e\right )} + 30 i \, A e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-48 i \, A + 8 \, B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} +{\left (60 i \, A + 10 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (18 i \, A - 8 \, 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 )}}{120 \, a^{3} c 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}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{5}{2}} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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