Optimal. Leaf size=60 \[ -\frac {4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 i a^2}{c f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3603, 3568, 45}
\begin {gather*} \frac {2 i a^2}{c f \sqrt {c-i c \tan (e+f x)}}-\frac {4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^2}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx\\ &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {c-x}{(c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \left (\frac {2 c}{(c+x)^{5/2}}-\frac {1}{(c+x)^{3/2}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=-\frac {4 i a^2}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 i a^2}{c f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.40, size = 93, normalized size = 1.55 \begin {gather*} \frac {2 a^2 \cos (e+f x) (i \cos (e+f x)+3 \sin (e+f x)) (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x))) \sqrt {c-i c \tan (e+f x)}}{3 c^2 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 47, normalized size = 0.78
method | result | size |
risch | \(-\frac {i a^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-2\right ) \sqrt {2}}{3 c \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(44\) |
derivativedivides | \(-\frac {2 i a^{2} \left (\frac {2 c}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f c}\) | \(47\) |
default | \(-\frac {2 i a^{2} \left (\frac {2 c}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f c}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 46, normalized size = 0.77 \begin {gather*} \frac {2 i \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} - 2 \, a^{2} c\right )}}{3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 65, normalized size = 1.08 \begin {gather*} \frac {\sqrt {2} {\left (-i \, a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \, c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - a^{2} \left (\int \frac {\tan ^{2}{\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {2 i \tan {\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \left (- \frac {1}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.03, size = 98, normalized size = 1.63 \begin {gather*} \frac {a^2\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )+2{}\mathrm {i}\right )}{3\,c^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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