Optimal. Leaf size=27 \[ -\frac {2 i a}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 32}
\begin {gather*} -\frac {2 i a}{3 f (c-i c \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{3/2}} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx\\ &=\frac {(i a) \text {Subst}\left (\int \frac {1}{(c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {2 i a}{3 f (c-i c \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(72\) vs. \(2(27)=54\).
time = 0.80, size = 72, normalized size = 2.67 \begin {gather*} \frac {2 a \cos ^2(e+f x) (\cos (f x)-i \sin (f x)) (-i \cos (2 e+3 f x)+\sin (2 e+3 f x)) \sqrt {c-i c \tan (e+f x)}}{3 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 22, normalized size = 0.81
method | result | size |
derivativedivides | \(-\frac {2 i a}{3 f \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\) | \(22\) |
default | \(-\frac {2 i a}{3 f \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\) | \(22\) |
risch | \(-\frac {i a \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {2}}{6 c \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 20, normalized size = 0.74 \begin {gather*} -\frac {2 i \, a}{3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 59 vs. \(2 (20) = 40\).
time = 1.55, size = 59, normalized size = 2.19 \begin {gather*} \frac {\sqrt {2} {\left (-i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 2 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.32, size = 46, normalized size = 1.70 \begin {gather*} \begin {cases} - \frac {2 i a}{3 f \left (- i c \tan {\left (e + f x \right )} + c\right )^{\frac {3}{2}}} & \text {for}\: f \neq 0 \\\frac {x \left (i a \tan {\left (e \right )} + a\right )}{\left (- i c \tan {\left (e \right )} + c\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \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 = 5.09, size = 93, normalized size = 3.44 \begin {gather*} \frac {a\,\sqrt {-\frac {c\,\left (-2\,{\cos \left (e+f\,x\right )}^2+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (e+f\,x\right )}^2}}\,\left (-{\cos \left (e+f\,x\right )}^2\,4{}\mathrm {i}-{\cos \left (2\,e+2\,f\,x\right )}^2\,2{}\mathrm {i}+2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )+2{}\mathrm {i}\right )}{6\,c^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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