Optimal. Leaf size=25 \[ \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \]
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Rubi [A]
time = 0.07, antiderivative size = 25, 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 \sqrt {c-i c \tan (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{\sqrt {c-i c \tan (e+f x)}} \, dx\\ &=\frac {(i a) \text {Subst}\left (\int \frac {1}{\sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 i a \sqrt {c-i c \tan (e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 22, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {2 i a \sqrt {c -i c \tan \left (f x +e \right )}}{f}\) | \(22\) |
default | \(\frac {2 i a \sqrt {c -i c \tan \left (f x +e \right )}}{f}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 0.80 \begin {gather*} \frac {2 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.92, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 i \, \sqrt {2} a \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 42 vs. \(2 (20) = 40\).
time = 1.69, size = 42, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {2 i a \sqrt {- i c \tan {\left (e + f x \right )} + c}}{f} & \text {for}\: f \neq 0 \\x \left (i a \tan {\left (e \right )} + a\right ) \sqrt {- i c \tan {\left (e \right )} + c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 20, normalized size = 0.80 \begin {gather*} \frac {2 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.46, size = 29, normalized size = 1.16 \begin {gather*} \frac {\sqrt {2}\,a\,\sqrt {\frac {c}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,2{}\mathrm {i}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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