Optimal. Leaf size=36 \[ -\frac {2 i \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d} \]
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Rubi [A]
time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3596, 3569}
\begin {gather*} -\frac {2 i \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3596
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx &=\left (\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx\\ &=-\frac {2 i \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 36, normalized size = 1.00 \begin {gather*} -\frac {2 i \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 45, normalized size = 1.25
method | result | size |
default | \(-\frac {2 i \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{d}\) | \(45\) |
risch | \(-\frac {2 i \sqrt {2}\, \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}{d}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 75 vs. \(2 (28) = 56\).
time = 0.56, size = 75, normalized size = 2.08 \begin {gather*} -\frac {2 i \, \sqrt {a} \sqrt {-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1} e^{\frac {1}{2}}}{d \sqrt {-\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 51, normalized size = 1.42 \begin {gather*} -\frac {2 i \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c + \frac {1}{2}\right )}}{d} \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*} \int \sqrt {e \cos {\left (c + d x \right )}} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 18, normalized size = 0.50 \begin {gather*} -\frac {2 i \, \sqrt {a} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c + \frac {1}{2}\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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