Optimal. Leaf size=80 \[ \frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {4 i \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{3 a d} \]
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Rubi [A]
time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3596, 3583,
3569} \begin {gather*} \frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {4 i \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3583
Rule 3596
Rubi steps
\begin {align*} \int \frac {\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 {1}{\sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}+\frac {\left (2 \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}{3 a}\\ &=\frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {4 i \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{3 a d}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 48, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {e \cos (c+d x)} (-i+2 \tan (c+d x))}{3 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 74, normalized size = 0.92
method | result | size |
risch | \(-\frac {i \sqrt {2}\, \sqrt {e \cos \left (d x +c \right )}\, \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(72\) |
default | \(\frac {2 \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (i \left (\cos ^{2}\left (d x +c \right )\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right )-2 i\right )}{3 d a}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.62, size = 79, normalized size = 0.99 \begin {gather*} \frac {{\left (i \, \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - 3 i \, \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right ) + \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right )\right )} e^{\frac {1}{2}}}{3 \, \sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 70, normalized size = 0.88 \begin {gather*} \frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\frac {1}{2}} - 3 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {3}{2} i \, d x - \frac {3}{2} i \, c\right )}}{3 \, a 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 \frac {\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 [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 = 0.73, size = 82, normalized size = 1.02 \begin {gather*} \frac {\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )-3{}\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}}{3\,a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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