Optimal. Leaf size=60 \[ \frac {2 i a}{d \sqrt {e \cos (c+d x)}}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3596, 3567,
3856, 2720} \begin {gather*} \frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}+\frac {2 i a}{d \sqrt {e \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3596
Rule 3856
Rubi steps
\begin {align*} \int \frac {a+i a \tan (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx &=\frac {\int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx}{\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 i a}{d \sqrt {e \cos (c+d x)}}+\frac {a \int \sqrt {e \sec (c+d x)} \, dx}{\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 i a}{d \sqrt {e \cos (c+d x)}}+\frac {\left (a \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{\sqrt {e \cos (c+d x)}}\\ &=\frac {2 i a}{d \sqrt {e \cos (c+d x)}}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 1.04, size = 143, normalized size = 2.38 \begin {gather*} -\frac {\sqrt {2} a \sqrt {e \cos (c+d x)} (-i+\cot (c)) \left (\sqrt {2} \sqrt {\csc ^2(c)}+i \cos (c+d x) \sqrt {1+\cos (2 d x-2 \text {ArcTan}(\cot (c)))} \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\text {ArcTan}(\cot (c)))\right ) \sec (d x-\text {ArcTan}(\cot (c)))\right ) \sin (c) (\cos (d x)-i \sin (d x)) (-i+\tan (c+d x))}{d e \sqrt {\csc ^2(c)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 94, normalized size = 1.57
method | result | size |
default | \(\frac {2 \left (-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{\sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.08, size = 85, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (-2 i \, \sqrt {\frac {1}{2}} a \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\right )} + {\left (i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{d e^{\frac {1}{2}} + d e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )}} \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*} i a \left (\int \left (- \frac {i}{\sqrt {e \cos {\left (c + d x \right )}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\sqrt {e \cos {\left (c + d x \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 = 0.56, size = 74, normalized size = 1.23 \begin {gather*} \frac {2\,a\,\sqrt {\cos \left (c+d\,x\right )}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d\,\sqrt {e\,\cos \left (c+d\,x\right )}}+\frac {a\,\cos \left (c+d\,x\right )\,\sqrt {e\,\cos \left (c+d\,x\right )}\,4{}\mathrm {i}}{d\,e\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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