Optimal. Leaf size=89 \[ \frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac {2 a \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3567,
3853, 3856, 2719} \begin {gather*} \frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac {2 a \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x) \cos (c+d x)}{d (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3567
Rule 3596
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {\int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}+\frac {a \int (e \sec (c+d x))^{3/2} \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{d (e \cos (c+d x))^{3/2}}-\frac {\left (a e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{d (e \cos (c+d x))^{3/2}}-\frac {\left (a \cos ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \, dx}{(e \cos (c+d x))^{3/2}}\\ &=\frac {2 i a}{3 d (e \cos (c+d x))^{3/2}}-\frac {2 a \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d (e \cos (c+d x))^{3/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{d (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 4.07, size = 369, normalized size = 4.15 \begin {gather*} \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-\frac {2 \sqrt {2} e^{-i d x} (-i+\cot (c)) \left (3+3 e^{2 i (c+d x)}+3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (-i+e^{i (c+d x)}\right )} E\left (\left .\text {ArcSin}\left (\sqrt {-i \cos (c+d x)+\sin (c+d x)}\right )\right |-1\right )-3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (-i+e^{i (c+d x)}\right )} F\left (\left .\text {ArcSin}\left (\sqrt {-i \cos (c+d x)+\sin (c+d x)}\right )\right |-1\right )+e^{2 i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right )}{3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}+\frac {2 (-i+\cot (c)) (2 i+3 \cot (c)+3 \cos (c+2 d x) \csc (c)) \sin (c)}{3 \cos ^{\frac {3}{2}}(c+d x)}\right ) (\cos (d x)-i \sin (d x)) (a+i a \tan (c+d x))}{2 d (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 213 vs. \(2 (104 ) = 208\).
time = 1.66, size = 214, normalized size = 2.40
method | result | size |
default | \(\frac {2 \left (12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-6 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e d}\) | \(214\) |
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 = 141, normalized size = 1.58 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (3 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \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 )} + 3 \, {\left (i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{3 \, {\left (d e^{\frac {3}{2}} + d e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\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}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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