Optimal. Leaf size=66 \[ \frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2761, 2721,
2720} \begin {gather*} \frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d \sqrt {e \cos (c+d x)}}+\frac {2 e \sqrt {e \cos (c+d x)}}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 2721
Rule 2761
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx &=\frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{a}\\ &=\frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {\left (e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{a \sqrt {e \cos (c+d x)}}\\ &=\frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a 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 = 0.07, size = 66, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{2} (e \cos (c+d x))^{5/2} \, _2F_1\left (\frac {3}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{5 a d e (1+\sin (c+d x))^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.53, size = 110, normalized size = 1.67
method | result | size |
default | \(-\frac {2 e^{2} \left (\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(110\) |
risch | \(\frac {\sqrt {2}\, e \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{d a}+\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) e \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{i \left (d x +c \right )}}}{d \sqrt {{\mathrm e}^{3 i \left (d x +c \right )} e +{\mathrm e}^{i \left (d x +c \right )} e}\, a \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}\) | \(217\) |
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 = 0.11, size = 70, normalized size = 1.06 \begin {gather*} \frac {-i \, \sqrt {2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {\cos \left (d x + c\right )} e^{\frac {3}{2}}}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.02 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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