Optimal. Leaf size=136 \[ -\frac {6 a^3 \sqrt {e \cos (c+d x)}}{d e}+\frac {6 a^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}{5 d e}-\frac {6 \sqrt {e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}{5 d e} \]
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Rubi [A]
time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2757, 2748,
2721, 2720} \begin {gather*} -\frac {6 a^3 \sqrt {e \cos (c+d x)}}{d e}-\frac {6 \left (a^3 \sin (c+d x)+a^3\right ) \sqrt {e \cos (c+d x)}}{5 d e}+\frac {6 a^3 \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 a (a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}{5 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 2721
Rule 2748
Rule 2757
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^3}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}{5 d e}+\frac {1}{5} (9 a) \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}{5 d e}-\frac {6 \sqrt {e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}{5 d e}+\left (3 a^2\right ) \int \frac {a+a \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {6 a^3 \sqrt {e \cos (c+d x)}}{d e}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}{5 d e}-\frac {6 \sqrt {e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}{5 d e}+\left (3 a^3\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {6 a^3 \sqrt {e \cos (c+d x)}}{d e}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}{5 d e}-\frac {6 \sqrt {e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}{5 d e}+\frac {\left (3 a^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{\sqrt {e \cos (c+d x)}}\\ &=-\frac {6 a^3 \sqrt {e \cos (c+d x)}}{d e}+\frac {6 a^3 \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 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2}{5 d e}-\frac {6 \sqrt {e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}{5 d e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.02, size = 64, normalized size = 0.47 \begin {gather*} -\frac {16 \sqrt [4]{2} a^3 \sqrt {e \cos (c+d x)} \, _2F_1\left (-\frac {9}{4},\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt [4]{1+\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.02, size = 178, normalized size = 1.31
method | result | size |
default | \(-\frac {2 a^{3} \left (8 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-34 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(178\) |
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 = 99, normalized size = 0.73 \begin {gather*} \frac {{\left (-15 i \, \sqrt {2} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \sin \left (d x + c\right ) - 20 \, a^{3}\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {1}{2}\right )}}{5 \, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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