Integrand size = 25, antiderivative size = 105 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=-\frac {14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {14 a^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}{5 d e} \]
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Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2757, 2748, 2721, 2719} \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=-\frac {14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}+\frac {14 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}} \]
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Rule 2719
Rule 2721
Rule 2748
Rule 2757
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}{5 d e}+\frac {1}{5} (7 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x)) \, dx \\ & = -\frac {14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}{5 d e}+\frac {1}{5} \left (7 a^2\right ) \int \sqrt {e \cos (c+d x)} \, dx \\ & = -\frac {14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}{5 d e}+\frac {\left (7 a^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}} \\ & = -\frac {14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {14 a^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )}{5 d e} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=-\frac {8\ 2^{3/4} a^2 (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (1+\sin (c+d x))^{3/4}} \]
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Time = 4.64 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.79
method | result | size |
default | \(-\frac {2 a^{2} e \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+40 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(188\) |
parts | \(\frac {2 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {4 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \left (4 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {4 a^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e}\) | \(363\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=\frac {21 i \, \sqrt {2} a^{2} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} a^{2} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 10 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{15 \, d} \]
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\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \sqrt {e \cos {\left (c + d x \right )}}\, dx + \int 2 \sqrt {e \cos {\left (c + d x \right )}} \sin {\left (c + d x \right )}\, dx + \int \sqrt {e \cos {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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