Integrand size = 25, antiderivative size = 105 \[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {10 a^2 \sqrt {e \cos (c+d x)}}{3 d e}+\frac {10 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}{3 d e} \]
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Time = 0.07 (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, 2720} \[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {10 a^2 \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) \sqrt {e \cos (c+d x)}}{3 d e}+\frac {10 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}} \]
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Rule 2720
Rule 2721
Rule 2748
Rule 2757
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}{3 d e}+\frac {1}{3} (5 a) \int \frac {a+a \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {10 a^2 \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}{3 d e}+\frac {1}{3} \left (5 a^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {10 a^2 \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}{3 d e}+\frac {\left (5 a^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 \sqrt {e \cos (c+d x)}} \\ & = -\frac {10 a^2 \sqrt {e \cos (c+d x)}}{3 d e}+\frac {10 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )}{3 d e} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.61 \[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {8 \sqrt [4]{2} a^2 \sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt [4]{1+\sin (c+d x)}} \]
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Time = 3.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {2 a^{2} \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+12 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(152\) |
parts | \(\frac {2 a^{2} \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| \sqrt {2}\right )}{d \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}}+\frac {4 a^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {4 a^{2} \sqrt {e \cos \left (d x +c \right )}}{d e}\) | \(269\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=\frac {-5 i \, \sqrt {2} a^{2} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} a^{2} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (a^{2} \sin \left (d x + c\right ) + 6 \, a^{2}\right )} \sqrt {e \cos \left (d x + c\right )}}{3 \, d e} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \cos \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]
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