Integrand size = 25, antiderivative size = 119 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 119, 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 = {2770, 2748, 2721, 2720} \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rule 2720
Rule 2721
Rule 2748
Rule 2770
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}-\frac {2 \int \frac {-\frac {a^2}{2}+b^2+\frac {1}{2} a b \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2} \\ & = \frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a^2-2 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2} \\ & = \frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (\left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2 \sqrt {e \cos (c+d x)}} \\ & = \frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (2 a b+\left (a^2-2 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (a^2+b^2\right ) \sin (c+d x)\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(131)=262\).
Time = 4.52 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.80
method | result | size |
default | \(-\frac {2 \left (2 F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-4 F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a b \right )}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d}\) | \(333\) |
parts | \(-\frac {2 a^{2} \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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 )-2 \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 ) \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 )}}{3 e^{2} \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 )}\, \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\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 b^{2} \left (2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \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 )-\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 ) \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 )}}{3 e^{2} \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\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 b}{3 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} e d}\) | \(509\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {e} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (2 \, a b + {\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{3 \, d e^{3} \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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