Integrand size = 25, antiderivative size = 118 \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}-\frac {2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2}}{d e^3} \]
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Time = 0.15 (sec) , antiderivative size = 118, 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, 2719} \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}} \]
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Rule 2719
Rule 2721
Rule 2748
Rule 2770
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}-\frac {2 \int \left (\frac {a^2}{2}+b^2+\frac {3}{2} a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)} \, dx}{e^2} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac {\left (a^2+2 b^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{e^2} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac {\left (\left (a^2+2 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{d e \sqrt {e \sin (c+d x)}}-\frac {2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a b (e \sin (c+d x))^{3/2}}{d e^3} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\frac {-4 a b-2 \left (a^2+b^2\right ) \cos (c+d x)+2 \left (a^2+2 b^2\right ) E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sqrt {\sin (c+d x)}}{d e \sqrt {e \sin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(140)=280\).
Time = 3.48 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.40
method | result | size |
default | \(\frac {2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+4 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 a^{2} \left (\cos ^{2}\left (d x +c \right )\right )-2 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )-4 \cos \left (d x +c \right ) a b}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(283\) |
parts | \(\frac {a^{2} \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (d x +c \right )\right )\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 b^{2} \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\left (\cos ^{2}\left (d x +c \right )\right )\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {4 a b}{\sqrt {e \sin \left (d x +c \right )}\, e d}\) | \(309\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (2 \, a b + {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {e \sin \left (d x + c\right )}}{d e^{2} \sin \left (d x + c\right )} \]
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\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{2}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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