Integrand size = 23, antiderivative size = 96 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 b}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}} \]
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Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2716, 2721, 2719} \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 b}{d e \sqrt {e \sin (c+d x)}} \]
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Rule 2716
Rule 2719
Rule 2721
Rule 2748
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b}{d e \sqrt {e \sin (c+d x)}}+a \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {2 b}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {a \int \sqrt {e \sin (c+d x)} \, dx}{e^2} \\ & = -\frac {2 b}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {\left (a \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 b}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a 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)}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (b+a \cos (c+d x)-a E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \sin (c+d x)}} \]
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Time = 2.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.59
method | result | size |
default | \(\frac {2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a -a \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 a \left (\cos ^{2}\left (d x +c \right )\right )-2 \cos \left (d x +c \right ) b}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(153\) |
parts | \(\frac {a \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (d x +c \right )\right )\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b}{d e \sqrt {e \sin \left (d x +c \right )}}\) | \(162\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\frac {-i \, \sqrt {2} a \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + i \, \sqrt {2} a \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (a \cos \left (d x + c\right ) + b\right )} \sqrt {e \sin \left (d x + c\right )}}{d e^{2} \sin \left (d x + c\right )} \]
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\[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {a + b \cos {\left (c + d x \right )}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {b \cos \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {b \cos \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {a+b\,\cos \left (c+d\,x\right )}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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