Integrand size = 25, antiderivative size = 79 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (A-B) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {2 B \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2830, 2728, 212} \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (A-B) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {2 B \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}} \]
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Rule 212
Rule 2728
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {2 B \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+(A-B) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = -\frac {2 B \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {(2 (A-B)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f} \\ & = -\frac {\sqrt {2} (A-B) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {2 B \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((1+i) (-1)^{3/4} (A-B) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )+B \left (-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f \sqrt {a (1+\sin (e+f x))}} \]
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Time = 2.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.62
method | result | size |
default | \(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) A -\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) B +2 B \sqrt {a -a \sin \left (f x +e \right )}\right )}{a \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(128\) |
parts | \(-\frac {A \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{\sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {B \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-2 \sqrt {a -a \sin \left (f x +e \right )}\right )}{a \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(171\) |
risch | \(-\frac {\left (-2 i A +i B +B \,{\mathrm e}^{i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \sqrt {2}\, {\mathrm e}^{-i \left (f x +e \right )}}{f \sqrt {-a \left (-2 \,{\mathrm e}^{i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )}-i\right ) {\mathrm e}^{-i \left (f x +e \right )}}}-\frac {2 i \left (A -B \right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \left (\arctan \left (\frac {\sqrt {-i {\mathrm e}^{i \left (f x +e \right )} a}}{\sqrt {a}}\right ) a \sqrt {-i {\mathrm e}^{i \left (f x +e \right )} a}+a^{\frac {3}{2}}\right ) \sqrt {2}\, {\mathrm e}^{-i \left (f x +e \right )}}{f \,a^{\frac {3}{2}} \sqrt {-a \left (-2 \,{\mathrm e}^{i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )}-i\right ) {\mathrm e}^{-i \left (f x +e \right )}}}\) | \(222\) |
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (68) = 136\).
Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.66 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\frac {\sqrt {2} {\left ({\left (A - B\right )} a \cos \left (f x + e\right ) + {\left (A - B\right )} a \sin \left (f x + e\right ) + {\left (A - B\right )} a\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {a}} + 4 \, {\left (B \cos \left (f x + e\right ) - B \sin \left (f x + e\right ) + B\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]
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\[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {A + B \sin {\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {B \sin \left (f x + e\right ) + A}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (68) = 136\).
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.81 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\frac {4 \, \sqrt {2} B \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (A \sqrt {a} - B \sqrt {a}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (A \sqrt {a} - B \sqrt {a}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{2 \, f} \]
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Time = 1.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.91 \[ \int \frac {A+B \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {A\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |1\right )\,\sqrt {\frac {2\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{a}}}{f\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}-\frac {B\,\left (4\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |1\right )-2\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |1\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {a+a\,\sin \left (e+f\,x\right )}{2\,a}}}{f\,\cos \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}} \]
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