Integrand size = 25, antiderivative size = 73 \[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\sqrt {\frac {2}{3}} (c-d) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{f}-\frac {2 d \cos (e+f x)}{f \sqrt {3+3 \sin (e+f x)}} \]
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Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08, 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} \[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\sqrt {2} (c-d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {a} f}-\frac {2 d \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}} \]
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Rule 212
Rule 2728
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+(c-d) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = -\frac {2 d \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {(2 (c-d)) \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} (c-d) \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 d \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.49 \[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left ((1+i) (-1)^{3/4} (c-d) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )+d \left (-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{\sqrt {3} f \sqrt {1+\sin (e+f x)}} \]
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Time = 3.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.75
method | result | size |
default | \(-\frac {\left (\sin \left (f x +e \right )+1\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) c -\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) d +2 \sqrt {a -a \sin \left (f x +e \right )}\, d \right )}{a \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(128\) |
parts | \(-\frac {c \left (\sin \left (f x +e \right )+1\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{\sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {d \left (\sin \left (f x +e \right )+1\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-2 \sqrt {a -a \sin \left (f x +e \right )}\right )}{a \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(171\) |
risch | \(-\frac {\left (-2 i c +i d +d \,{\mathrm e}^{i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \sqrt {2}\, {\mathrm e}^{-i \left (f x +e \right )}}{f \sqrt {-a \left (i {\mathrm e}^{2 i \left (f x +e \right )}-i-2 \,{\mathrm e}^{i \left (f x +e \right )}\right ) {\mathrm e}^{-i \left (f x +e \right )}}}-\frac {2 i \left (c -d \right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) \left (\arctan \left (\frac {\sqrt {-i a \,{\mathrm e}^{i \left (f x +e \right )}}}{\sqrt {a}}\right ) a \sqrt {-i a \,{\mathrm e}^{i \left (f x +e \right )}}+a^{\frac {3}{2}}\right ) \sqrt {2}\, {\mathrm e}^{-i \left (f x +e \right )}}{f \,a^{\frac {3}{2}} \sqrt {-a \left (i {\mathrm e}^{2 i \left (f x +e \right )}-i-2 \,{\mathrm e}^{i \left (f x +e \right )}\right ) {\mathrm e}^{-i \left (f x +e \right )}}}\) | \(222\) |
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (68) = 136\).
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.93 \[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\frac {\sqrt {2} {\left (a c - a d + {\left (a c - a d\right )} \cos \left (f x + e\right ) + {\left (a c - a d\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {a}} + 4 \, {\left (d \cos \left (f x + e\right ) - d \sin \left (f x + e\right ) + d\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \]
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\[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {c + d \sin {\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {d \sin \left (f x + e\right ) + c}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (68) = 136\).
Time = 0.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.96 \[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {\frac {4 \, \sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (\sqrt {a} c - \sqrt {a} d\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (\sqrt {a} c - \sqrt {a} d\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{2 \, f} \]
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Time = 9.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.07 \[ \int \frac {c+d \sin (e+f x)}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {c\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |1\right )\,\sqrt {\frac {2\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{a}}}{f\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}-\frac {d\,\left (4\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |1\right )-2\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |1\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {a+a\,\sin \left (e+f\,x\right )}{2\,a}}}{f\,\cos \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}} \]
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