Integrand size = 23, antiderivative size = 62 \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\frac {3 x}{d}-\frac {6 (c-d) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d \sqrt {c^2-d^2} f} \]
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Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2814, 2739, 632, 210} \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\frac {a x}{d}-\frac {2 a (c-d) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d f \sqrt {c^2-d^2}} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rubi steps \begin{align*} \text {integral}& = \frac {a x}{d}-\frac {(a (c-d)) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d} \\ & = \frac {a x}{d}-\frac {(2 a (c-d)) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d f} \\ & = \frac {a x}{d}+\frac {(4 a (c-d)) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d f} \\ & = \frac {a x}{d}-\frac {2 a (c-d) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d \sqrt {c^2-d^2} f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.94 \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\frac {3 \left (-2 (c-d) \arctan \left (\frac {\sec \left (\frac {f x}{2}\right ) (\cos (e)-i \sin (e)) \left (d \cos \left (e+\frac {f x}{2}\right )+c \sin \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (\cos (e)-i \sin (e))+\sqrt {c^2-d^2} f x \sqrt {(\cos (e)-i \sin (e))^2}\right ) (1+\sin (e+f x))}{d \sqrt {c^2-d^2} f \sqrt {(\cos (e)-i \sin (e))^2} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
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Time = 0.76 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {2 a \left (\frac {\left (-c +d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{d \sqrt {c^{2}-d^{2}}}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d}\right )}{f}\) | \(72\) |
default | \(\frac {2 a \left (\frac {\left (-c +d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{d \sqrt {c^{2}-d^{2}}}+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d}\right )}{f}\) | \(72\) |
risch | \(\frac {a x}{d}+\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c -\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{\left (c +d \right ) f d}-\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{\left (c +d \right ) f d}\) | \(125\) |
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Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.68 \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\left [\frac {2 \, a f x + a \sqrt {-\frac {c - d}{c + d}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right )}{2 \, d f}, \frac {a f x + a \sqrt {\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (c \sin \left (f x + e\right ) + d\right )} \sqrt {\frac {c - d}{c + d}}}{{\left (c - d\right )} \cos \left (f x + e\right )}\right )}{d f}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (49) = 98\).
Time = 13.08 (sec) , antiderivative size = 314, normalized size of antiderivative = 5.06 \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\begin {cases} \frac {\tilde {\infty } x \left (a \sin {\left (e \right )} + a\right )}{\sin {\left (e \right )}} & \text {for}\: c = 0 \wedge d = 0 \wedge f = 0 \\\frac {a x + \frac {a \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} \right )}}{f}}{d} & \text {for}\: c = 0 \\\frac {a x - \frac {a \cos {\left (e + f x \right )}}{f}}{c} & \text {for}\: d = 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )}{c + d \sin {\left (e \right )}} & \text {for}\: f = 0 \\\frac {a f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{d f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - d f} - \frac {a f x}{d f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - d f} + \frac {4 a}{d f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - d f} & \text {for}\: c = - d \\\frac {a x}{d} & \text {for}\: c = d \\- \frac {a c \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} - \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )}}{d f \sqrt {- c^{2} + d^{2}}} + \frac {a c \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} + \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )}}{d f \sqrt {- c^{2} + d^{2}}} + \frac {a \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} - \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )}}{f \sqrt {- c^{2} + d^{2}}} - \frac {a \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} + \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )}}{f \sqrt {- c^{2} + d^{2}}} + \frac {a x}{d} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.85 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.34 \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\frac {\frac {{\left (f x + e\right )} a}{d} - \frac {2 \, {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )} {\left (a c - a d\right )}}{\sqrt {c^{2} - d^{2}} d}}{f} \]
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Time = 7.17 (sec) , antiderivative size = 449, normalized size of antiderivative = 7.24 \[ \int \frac {3+3 \sin (e+f x)}{c+d \sin (e+f x)} \, dx=\frac {2\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{f\,\left (c+d\right )}-\frac {2\,a\,\mathrm {atanh}\left (\frac {3\,d^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (d^2-c^2\right )}^{3/2}-2\,c^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}-2\,c^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (d^2-c^2\right )}^{3/2}+d^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}+2\,c^2\,d^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}+3\,c^2\,d^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}+c\,d\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (d^2-c^2\right )}^{3/2}+c\,d^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}+c^3\,d\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}+4\,c\,d^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}-2\,c^3\,d\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\sqrt {d^2-c^2}}{2\,\left (d^2+c\,d\right )\,\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3+2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d-\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^2-2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^3\right )}\right )\,\sqrt {d^2-c^2}}{d\,f\,\left (c+d\right )}+\frac {2\,a\,c\,\mathrm {atan}\left (\frac {\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}\right )}{d\,f\,\left (c+d\right )} \]
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