Integrand size = 19, antiderivative size = 57 \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {x}{b}-\frac {2 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2} d} \]
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Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2814, 2739, 632, 210} \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {x}{b}-\frac {2 a \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rubi steps \begin{align*} \text {integral}& = \frac {x}{b}-\frac {a \int \frac {1}{a+b \sin (c+d x)} \, dx}{b} \\ & = \frac {x}{b}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d} \\ & = \frac {x}{b}+\frac {(4 a) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d} \\ & = \frac {x}{b}-\frac {2 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2} d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04 \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {c}{d}+x-\frac {2 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}}{b} \]
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Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b \sqrt {a^{2}-b^{2}}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}}{d}\) | \(68\) |
default | \(\frac {-\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b \sqrt {a^{2}-b^{2}}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b}}{d}\) | \(68\) |
risch | \(\frac {x}{b}-\frac {i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, d b}+\frac {i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, d b}\) | \(149\) |
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Time = 0.31 (sec) , antiderivative size = 237, normalized size of antiderivative = 4.16 \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\left [\frac {2 \, {\left (a^{2} - b^{2}\right )} d x - \sqrt {-a^{2} + b^{2}} a \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right )}{2 \, {\left (a^{2} b - b^{3}\right )} d}, \frac {{\left (a^{2} - b^{2}\right )} d x + \sqrt {a^{2} - b^{2}} a \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right )}{{\left (a^{2} b - b^{3}\right )} d}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (44) = 88\).
Time = 11.96 (sec) , antiderivative size = 253, normalized size of antiderivative = 4.44 \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {x}{b} & \text {for}\: a = 0 \\- \frac {\cos {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {x \sin {\left (c \right )}}{a + b \sin {\left (c \right )}} & \text {for}\: d = 0 \\\frac {d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - b d} - \frac {d x}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - b d} + \frac {2}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - b d} & \text {for}\: a = - b \\\frac {d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + b d} + \frac {d x}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + b d} + \frac {2}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + b d} & \text {for}\: a = b \\- \frac {a \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{b d \sqrt {- a^{2} + b^{2}}} + \frac {a \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{b d \sqrt {- a^{2} + b^{2}}} + \frac {x}{b} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.35 \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a}{\sqrt {a^{2} - b^{2}} b} - \frac {d x + c}{b}}{d} \]
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Time = 1.70 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.44 \[ \int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {x}{b}-\frac {2\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^3+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^4}{{\left (b^2-a^2\right )}^{3/2}\,\left (a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{b\,d\,\sqrt {b^2-a^2}} \]
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