Integrand size = 23, antiderivative size = 41 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{\sqrt {b} f} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 223, 209} \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{\sqrt {b} f} \]
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Rule 209
Rule 223
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{f} \\ & = -\frac {\arctan \left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{\sqrt {b} f} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\log \left (\sqrt {2} \sqrt {-b} \cos (e+f x)+\sqrt {2 a+b-b \cos (2 (e+f x))}\right )}{\sqrt {-b} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(35)=70\).
Time = 0.65 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.41
method | result | size |
default | \(\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, \arctan \left (\frac {2 b \left (\sin ^{2}\left (f x +e \right )\right )+a -b}{2 \sqrt {b}\, \sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}}\right )}{2 \sqrt {b}\, \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(99\) |
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (35) = 70\).
Time = 0.35 (sec) , antiderivative size = 370, normalized size of antiderivative = 9.02 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\left [-\frac {\sqrt {-b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{6} + 160 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{5} + 10 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}\right )}{8 \, b f}, \frac {\arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{5} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )}}\right )}{4 \, \sqrt {b} f}\right ] \]
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\[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\sin {\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
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none
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.59 \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {b} f} \]
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\[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\sin \left (e+f\,x\right )}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]
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