Integrand size = 23, antiderivative size = 71 \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {b} (a+b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{5/2} f}-\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f} \]
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Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4218, 470, 327, 211} \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {b} (a+b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{5/2} f}-\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f} \]
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Rule 211
Rule 327
Rule 470
Rule 4218
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^2 \left (1-x^2\right )}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x)}{3 a f}-\frac {(a+b) \text {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{a f} \\ & = -\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f}+\frac {(b (a+b)) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cos (e+f x)\right )}{a^2 f} \\ & = \frac {\sqrt {b} (a+b) \arctan \left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {b}}\right )}{a^{5/2} f}-\frac {(a+b) \cos (e+f x)}{a^2 f}+\frac {\cos ^3(e+f x)}{3 a f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.21 (sec) , antiderivative size = 376, normalized size of antiderivative = 5.30 \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \left (3 \left (a^2+8 a b+8 b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}-i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}-\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )+3 \left (a^2+8 a b+8 b^2\right ) \arctan \left (\frac {\left (-\sqrt {a}+i \sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2}\right ) \sin (e) \tan \left (\frac {f x}{2}\right )+\cos (e) \left (\sqrt {a}+\sqrt {a+b} \sqrt {(\cos (e)-i \sin (e))^2} \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {b}}\right )-3 a^2 \arctan \left (\frac {\sqrt {a}-\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )-3 a^2 \arctan \left (\frac {\sqrt {a}+\sqrt {a+b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {b}}\right )+4 \sqrt {a} \sqrt {b} \cos (e+f x) (-5 a-6 b+a \cos (2 (e+f x)))\right ) \sec ^2(e+f x)}{48 a^{5/2} \sqrt {b} f \left (a+b \sec ^2(e+f x)\right )} \]
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Time = 0.49 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -b \cos \left (f x +e \right )}{a^{2}}+\frac {b \left (a +b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}}{f}\) | \(67\) |
default | \(\frac {\frac {\frac {a \cos \left (f x +e \right )^{3}}{3}-\cos \left (f x +e \right ) a -b \cos \left (f x +e \right )}{a^{2}}+\frac {b \left (a +b \right ) \arctan \left (\frac {a \cos \left (f x +e \right )}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}}{f}\) | \(67\) |
risch | \(-\frac {3 \,{\mathrm e}^{i \left (f x +e \right )}}{8 a f}-\frac {{\mathrm e}^{i \left (f x +e \right )} b}{2 a^{2} f}-\frac {3 \,{\mathrm e}^{-i \left (f x +e \right )}}{8 a f}-\frac {{\mathrm e}^{-i \left (f x +e \right )} b}{2 a^{2} f}+\frac {i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{2 a^{2} f}+\frac {i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{2 a^{3} f}-\frac {i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right )}{2 a^{2} f}-\frac {i \sqrt {a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i \left (f x +e \right )}}{a}+1\right ) b}{2 a^{3} f}+\frac {\cos \left (3 f x +3 e \right )}{12 f a}\) | \(275\) |
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Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.17 \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\left [\frac {2 \, a \cos \left (f x + e\right )^{3} + 3 \, {\left (a + b\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, a \sqrt {-\frac {b}{a}} \cos \left (f x + e\right ) - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 6 \, {\left (a + b\right )} \cos \left (f x + e\right )}{6 \, a^{2} f}, \frac {a \cos \left (f x + e\right )^{3} + 3 \, {\left (a + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}} \cos \left (f x + e\right )}{b}\right ) - 3 \, {\left (a + b\right )} \cos \left (f x + e\right )}{3 \, a^{2} f}\right ] \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\frac {3 \, {\left (a b + b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a \cos \left (f x + e\right )^{3} - 3 \, {\left (a + b\right )} \cos \left (f x + e\right )}{a^{2}}}{3 \, f} \]
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Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20 \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {{\left (a b + b^{2}\right )} \arctan \left (\frac {a \cos \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} f} + \frac {a^{2} f^{5} \cos \left (f x + e\right )^{3} - 3 \, a^{2} f^{5} \cos \left (f x + e\right ) - 3 \, a b f^{5} \cos \left (f x + e\right )}{3 \, a^{3} f^{6}} \]
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Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {{\cos \left (e+f\,x\right )}^3}{3\,a\,f}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {b}{a^2}+\frac {1}{a}\right )}{f}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\cos \left (e+f\,x\right )\,\left (a+b\right )}{b^2+a\,b}\right )\,\left (a+b\right )}{a^{5/2}\,f} \]
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