Integrand size = 23, antiderivative size = 76 \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b} f}+\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f} \]
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Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4232, 398, 214} \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{5/2} f \sqrt {a+b}}+\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f} \]
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Rule 214
Rule 398
Rule 4232
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a-b}{a^2}-\frac {x^2}{a}+\frac {b^2}{a^2 \left (a+b-a x^2\right )}\right ) \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{a^2 f} \\ & = \frac {b^2 \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b} f}+\frac {(a-b) \sin (e+f x)}{a^2 f}-\frac {\sin ^3(e+f x)}{3 a f} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\frac {6 b^2 \left (-\log \left (\sqrt {a+b}-\sqrt {a} \sin (e+f x)\right )+\log \left (\sqrt {a+b}+\sqrt {a} \sin (e+f x)\right )\right )}{\sqrt {a+b}}+3 \sqrt {a} (3 a-4 b) \sin (e+f x)+a^{3/2} \sin (3 (e+f x))}{12 a^{5/2} f} \]
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Time = 0.99 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {a \sin \left (f x +e \right )^{3}}{3}-\sin \left (f x +e \right ) a +\sin \left (f x +e \right ) b}{a^{2}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}}{f}\) | \(70\) |
default | \(\frac {-\frac {\frac {a \sin \left (f x +e \right )^{3}}{3}-\sin \left (f x +e \right ) a +\sin \left (f x +e \right ) b}{a^{2}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}}{f}\) | \(70\) |
risch | \(-\frac {3 i {\mathrm e}^{i \left (f x +e \right )}}{8 a f}+\frac {i {\mathrm e}^{i \left (f x +e \right )} b}{2 a^{2} f}+\frac {3 i {\mathrm e}^{-i \left (f x +e \right )}}{8 a f}-\frac {i {\mathrm e}^{-i \left (f x +e \right )} b}{2 a^{2} f}+\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, f \,a^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, f \,a^{2}}+\frac {\sin \left (3 f x +3 e \right )}{12 a f}\) | \(205\) |
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Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 3.03 \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\left [\frac {3 \, \sqrt {a^{2} + a b} b^{2} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2} + {\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{4} + a^{3} b\right )} f}, -\frac {3 \, \sqrt {-a^{2} - a b} b^{2} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) - {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2} + {\left (a^{3} + a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{4} + a^{3} b\right )} f}\right ] \]
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Timed out. \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\frac {3 \, b^{2} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{2}} + \frac {2 \, {\left (a \sin \left (f x + e\right )^{3} - 3 \, {\left (a - b\right )} \sin \left (f x + e\right )\right )}}{a^{2}}}{6 \, f} \]
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Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\frac {3 \, b^{2} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} a^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3} - 3 \, a^{2} \sin \left (f x + e\right ) + 3 \, a b \sin \left (f x + e\right )}{a^{3}}}{3 \, f} \]
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Time = 18.60 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )}{a^{5/2}\,f\,\sqrt {a+b}}-\frac {{\sin \left (e+f\,x\right )}^3}{3\,a\,f}-\frac {\sin \left (e+f\,x\right )\,\left (\frac {a+b}{a^2}-\frac {2}{a}\right )}{f} \]
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