Integrand size = 23, antiderivative size = 126 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {b^2 (6 a+5 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{7/2} (a+b)^{3/2} f}+\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}-\frac {b^3 \sin (e+f x)}{2 a^3 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )} \]
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Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4232, 398, 393, 214} \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {b^2 (6 a+5 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{7/2} f (a+b)^{3/2}}-\frac {b^3 \sin (e+f x)}{2 a^3 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f} \]
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Rule 214
Rule 393
Rule 398
Rule 4232
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a-2 b}{a^3}-\frac {x^2}{a^2}+\frac {b^2 (3 a+2 b)-3 a b^2 x^2}{a^3 \left (a+b-a x^2\right )^2}\right ) \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}+\frac {\text {Subst}\left (\int \frac {b^2 (3 a+2 b)-3 a b^2 x^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{a^3 f} \\ & = \frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}-\frac {b^3 \sin (e+f x)}{2 a^3 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {\left (b^2 (6 a+5 b)\right ) \text {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{2 a^3 (a+b) f} \\ & = \frac {b^2 (6 a+5 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{7/2} (a+b)^{3/2} f}+\frac {(a-2 b) \sin (e+f x)}{a^3 f}-\frac {\sin ^3(e+f x)}{3 a^2 f}-\frac {b^3 \sin (e+f x)}{2 a^3 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {-\frac {3 b^2 (6 a+5 b) \left (\log \left (\sqrt {a+b}-\sqrt {a} \sin (e+f x)\right )-\log \left (\sqrt {a+b}+\sqrt {a} \sin (e+f x)\right )\right )}{(a+b)^{3/2}}+3 \sqrt {a} \left (3 a-8 b-\frac {4 b^3}{(a+b) (a+2 b+a \cos (2 (e+f x)))}\right ) \sin (e+f x)+a^{3/2} \sin (3 (e+f x))}{12 a^{7/2} f} \]
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Time = 3.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {a \sin \left (f x +e \right )^{3}}{3}-\sin \left (f x +e \right ) a +2 \sin \left (f x +e \right ) b}{a^{3}}-\frac {b^{2} \left (-\frac {b \sin \left (f x +e \right )}{2 \left (a +b \right ) \left (a \sin \left (f x +e \right )^{2}-a -b \right )}-\frac {\left (6 a +5 b \right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 \left (a +b \right ) \sqrt {a \left (a +b \right )}}\right )}{a^{3}}}{f}\) | \(120\) |
default | \(\frac {-\frac {\frac {a \sin \left (f x +e \right )^{3}}{3}-\sin \left (f x +e \right ) a +2 \sin \left (f x +e \right ) b}{a^{3}}-\frac {b^{2} \left (-\frac {b \sin \left (f x +e \right )}{2 \left (a +b \right ) \left (a \sin \left (f x +e \right )^{2}-a -b \right )}-\frac {\left (6 a +5 b \right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 \left (a +b \right ) \sqrt {a \left (a +b \right )}}\right )}{a^{3}}}{f}\) | \(120\) |
risch | \(-\frac {i {\mathrm e}^{3 i \left (f x +e \right )}}{24 a^{2} f}-\frac {3 i {\mathrm e}^{i \left (f x +e \right )}}{8 a^{2} f}+\frac {i {\mathrm e}^{i \left (f x +e \right )} b}{a^{3} f}+\frac {3 i {\mathrm e}^{-i \left (f x +e \right )}}{8 a^{2} f}-\frac {i {\mathrm e}^{-i \left (f x +e \right )} b}{a^{3} f}+\frac {i {\mathrm e}^{-3 i \left (f x +e \right )}}{24 a^{2} f}+\frac {i b^{3} \left ({\mathrm e}^{3 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{3} \left (a +b \right ) f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right ) f \,a^{2}}+\frac {5 b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) f \,a^{3}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right ) f \,a^{2}}-\frac {5 b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) f \,a^{3}}\) | \(433\) |
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (115) = 230\).
Time = 0.31 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.89 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [\frac {3 \, {\left (6 \, a b^{3} + 5 \, b^{4} + {\left (6 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4} + 2 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{5} - a^{4} b - 8 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{12 \, {\left ({\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f\right )}}, -\frac {3 \, {\left (6 \, a b^{3} + 5 \, b^{4} + {\left (6 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) - {\left (4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4} + 2 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{5} - a^{4} b - 8 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{6 \, {\left ({\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {6 \, b^{3} \sin \left (f x + e\right )}{a^{5} + 2 \, a^{4} b + a^{3} b^{2} - {\left (a^{5} + a^{4} b\right )} \sin \left (f x + e\right )^{2}} + \frac {3 \, {\left (6 \, a b^{2} + 5 \, b^{3}\right )} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} a}} + \frac {4 \, {\left (a \sin \left (f x + e\right )^{3} - 3 \, {\left (a - 2 \, b\right )} \sin \left (f x + e\right )\right )}}{a^{3}}}{12 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {3 \, b^{3} \sin \left (f x + e\right )}{{\left (a^{4} + a^{3} b\right )} {\left (a \sin \left (f x + e\right )^{2} - a - b\right )}} - \frac {3 \, {\left (6 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {-a^{2} - a b}} - \frac {2 \, {\left (a^{4} \sin \left (f x + e\right )^{3} - 3 \, a^{4} \sin \left (f x + e\right ) + 6 \, a^{3} b \sin \left (f x + e\right )\right )}}{a^{6}}}{6 \, f} \]
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Time = 19.79 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (6\,a+5\,b\right )}{2\,a^{7/2}\,f\,{\left (a+b\right )}^{3/2}}-\frac {{\sin \left (e+f\,x\right )}^3}{3\,a^2\,f}-\frac {b^3\,\sin \left (e+f\,x\right )}{2\,f\,\left (a+b\right )\,\left (-a^4\,{\sin \left (e+f\,x\right )}^2+a^4+b\,a^3\right )}-\frac {\sin \left (e+f\,x\right )\,\left (\frac {2\,\left (a+b\right )}{a^3}-\frac {3}{a^2}\right )}{f} \]
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