Integrand size = 21, antiderivative size = 101 \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {b (4 a+3 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{5/2} (a+b)^{3/2} f}+\frac {\sin (e+f x)}{a^2 f}+\frac {b^2 \sin (e+f x)}{2 a^2 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )} \]
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Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4232, 398, 393, 214} \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {b (4 a+3 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{5/2} f (a+b)^{3/2}}+\frac {b^2 \sin (e+f x)}{2 a^2 f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )}+\frac {\sin (e+f x)}{a^2 f} \]
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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 )^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {b (2 a+b)-2 a b x^2}{a^2 \left (a+b-a x^2\right )^2}\right ) \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\sin (e+f x)}{a^2 f}-\frac {\text {Subst}\left (\int \frac {b (2 a+b)-2 a b x^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{a^2 f} \\ & = \frac {\sin (e+f x)}{a^2 f}+\frac {b^2 \sin (e+f x)}{2 a^2 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}-\frac {(b (4 a+3 b)) \text {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{2 a^2 (a+b) f} \\ & = -\frac {b (4 a+3 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{5/2} (a+b)^{3/2} f}+\frac {\sin (e+f x)}{a^2 f}+\frac {b^2 \sin (e+f x)}{2 a^2 (a+b) f \left (a+b-a \sin ^2(e+f x)\right )} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {-\frac {b (4 a+3 b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\sqrt {a} \sin (e+f x) \left (2+\frac {b^2}{(a+b) \left (a+b-a \sin ^2(e+f x)\right )}\right )}{2 a^{5/2} f} \]
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Time = 1.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (f x +e \right )}{a^{2}}+\frac {b \left (-\frac {b \sin \left (f x +e \right )}{2 \left (a +b \right ) \left (a \sin \left (f x +e \right )^{2}-a -b \right )}-\frac {\left (4 a +3 b \right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 \left (a +b \right ) \sqrt {a \left (a +b \right )}}\right )}{a^{2}}}{f}\) | \(92\) |
default | \(\frac {\frac {\sin \left (f x +e \right )}{a^{2}}+\frac {b \left (-\frac {b \sin \left (f x +e \right )}{2 \left (a +b \right ) \left (a \sin \left (f x +e \right )^{2}-a -b \right )}-\frac {\left (4 a +3 b \right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 \left (a +b \right ) \sqrt {a \left (a +b \right )}}\right )}{a^{2}}}{f}\) | \(92\) |
risch | \(-\frac {i {\mathrm e}^{i \left (f x +e \right )}}{2 a^{2} f}+\frac {i {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{2} f}-\frac {i b^{2} \left ({\mathrm e}^{3 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} \left (a +b \right ) f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b}{\sqrt {a^{2}+a b}\, \left (a +b \right ) f a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) f \,a^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b}{\sqrt {a^{2}+a b}\, \left (a +b \right ) f a}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) f \,a^{2}}\) | \(354\) |
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Time = 0.28 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.87 \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [\frac {{\left (4 \, a b^{2} + 3 \, b^{3} + {\left (4 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) + 2 \, {\left (2 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, a b^{3} + 2 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f\right )}}, \frac {{\left (4 \, a b^{2} + 3 \, b^{3} + {\left (4 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) + {\left (2 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, a b^{3} + 2 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f\right )}}\right ] \]
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\[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\int \frac {\cos {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.32 \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {2 \, b^{2} \sin \left (f x + e\right )}{a^{4} + 2 \, a^{3} b + a^{2} b^{2} - {\left (a^{4} + a^{3} b\right )} \sin \left (f x + e\right )^{2}} + \frac {{\left (4 \, a b + 3 \, b^{2}\right )} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{3} + a^{2} b\right )} \sqrt {{\left (a + b\right )} a}} + \frac {4 \, \sin \left (f x + e\right )}{a^{2}}}{4 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.12 \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {b^{2} \sin \left (f x + e\right )}{{\left (a^{3} + a^{2} b\right )} {\left (a \sin \left (f x + e\right )^{2} - a - b\right )}} - \frac {{\left (4 \, a b + 3 \, b^{2}\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a^{3} + a^{2} b\right )} \sqrt {-a^{2} - a b}} - \frac {2 \, \sin \left (f x + e\right )}{a^{2}}}{2 \, f} \]
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Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.93 \[ \int \frac {\cos (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\sin \left (e+f\,x\right )}{a^2\,f}+\frac {b^2\,\sin \left (e+f\,x\right )}{2\,f\,\left (a+b\right )\,\left (-a^3\,{\sin \left (e+f\,x\right )}^2+a^3+b\,a^2\right )}-\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (4\,a+3\,b\right )}{2\,a^{5/2}\,f\,{\left (a+b\right )}^{3/2}} \]
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