Integrand size = 21, antiderivative size = 83 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2} f}-\frac {b \sin (e+f x)}{2 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )} \]
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Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4232, 393, 214} \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{3/2} f (a+b)^{3/2}}-\frac {b \sin (e+f x)}{2 a f (a+b) \left (-a \sin ^2(e+f x)+a+b\right )} \]
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Rule 214
Rule 393
Rule 4232
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a+b-a x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {b \sin (e+f x)}{2 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )}+\frac {(2 a+b) \text {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{2 a (a+b) f} \\ & = \frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2} f}-\frac {b \sin (e+f x)}{2 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right )} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {2 \sqrt {a} b \sin (e+f x)}{(a+b) (a+2 b+a \cos (2 (e+f x)))}}{2 a^{3/2} f} \]
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Time = 0.55 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {b \sin \left (f x +e \right )}{2 a \left (a +b \right ) \left (a \sin \left (f x +e \right )^{2}-a -b \right )}+\frac {\left (2 a +b \right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 a \left (a +b \right ) \sqrt {a \left (a +b \right )}}}{f}\) | \(80\) |
default | \(\frac {\frac {b \sin \left (f x +e \right )}{2 a \left (a +b \right ) \left (a \sin \left (f x +e \right )^{2}-a -b \right )}+\frac {\left (2 a +b \right ) \operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 a \left (a +b \right ) \sqrt {a \left (a +b \right )}}}{f}\) | \(80\) |
risch | \(\frac {i b \left ({\mathrm e}^{3 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{a f \left (a +b \right ) \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right ) f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) f a}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right ) f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right ) b}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) f a}\) | \(305\) |
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Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.63 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [\frac {{\left ({\left (2 \, a^{2} + a b\right )} \cos \left (f x + e\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt {a^{2} + a b} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right ) - 2 \, {\left (a^{2} b + a b^{2}\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f\right )}}, -\frac {{\left ({\left (2 \, a^{2} + a b\right )} \cos \left (f x + e\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right ) + {\left (a^{2} b + a b^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f\right )}}\right ] \]
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\[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.34 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {2 \, b \sin \left (f x + e\right )}{a^{3} + 2 \, a^{2} b + a b^{2} - {\left (a^{3} + a^{2} b\right )} \sin \left (f x + e\right )^{2}} + \frac {{\left (2 \, a + b\right )} \log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} {\left (a^{2} + a b\right )}}}{4 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {{\left (2 \, a + b\right )} \arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{{\left (a^{2} + a b\right )} \sqrt {-a^{2} - a b}} - \frac {b \sin \left (f x + e\right )}{{\left (a \sin \left (f x + e\right )^{2} - a - b\right )} {\left (a^{2} + a b\right )}}}{2 \, f} \]
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Time = 19.74 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.86 \[ \int \frac {\sec (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (2\,a+b\right )}{2\,a^{3/2}\,f\,{\left (a+b\right )}^{3/2}}-\frac {b\,\sin \left (e+f\,x\right )}{2\,a\,f\,\left (a+b\right )\,\left (-a\,{\sin \left (e+f\,x\right )}^2+a+b\right )} \]
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