Integrand size = 13, antiderivative size = 73 \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^2}+\frac {\text {arctanh}(\sin (x))}{(a+b)^2}+\frac {b \sin (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3269, 425, 536, 212, 211} \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^2}+\frac {\text {arctanh}(\sin (x))}{(a+b)^2}+\frac {b \sin (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )} \]
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Rule 211
Rule 212
Rule 425
Rule 536
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\sin (x)\right ) \\ & = \frac {b \sin (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}-\frac {\text {Subst}\left (\int \frac {b-2 (a+b)+b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\sin (x)\right )}{2 a (a+b)} \\ & = \frac {b \sin (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )}{(a+b)^2}+\frac {(b (3 a+b)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a (a+b)^2} \\ & = \frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^2}+\frac {\text {arctanh}(\sin (x))}{(a+b)^2}+\frac {b \sin (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.78 \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {-\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{a^{3/2}}+\frac {\sqrt {b} (3 a+b) \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{a^{3/2}}+4 \left (-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {b (a+b) \sin (x)}{a (2 a+b-b \cos (2 x))}\right )}{4 (a+b)^2} \]
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Time = 0.84 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\ln \left (1+\sin \left (x \right )\right )}{2 \left (a +b \right )^{2}}+\frac {b \left (\frac {\left (a +b \right ) \sin \left (x \right )}{2 a \left (a +b \left (\sin ^{2}\left (x \right )\right )\right )}+\frac {\left (3 a +b \right ) \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\sin \left (x \right )-1\right )}{2 \left (a +b \right )^{2}}\) | \(79\) |
risch | \(\frac {i b \left ({\mathrm e}^{3 i x}-{\mathrm e}^{i x}\right )}{a \left (a +b \right ) \left (b \,{\mathrm e}^{4 i x}-4 a \,{\mathrm e}^{2 i x}-2 b \,{\mathrm e}^{2 i x}+b \right )}+\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{a^{2}+2 a b +b^{2}}-\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{a^{2}+2 a b +b^{2}}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a \left (a +b \right )^{2}}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right ) b}{4 a^{2} \left (a +b \right )^{2}}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{4 a \left (a +b \right )^{2}}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right ) b}{4 a^{2} \left (a +b \right )^{2}}\) | \(267\) |
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (61) = 122\).
Time = 0.35 (sec) , antiderivative size = 354, normalized size of antiderivative = 4.85 \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\left [-\frac {{\left ({\left (3 \, a b + b^{2}\right )} \cos \left (x\right )^{2} - 3 \, a^{2} - 4 \, a b - b^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (-\frac {b \cos \left (x\right )^{2} - 2 \, a \sqrt {-\frac {b}{a}} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) + 2 \, {\left (a b \cos \left (x\right )^{2} - a^{2} - a b\right )} \log \left (\sin \left (x\right ) + 1\right ) - 2 \, {\left (a b \cos \left (x\right )^{2} - a^{2} - a b\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (a b + b^{2}\right )} \sin \left (x\right )}{4 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (x\right )^{2}\right )}}, -\frac {{\left ({\left (3 \, a b + b^{2}\right )} \cos \left (x\right )^{2} - 3 \, a^{2} - 4 \, a b - b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \sin \left (x\right )\right ) + {\left (a b \cos \left (x\right )^{2} - a^{2} - a b\right )} \log \left (\sin \left (x\right ) + 1\right ) - {\left (a b \cos \left (x\right )^{2} - a^{2} - a b\right )} \log \left (-\sin \left (x\right ) + 1\right ) - {\left (a b + b^{2}\right )} \sin \left (x\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (x\right )^{2}\right )}}\right ] \]
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\[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\int \frac {\sec {\left (x \right )}}{\left (a + b \sin ^{2}{\left (x \right )}\right )^{2}}\, dx \]
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Time = 0.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.58 \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {b \sin \left (x\right )}{2 \, {\left (a^{3} + a^{2} b + {\left (a^{2} b + a b^{2}\right )} \sin \left (x\right )^{2}\right )}} + \frac {{\left (3 \, a b + b^{2}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b}} + \frac {\log \left (\sin \left (x\right ) + 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {\log \left (\sin \left (x\right ) - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.49 \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\frac {{\left (3 \, a b + b^{2}\right )} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {a b}} + \frac {\log \left (\sin \left (x\right ) + 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {\log \left (-\sin \left (x\right ) + 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {b \sin \left (x\right )}{2 \, {\left (b \sin \left (x\right )^{2} + a\right )} {\left (a^{2} + a b\right )}} \]
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Time = 14.92 (sec) , antiderivative size = 2213, normalized size of antiderivative = 30.32 \[ \int \frac {\sec (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx=\text {Too large to display} \]
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