Optimal. Leaf size=56 \[ -\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{a^{5/2} \sqrt{a+b}}+\frac{(a-b) \tan (x)}{a^2}+\frac{\tan ^3(x)}{3 a} \]
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Rubi [A] time = 0.0860343, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3187, 461, 205} \[ -\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{a^{5/2} \sqrt{a+b}}+\frac{(a-b) \tan (x)}{a^2}+\frac{\tan ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^4(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{a x^4}+\frac{a-b}{a^2 x^2}+\frac{b^2}{a^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=\frac{(a-b) \tan (x)}{a^2}+\frac{\tan ^3(x)}{3 a}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{a^2}\\ &=-\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{a^{5/2} \sqrt{a+b}}+\frac{(a-b) \tan (x)}{a^2}+\frac{\tan ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.135363, size = 55, normalized size = 0.98 \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{a^{5/2} \sqrt{a+b}}+\frac{\tan (x) \left (a \sec ^2(x)+2 a-3 b\right )}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 51, normalized size = 0.9 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3\,a}}+{\frac{\tan \left ( x \right ) }{a}}-{\frac{\tan \left ( x \right ) b}{{a}^{2}}}+{\frac{{b}^{2}}{{a}^{2}}\arctan \left ({\tan \left ( x \right ) a{\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89319, size = 672, normalized size = 12. \begin{align*} \left [-\frac{3 \, \sqrt{-a^{2} - a b} b^{2} \cos \left (x\right )^{3} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt{-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right ) - 4 \,{\left (a^{3} + a^{2} b +{\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{12 \,{\left (a^{4} + a^{3} b\right )} \cos \left (x\right )^{3}}, -\frac{3 \, \sqrt{a^{2} + a b} b^{2} \arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt{a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right ) \cos \left (x\right )^{3} - 2 \,{\left (a^{3} + a^{2} b +{\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \,{\left (a^{4} + a^{3} b\right )} \cos \left (x\right )^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19087, size = 96, normalized size = 1.71 \begin{align*} \frac{{\left (\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )\right )} b^{2}}{\sqrt{a^{2} + a b} a^{2}} + \frac{a^{2} \tan \left (x\right )^{3} + 3 \, a^{2} \tan \left (x\right ) - 3 \, a b \tan \left (x\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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