Optimal. Leaf size=79 \[ \frac{\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a+b}}+\frac{(2 a-b) \tan ^3(x)}{3 a^2}+\frac{\tan ^5(x)}{5 a} \]
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Rubi [A] time = 0.101229, antiderivative size = 79, 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} \[ \frac{\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a+b}}+\frac{(2 a-b) \tan ^3(x)}{3 a^2}+\frac{\tan ^5(x)}{5 a} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^6(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^6 \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{a x^6}+\frac{2 a-b}{a^2 x^4}+\frac{a^2-a b+b^2}{a^3 x^2}+\frac{b^3}{a^3 \left (-a-(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=\frac{\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac{(2 a-b) \tan ^3(x)}{3 a^2}+\frac{\tan ^5(x)}{5 a}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{-a-(a+b) x^2} \, dx,x,\cot (x)\right )}{a^3}\\ &=\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a+b}}+\frac{\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac{(2 a-b) \tan ^3(x)}{3 a^2}+\frac{\tan ^5(x)}{5 a}\\ \end{align*}
Mathematica [A] time = 0.319061, size = 80, normalized size = 1.01 \[ \frac{\tan (x) \left (3 a^2 \sec ^4(x)+8 a^2+a (4 a-5 b) \sec ^2(x)-10 a b+15 b^2\right )}{15 a^3}-\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{a^{7/2} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 80, normalized size = 1. \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}}{5\,a}}+{\frac{2\, \left ( \tan \left ( x \right ) \right ) ^{3}}{3\,a}}-{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}b}{3\,{a}^{2}}}+{\frac{\tan \left ( x \right ) }{a}}-{\frac{\tan \left ( x \right ) b}{{a}^{2}}}+{\frac{{b}^{2}\tan \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{3}}{{a}^{3}}\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 = 1.8482, size = 832, normalized size = 10.53 \begin{align*} \left [-\frac{15 \, \sqrt{-a^{2} - a b} b^{3} \cos \left (x\right )^{5} \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 ({\left (8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \left (x\right )^{4} + 3 \, a^{4} + 3 \, a^{3} b +{\left (4 \, a^{4} - a^{3} b - 5 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{60 \,{\left (a^{5} + a^{4} b\right )} \cos \left (x\right )^{5}}, \frac{15 \, \sqrt{a^{2} + a b} b^{3} \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 )^{5} + 2 \,{\left ({\left (8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \left (x\right )^{4} + 3 \, a^{4} + 3 \, a^{3} b +{\left (4 \, a^{4} - a^{3} b - 5 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \,{\left (a^{5} + a^{4} b\right )} \cos \left (x\right )^{5}}\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 = 1.18315, size = 140, normalized size = 1.77 \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^{3}}{\sqrt{a^{2} + a b} a^{3}} + \frac{3 \, a^{4} \tan \left (x\right )^{5} + 10 \, a^{4} \tan \left (x\right )^{3} - 5 \, a^{3} b \tan \left (x\right )^{3} + 15 \, a^{4} \tan \left (x\right ) - 15 \, a^{3} b \tan \left (x\right ) + 15 \, a^{2} b^{2} \tan \left (x\right )}{15 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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