Optimal. Leaf size=87 \[ \frac{\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}}+\frac{\tan ^5(x)}{5 (a+b)}+\frac{(2 a+3 b) \tan ^3(x)}{3 (a+b)^2} \]
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Rubi [A] time = 0.113406, antiderivative size = 87, 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 = {3191, 390, 205} \[ \frac{\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}}+\frac{\tan ^5(x)}{5 (a+b)}+\frac{(2 a+3 b) \tan ^3(x)}{3 (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 390
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^6(x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{a+(a+b) x^2} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{a^2+3 a b+3 b^2}{(a+b)^3}+\frac{(2 a+3 b) x^2}{(a+b)^2}+\frac{x^4}{a+b}+\frac{b^3}{(a+b)^3 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\tan (x)\right )\\ &=\frac{\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac{(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}+\frac{\tan ^5(x)}{5 (a+b)}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (x)\right )}{(a+b)^3}\\ &=\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}}+\frac{\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac{(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}+\frac{\tan ^5(x)}{5 (a+b)}\\ \end{align*}
Mathematica [A] time = 0.370925, size = 90, normalized size = 1.03 \[ \frac{\tan (x) \left (\left (4 a^2+13 a b+9 b^2\right ) \sec ^2(x)+8 a^2+3 (a+b)^2 \sec ^4(x)+26 a b+33 b^2\right )}{15 (a+b)^3}+\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 147, normalized size = 1.7 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}{a}^{2}}{5\, \left ( a+b \right ) ^{3}}}+{\frac{2\, \left ( \tan \left ( x \right ) \right ) ^{5}ab}{5\, \left ( a+b \right ) ^{3}}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{5}{b}^{2}}{5\, \left ( a+b \right ) ^{3}}}+{\frac{2\, \left ( \tan \left ( x \right ) \right ) ^{3}{a}^{2}}{3\, \left ( a+b \right ) ^{3}}}+{\frac{5\, \left ( \tan \left ( x \right ) \right ) ^{3}ab}{3\, \left ( a+b \right ) ^{3}}}+{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}{b}^{2}}{ \left ( a+b \right ) ^{3}}}+{\frac{{a}^{2}\tan \left ( x \right ) }{ \left ( a+b \right ) ^{3}}}+3\,{\frac{ab\tan \left ( x \right ) }{ \left ( a+b \right ) ^{3}}}+3\,{\frac{{b}^{2}\tan \left ( x \right ) }{ \left ( a+b \right ) ^{3}}}+{\frac{{b}^{3}}{ \left ( a+b \right ) ^{3}}\arctan \left ({ \left ( a+b \right ) \tan \left ( x \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \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 = 2.32889, size = 1098, normalized size = 12.62 \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} + 5 \, a b + b^{2}\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} -{\left (a + b\right )} \cos \left (x\right )\right )} \sqrt{-a^{2} - a b} \sin \left (x\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (x\right )^{4} - 2 \,{\left (a b + b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) - 4 \,{\left ({\left (8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right )} \cos \left (x\right )^{4} + 3 \, a^{4} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3} +{\left (4 \, a^{4} + 17 \, a^{3} b + 22 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{60 \,{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\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 - b}{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} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right )} \cos \left (x\right )^{4} + 3 \, a^{4} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3} +{\left (4 \, a^{4} + 17 \, a^{3} b + 22 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \,{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\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 [B] time = 1.11407, size = 343, normalized size = 3.94 \begin{align*} \frac{{\left (\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac{a \tan \left (x\right ) + b \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )\right )} b^{3}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt{a^{2} + a b}} + \frac{3 \, a^{4} \tan \left (x\right )^{5} + 12 \, a^{3} b \tan \left (x\right )^{5} + 18 \, a^{2} b^{2} \tan \left (x\right )^{5} + 12 \, a b^{3} \tan \left (x\right )^{5} + 3 \, b^{4} \tan \left (x\right )^{5} + 10 \, a^{4} \tan \left (x\right )^{3} + 45 \, a^{3} b \tan \left (x\right )^{3} + 75 \, a^{2} b^{2} \tan \left (x\right )^{3} + 55 \, a b^{3} \tan \left (x\right )^{3} + 15 \, b^{4} \tan \left (x\right )^{3} + 15 \, a^{4} \tan \left (x\right ) + 75 \, a^{3} b \tan \left (x\right ) + 150 \, a^{2} b^{2} \tan \left (x\right ) + 135 \, a b^{3} \tan \left (x\right ) + 45 \, b^{4} \tan \left (x\right )}{15 \,{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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