Optimal. Leaf size=39 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{3/2}}+\frac{\tan (x)}{a+b} \]
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Rubi [A] time = 0.0606788, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3191, 388, 205} \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{3/2}}+\frac{\tan (x)}{a+b} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{a+(a+b) x^2} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{a+b}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (x)\right )}{a+b}\\ &=\frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{3/2}}+\frac{\tan (x)}{a+b}\\ \end{align*}
Mathematica [A] time = 0.0804537, size = 39, normalized size = 1. \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{3/2}}+\frac{\tan (x)}{a+b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 38, normalized size = 1. \begin{align*}{\frac{\tan \left ( x \right ) }{a+b}}+{\frac{b}{a+b}\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 = 2.38485, size = 641, normalized size = 16.44 \begin{align*} \left [-\frac{\sqrt{-a^{2} - a b} b \cos \left (x\right ) \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 (a^{2} + a b\right )} \sin \left (x\right )}{4 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (x\right )}, -\frac{\sqrt{a^{2} + a b} b \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 ) - 2 \,{\left (a^{2} + a b\right )} \sin \left (x\right )}{2 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (x\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (x \right )}}{a + b \sin ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12551, size = 61, normalized size = 1.56 \begin{align*} \frac{b \arctan \left (\frac{a \tan \left (x\right ) + b \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )}{\sqrt{a^{2} + a b}{\left (a + b\right )}} + \frac{\tan \left (x\right )}{a + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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