Optimal. Leaf size=29 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}} \]
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Rubi [A] time = 0.030828, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3186, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos (x)}{a+b \cos ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.0112525, size = 29, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 21, normalized size = 0.7 \begin{align*}{{\it Artanh} \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \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 = 1.78319, size = 228, normalized size = 7.86 \begin{align*} \left [\frac{\log \left (-\frac{b \cos \left (x\right )^{2} - 2 \, \sqrt{a b + b^{2}} \sin \left (x\right ) - a - 2 \, b}{b \cos \left (x\right )^{2} + a}\right )}{2 \, \sqrt{a b + b^{2}}}, -\frac{\sqrt{-a b - b^{2}} \arctan \left (\frac{\sqrt{-a b - b^{2}} \sin \left (x\right )}{a + b}\right )}{a b + b^{2}}\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 = 1.12551, size = 42, normalized size = 1.45 \begin{align*} -\frac{\arctan \left (\frac{b \sin \left (x\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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