Optimal. Leaf size=65 \[ -\frac {(2 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \sin (x) \cos (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3184, 12, 3181, 205} \[ -\frac {(2 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \sin (x) \cos (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 3181
Rule 3184
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^2(x)\right )^2} \, dx &=-\frac {b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}-\frac {\int \frac {-2 a-b}{a+b \cos ^2(x)} \, dx}{2 a (a+b)}\\ &=-\frac {b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}+\frac {(2 a+b) \int \frac {1}{a+b \cos ^2(x)} \, dx}{2 a (a+b)}\\ &=-\frac {b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}-\frac {(2 a+b) \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{2 a (a+b)}\\ &=-\frac {(2 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \cos (x) \sin (x)}{2 a (a+b) \left (a+b \cos ^2(x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 70, normalized size = 1.08 \[ -\frac {(-2 a-b) \tan ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \sin (2 x)}{2 a (a+b) (2 a+b \cos (2 x)+b)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 326, normalized size = 5.02 \[ \left [-\frac {4 \, {\left (a^{2} b + a b^{2}\right )} \cos \relax (x) \sin \relax (x) + {\left ({\left (2 \, a b + b^{2}\right )} \cos \relax (x)^{2} + 2 \, a^{2} + a b\right )} \sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \relax (x)^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \relax (x)^{3} - a \cos \relax (x)\right )} \sqrt {-a^{2} - a b} \sin \relax (x) + a^{2}}{b^{2} \cos \relax (x)^{4} + 2 \, a b \cos \relax (x)^{2} + a^{2}}\right )}{8 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cos \relax (x)^{2}\right )}}, -\frac {2 \, {\left (a^{2} b + a b^{2}\right )} \cos \relax (x) \sin \relax (x) + {\left ({\left (2 \, a b + b^{2}\right )} \cos \relax (x)^{2} + 2 \, a^{2} + a b\right )} \sqrt {a^{2} + a b} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \relax (x)^{2} - a}{2 \, \sqrt {a^{2} + a b} \cos \relax (x) \sin \relax (x)}\right )}{4 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cos \relax (x)^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 69, normalized size = 1.06 \[ \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} {\left (2 \, a + b\right )}}{2 \, {\left (a^{2} + a b\right )}^{\frac {3}{2}}} - \frac {b \tan \relax (x)}{2 \, {\left (a \tan \relax (x)^{2} + a + b\right )} {\left (a^{2} + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 81, normalized size = 1.25 \[ -\frac {b \tan \relax (x )}{2 \left (a +b \right ) a \left (a \left (\tan ^{2}\relax (x )\right )+a +b \right )}+\frac {\arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right )}{\left (a +b \right ) \sqrt {\left (a +b \right ) a}}+\frac {\arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right ) b}{2 \left (a +b \right ) a \sqrt {\left (a +b \right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 72, normalized size = 1.11 \[ -\frac {b \tan \relax (x)}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{3} + a^{2} b\right )} \tan \relax (x)^{2}\right )}} + \frac {{\left (2 \, a + b\right )} \arctan \left (\frac {a \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{2 \, \sqrt {{\left (a + b\right )} a} {\left (a^{2} + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 52, normalized size = 0.80 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\relax (x)}{\sqrt {a+b}}\right )\,\left (2\,a+b\right )}{2\,a^{3/2}\,{\left (a+b\right )}^{3/2}}-\frac {b\,\mathrm {tan}\relax (x)}{2\,a\,\left (a+b\right )\,\left (a\,{\mathrm {tan}\relax (x)}^2+a+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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