Optimal. Leaf size=87 \[ \frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{b^3 \sqrt {a+b}}+\frac {x \left (8 a^2-4 a b+3 b^2\right )}{8 b^3}-\frac {(4 a-3 b) \sin (x) \cos (x)}{8 b^2}+\frac {\sin (x) \cos ^3(x)}{4 b} \]
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Rubi [A] time = 0.20, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3187, 470, 578, 522, 203, 205} \[ \frac {x \left (8 a^2-4 a b+3 b^2\right )}{8 b^3}+\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{b^3 \sqrt {a+b}}-\frac {(4 a-3 b) \sin (x) \cos (x)}{8 b^2}+\frac {\sin (x) \cos ^3(x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 470
Rule 522
Rule 578
Rule 3187
Rubi steps
\begin {align*} \int \frac {\cos ^6(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3 \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )\\ &=\frac {\cos ^3(x) \sin (x)}{4 b}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a+(-a+3 b) x^2\right )}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )}{4 b}\\ &=-\frac {(4 a-3 b) \cos (x) \sin (x)}{8 b^2}+\frac {\cos ^3(x) \sin (x)}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {a (4 a-3 b)+\left (-4 a^2+a b-3 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )}{8 b^2}\\ &=-\frac {(4 a-3 b) \cos (x) \sin (x)}{8 b^2}+\frac {\cos ^3(x) \sin (x)}{4 b}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{b^3}-\frac {\left (8 a^2-4 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (x)\right )}{8 b^3}\\ &=\frac {\left (8 a^2-4 a b+3 b^2\right ) x}{8 b^3}+\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{b^3 \sqrt {a+b}}-\frac {(4 a-3 b) \cos (x) \sin (x)}{8 b^2}+\frac {\cos ^3(x) \sin (x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 76, normalized size = 0.87 \[ \frac {-\frac {32 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+4 x \left (8 a^2-4 a b+3 b^2\right )-8 b (a-b) \sin (2 x)+b^2 \sin (4 x)}{32 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 273, normalized size = 3.14 \[ \left [\frac {2 \, a^{2} \sqrt {-\frac {a}{a + b}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \relax (x)^{2} + 4 \, {\left ({\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \cos \relax (x)^{3} - {\left (a^{2} + a b\right )} \cos \relax (x)\right )} \sqrt {-\frac {a}{a + b}} \sin \relax (x) + a^{2}}{b^{2} \cos \relax (x)^{4} + 2 \, a b \cos \relax (x)^{2} + a^{2}}\right ) + {\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} x + {\left (2 \, b^{2} \cos \relax (x)^{3} - {\left (4 \, a b - 3 \, b^{2}\right )} \cos \relax (x)\right )} \sin \relax (x)}{8 \, b^{3}}, \frac {4 \, a^{2} \sqrt {\frac {a}{a + b}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \relax (x)^{2} - a\right )} \sqrt {\frac {a}{a + b}}}{2 \, a \cos \relax (x) \sin \relax (x)}\right ) + {\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} x + {\left (2 \, b^{2} \cos \relax (x)^{3} - {\left (4 \, a b - 3 \, b^{2}\right )} \cos \relax (x)\right )} \sin \relax (x)}{8 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 104, normalized size = 1.20 \[ -\frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} a^{3}}{\sqrt {a^{2} + a b} b^{3}} + \frac {{\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} x}{8 \, b^{3}} - \frac {4 \, a \tan \relax (x)^{3} - 3 \, b \tan \relax (x)^{3} + 4 \, a \tan \relax (x) - 5 \, b \tan \relax (x)}{8 \, {\left (\tan \relax (x)^{2} + 1\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 122, normalized size = 1.40 \[ -\frac {\arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right ) a^{3}}{b^{3} \sqrt {\left (a +b \right ) a}}-\frac {\left (\tan ^{3}\relax (x )\right ) a}{2 b^{2} \left (\tan ^{2}\relax (x )+1\right )^{2}}+\frac {3 \left (\tan ^{3}\relax (x )\right )}{8 b \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {\tan \relax (x ) a}{2 b^{2} \left (\tan ^{2}\relax (x )+1\right )^{2}}+\frac {5 \tan \relax (x )}{8 b \left (\tan ^{2}\relax (x )+1\right )^{2}}+\frac {\arctan \left (\tan \relax (x )\right ) a^{2}}{b^{3}}-\frac {\arctan \left (\tan \relax (x )\right ) a}{2 b^{2}}+\frac {3 \arctan \left (\tan \relax (x )\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 97, normalized size = 1.11 \[ -\frac {a^{3} \arctan \left (\frac {a \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} b^{3}} - \frac {{\left (4 \, a - 3 \, b\right )} \tan \relax (x)^{3} + {\left (4 \, a - 5 \, b\right )} \tan \relax (x)}{8 \, {\left (b^{2} \tan \relax (x)^{4} + 2 \, b^{2} \tan \relax (x)^{2} + b^{2}\right )}} + \frac {{\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} x}{8 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 1036, normalized size = 11.91 \[ \text {result too large to display} \]
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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