Optimal. Leaf size=79 \[ \frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a+b}}+\frac {(2 a-b) \tan ^3(x)}{3 a^2}+\frac {\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac {\tan ^5(x)}{5 a} \]
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Rubi [A] time = 0.10, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3187, 461, 205} \[ \frac {\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a+b}}+\frac {(2 a-b) \tan ^3(x)}{3 a^2}+\frac {\tan ^5(x)}{5 a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 461
Rule 3187
Rubi steps
\begin {align*} \int \frac {\sec ^6(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^6 \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{a x^6}+\frac {2 a-b}{a^2 x^4}+\frac {a^2-a b+b^2}{a^3 x^2}+\frac {b^3}{a^3 \left (-a-(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=\frac {\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac {(2 a-b) \tan ^3(x)}{3 a^2}+\frac {\tan ^5(x)}{5 a}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{-a-(a+b) x^2} \, dx,x,\cot (x)\right )}{a^3}\\ &=\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a+b}}+\frac {\left (a^2-a b+b^2\right ) \tan (x)}{a^3}+\frac {(2 a-b) \tan ^3(x)}{3 a^2}+\frac {\tan ^5(x)}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 80, normalized size = 1.01 \[ \frac {\tan (x) \left (3 a^2 \sec ^4(x)+8 a^2+a (4 a-5 b) \sec ^2(x)-10 a b+15 b^2\right )}{15 a^3}-\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{a^{7/2} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.20, size = 348, normalized size = 4.41 \[ \left [-\frac {15 \, \sqrt {-a^{2} - a b} b^{3} \cos \relax (x)^{5} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \relax (x)^{2} - 4 \, {\left ({\left (2 \, a + b\right )} \cos \relax (x)^{3} - a \cos \relax (x)\right )} \sqrt {-a^{2} - a b} \sin \relax (x) + a^{2}}{b^{2} \cos \relax (x)^{4} + 2 \, a b \cos \relax (x)^{2} + a^{2}}\right ) - 4 \, {\left ({\left (8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \relax (x)^{4} + 3 \, a^{4} + 3 \, a^{3} b + {\left (4 \, a^{4} - a^{3} b - 5 \, a^{2} b^{2}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{60 \, {\left (a^{5} + a^{4} b\right )} \cos \relax (x)^{5}}, \frac {15 \, \sqrt {a^{2} + a b} b^{3} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \relax (x)^{2} - a}{2 \, \sqrt {a^{2} + a b} \cos \relax (x) \sin \relax (x)}\right ) \cos \relax (x)^{5} + 2 \, {\left ({\left (8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \relax (x)^{4} + 3 \, a^{4} + 3 \, a^{3} b + {\left (4 \, a^{4} - a^{3} b - 5 \, a^{2} b^{2}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{30 \, {\left (a^{5} + a^{4} b\right )} \cos \relax (x)^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 104, normalized size = 1.32 \[ -\frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} b^{3}}{\sqrt {a^{2} + a b} a^{3}} + \frac {3 \, a^{4} \tan \relax (x)^{5} + 10 \, a^{4} \tan \relax (x)^{3} - 5 \, a^{3} b \tan \relax (x)^{3} + 15 \, a^{4} \tan \relax (x) - 15 \, a^{3} b \tan \relax (x) + 15 \, a^{2} b^{2} \tan \relax (x)}{15 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 80, normalized size = 1.01 \[ \frac {\tan ^{5}\relax (x )}{5 a}+\frac {2 \left (\tan ^{3}\relax (x )\right )}{3 a}-\frac {\left (\tan ^{3}\relax (x )\right ) b}{3 a^{2}}+\frac {\tan \relax (x )}{a}-\frac {\tan \relax (x ) b}{a^{2}}+\frac {b^{2} \tan \relax (x )}{a^{3}}-\frac {b^{3} \arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right )}{a^{3} \sqrt {\left (a +b \right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 74, normalized size = 0.94 \[ -\frac {b^{3} \arctan \left (\frac {a \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{3}} + \frac {3 \, a^{2} \tan \relax (x)^{5} + 5 \, {\left (2 \, a^{2} - a b\right )} \tan \relax (x)^{3} + 15 \, {\left (a^{2} - a b + b^{2}\right )} \tan \relax (x)}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.30, size = 84, normalized size = 1.06 \[ \frac {{\mathrm {tan}\relax (x)}^5}{5\,a}-{\mathrm {tan}\relax (x)}^3\,\left (\frac {a+b}{3\,a^2}-\frac {1}{a}\right )+\mathrm {tan}\relax (x)\,\left (\frac {3}{a}+\frac {\left (a+b\right )\,\left (\frac {a+b}{a^2}-\frac {3}{a}\right )}{a}\right )-\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\relax (x)}{\sqrt {a+b}}\right )}{a^{7/2}\,\sqrt {a+b}} \]
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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