Optimal. Leaf size=78 \[ \frac {\left (a^2+3 a b+3 b^2\right ) \cos (x)}{b^3}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {(a+3 b) \cos ^3(x)}{3 b^2}+\frac {\cos ^5(x)}{5 b} \]
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Rubi [A] time = 0.09, antiderivative size = 78, 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 = {3190, 390, 205} \[ \frac {\left (a^2+3 a b+3 b^2\right ) \cos (x)}{b^3}-\frac {(a+3 b) \cos ^3(x)}{3 b^2}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}+\frac {\cos ^5(x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rule 3190
Rubi steps
\begin {align*} \int \frac {\sin ^7(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-\frac {a^2+3 a b+3 b^2}{b^3}+\frac {(a+3 b) x^2}{b^2}-\frac {x^4}{b}+\frac {a^3+3 a^2 b+3 a b^2+b^3}{b^3 \left (a+b x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\frac {\left (a^2+3 a b+3 b^2\right ) \cos (x)}{b^3}-\frac {(a+3 b) \cos ^3(x)}{3 b^2}+\frac {\cos ^5(x)}{5 b}-\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cos (x)\right )}{b^3}\\ &=-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}+\frac {\left (a^2+3 a b+3 b^2\right ) \cos (x)}{b^3}-\frac {(a+3 b) \cos ^3(x)}{3 b^2}+\frac {\cos ^5(x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 143, normalized size = 1.83 \[ \frac {\left (8 a^2+22 a b+19 b^2\right ) \cos (x)}{8 b^3}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {x}{2}\right )+\sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {(4 a+9 b) \cos (3 x)}{48 b^2}+\frac {\cos (5 x)}{80 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 225, normalized size = 2.88 \[ \left [\frac {6 \, a b^{3} \cos \relax (x)^{5} - 10 \, {\left (a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \relax (x)^{3} - 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \cos \relax (x) - a}{b \cos \relax (x)^{2} + a}\right ) + 30 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \relax (x)}{30 \, a b^{4}}, \frac {3 \, a b^{3} \cos \relax (x)^{5} - 5 \, {\left (a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \relax (x)^{3} - 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \cos \relax (x)}{a}\right ) + 15 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \relax (x)}{15 \, a b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 99, normalized size = 1.27 \[ -\frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{4} \cos \relax (x)^{5} - 5 \, a b^{3} \cos \relax (x)^{3} - 15 \, b^{4} \cos \relax (x)^{3} + 15 \, a^{2} b^{2} \cos \relax (x) + 45 \, a b^{3} \cos \relax (x) + 45 \, b^{4} \cos \relax (x)}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 138, normalized size = 1.77 \[ \frac {\cos ^{5}\relax (x )}{5 b}-\frac {\left (\cos ^{3}\relax (x )\right ) a}{3 b^{2}}-\frac {\cos ^{3}\relax (x )}{b}+\frac {a^{2} \cos \relax (x )}{b^{3}}+\frac {3 a \cos \relax (x )}{b^{2}}+\frac {3 \cos \relax (x )}{b}-\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right ) a^{3}}{b^{3} \sqrt {a b}}-\frac {3 \arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right ) a^{2}}{b^{2} \sqrt {a b}}-\frac {3 \arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right ) a}{b \sqrt {a b}}-\frac {\arctan \left (\frac {\cos \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 87, normalized size = 1.12 \[ -\frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \cos \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{2} \cos \relax (x)^{5} - 5 \, {\left (a b + 3 \, b^{2}\right )} \cos \relax (x)^{3} + 15 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \cos \relax (x)}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 100, normalized size = 1.28 \[ \cos \relax (x)\,\left (\frac {3}{b}+\frac {a\,\left (\frac {a}{b^2}+\frac {3}{b}\right )}{b}\right )-{\cos \relax (x)}^3\,\left (\frac {a}{3\,b^2}+\frac {1}{b}\right )+\frac {{\cos \relax (x)}^5}{5\,b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\cos \relax (x)\,{\left (a+b\right )}^3}{\sqrt {a}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )\,{\left (a+b\right )}^3}{\sqrt {a}\,b^{7/2}} \]
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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