Optimal. Leaf size=78 \[ -\frac {\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}+\frac {(a+3 b) \sin ^3(x)}{3 b^2}-\frac {\sin ^5(x)}{5 b} \]
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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} \[ -\frac {\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac {(a+3 b) \sin ^3(x)}{3 b^2}+\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {\sin ^5(x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos ^7(x)}{a+b \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a+b x^2} \, dx,x,\sin (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,\sin (x)\right )\\ &=-\frac {\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac {(a+3 b) \sin ^3(x)}{3 b^2}-\frac {\sin ^5(x)}{5 b}+\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{b^3}\\ &=\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {\left (a^2+3 a b+3 b^2\right ) \sin (x)}{b^3}+\frac {(a+3 b) \sin ^3(x)}{3 b^2}-\frac {\sin ^5(x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 109, normalized size = 1.40 \[ \frac {-2 \sqrt {a} \sqrt {b} \sin (x) \left (120 a^2+4 b (5 a+12 b) \cos (2 x)+340 a b+3 b^2 \cos (4 x)+309 b^2\right )+120 (a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )-120 (a+b)^3 \tan ^{-1}\left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{240 \sqrt {a} b^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 233, normalized size = 2.99 \[ \left [-\frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right ) + 2 \, {\left (3 \, a b^{3} \cos \relax (x)^{4} + 15 \, a^{3} b + 40 \, a^{2} b^{2} + 33 \, a b^{3} + {\left (5 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{30 \, a b^{4}}, \frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \relax (x)}{a}\right ) - {\left (3 \, a b^{3} \cos \relax (x)^{4} + 15 \, a^{3} b + 40 \, a^{2} b^{2} + 33 \, a b^{3} + {\left (5 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{15 \, a b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 98, normalized size = 1.26 \[ \frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} - \frac {3 \, b^{4} \sin \relax (x)^{5} - 5 \, a b^{3} \sin \relax (x)^{3} - 15 \, b^{4} \sin \relax (x)^{3} + 15 \, a^{2} b^{2} \sin \relax (x) + 45 \, a b^{3} \sin \relax (x) + 45 \, b^{4} \sin \relax (x)}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 136, normalized size = 1.74 \[ -\frac {\sin ^{5}\relax (x )}{5 b}+\frac {a \left (\sin ^{3}\relax (x )\right )}{3 b^{2}}+\frac {\sin ^{3}\relax (x )}{b}-\frac {a^{2} \sin \relax (x )}{b^{3}}-\frac {3 a \sin \relax (x )}{b^{2}}-\frac {3 \sin \relax (x )}{b}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right ) a^{3}}{b^{3} \sqrt {a b}}+\frac {3 \arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right ) a^{2}}{b^{2} \sqrt {a b}}+\frac {3 \arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right ) a}{b \sqrt {a b}}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 86, normalized size = 1.10 \[ \frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} - \frac {3 \, b^{2} \sin \relax (x)^{5} - 5 \, {\left (a b + 3 \, b^{2}\right )} \sin \relax (x)^{3} + 15 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \sin \relax (x)}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 99, normalized size = 1.27 \[ {\sin \relax (x)}^3\,\left (\frac {a}{3\,b^2}+\frac {1}{b}\right )-\sin \relax (x)\,\left (\frac {3}{b}+\frac {a\,\left (\frac {a}{b^2}+\frac {3}{b}\right )}{b}\right )-\frac {{\sin \relax (x)}^5}{5\,b}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \relax (x)\,{\left (a+b\right )}^3}{\sqrt {a}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )\,{\left (a+b\right )}^3}{\sqrt {a}\,b^{7/2}} \]
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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