Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sin (x)}{2 a \left (a+b \sin ^2(x)\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3190, 199, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sin (x)}{2 a \left (a+b \sin ^2(x)\right )} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=\frac {\sin (x)}{2 a \left (a+b \sin ^2(x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sin (x)}{2 a \left (a+b \sin ^2(x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 48, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sin (x)}{2 a \left (a+b \sin ^2(x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 165, normalized size = 3.44 \[ \left [-\frac {2 \, a b \sin \relax (x) + {\left (b \cos \relax (x)^{2} - a - b\right )} \sqrt {-a b} \log \left (-\frac {b \cos \relax (x)^{2} + 2 \, \sqrt {-a b} \sin \relax (x) + a - b}{b \cos \relax (x)^{2} - a - b}\right )}{4 \, {\left (a^{2} b^{2} \cos \relax (x)^{2} - a^{3} b - a^{2} b^{2}\right )}}, -\frac {a b \sin \relax (x) - {\left (b \cos \relax (x)^{2} - a - b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \relax (x)}{a}\right )}{2 \, {\left (a^{2} b^{2} \cos \relax (x)^{2} - a^{3} b - a^{2} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 38, normalized size = 0.79 \[ \frac {\arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} + \frac {\sin \relax (x)}{2 \, {\left (b \sin \relax (x)^{2} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 39, normalized size = 0.81 \[ \frac {\sin \relax (x )}{2 a \left (a +b \left (\sin ^{2}\relax (x )\right )\right )}+\frac {\arctan \left (\frac {\sin \relax (x ) b}{\sqrt {a b}}\right )}{2 a \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 38, normalized size = 0.79 \[ \frac {\sin \relax (x)}{2 \, {\left (a b \sin \relax (x)^{2} + a^{2}\right )}} + \frac {\arctan \left (\frac {b \sin \relax (x)}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.38, size = 36, normalized size = 0.75 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \relax (x)}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}}+\frac {\sin \relax (x)}{2\,a\,\left (b\,{\sin \relax (x)}^2+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.38, size = 340, normalized size = 7.08 \[ \begin {cases} \frac {\tilde {\infty }}{\sin ^{3}{\relax (x )}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 b^{2} \sin ^{3}{\relax (x )}} & \text {for}\: a = 0 \\\frac {\sin {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \\\frac {2 i \sqrt {a} b \sqrt {\frac {1}{b}} \sin {\relax (x )}}{4 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 4 i a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} \sin ^{2}{\relax (x )}} + \frac {a \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sin {\relax (x )} \right )}}{4 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 4 i a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} \sin ^{2}{\relax (x )}} - \frac {a \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sin {\relax (x )} \right )}}{4 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 4 i a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} \sin ^{2}{\relax (x )}} + \frac {b \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sin {\relax (x )} \right )} \sin ^{2}{\relax (x )}}{4 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 4 i a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} \sin ^{2}{\relax (x )}} - \frac {b \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sin {\relax (x )} \right )} \sin ^{2}{\relax (x )}}{4 i a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 4 i a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} \sin ^{2}{\relax (x )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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