Optimal. Leaf size=35 \[ -\frac {2 \tanh ^{-1}\left (\frac {b+2 c \sin (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3258, 618, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {b+2 c \sin (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 3258
Rubi steps
\begin {align*} \int \frac {\cos (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,\sin (x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c \sin (x)\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {b+2 c \sin (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {b+2 c \sin (x)}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 139, normalized size = 3.97 \[ \left [\frac {\log \left (-\frac {2 \, c^{2} \cos \relax (x)^{2} - 2 \, b c \sin \relax (x) - b^{2} + 2 \, a c - 2 \, c^{2} + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c \sin \relax (x) + b\right )}}{c \cos \relax (x)^{2} - b \sin \relax (x) - a - c}\right )}{\sqrt {b^{2} - 4 \, a c}}, -\frac {2 \, \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c \sin \relax (x) + b\right )}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 35, normalized size = 1.00 \[ \frac {2 \, \arctan \left (\frac {2 \, c \sin \relax (x) + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 36, normalized size = 1.03 \[ \frac {2 \arctan \left (\frac {b +2 c \sin \relax (x )}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.07, size = 47, normalized size = 1.34 \[ \frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,\sin \relax (x)}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.23, size = 99, normalized size = 2.83 \[ \begin {cases} \frac {\log {\left (\frac {a}{b} + \sin {\relax (x )} \right )}}{b} & \text {for}\: c = 0 \\- \frac {2}{b + 2 c \sin {\relax (x )}} & \text {for}\: a = \frac {b^{2}}{4 c} \\\frac {\log {\left (\frac {b}{2 c} + \sin {\relax (x )} - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt {- 4 a c + b^{2}}} - \frac {\log {\left (\frac {b}{2 c} + \sin {\relax (x )} + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{\sqrt {- 4 a c + b^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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