Optimal. Leaf size=48 \[ \frac {2 \tan ^{-1}\left (\frac {2 a \tan (c+d x)+b}{\sqrt {4 a^2-b^2}}\right )}{d \sqrt {4 a^2-b^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2666, 2660, 618, 204} \[ \frac {2 \tan ^{-1}\left (\frac {2 a \tan (c+d x)+b}{\sqrt {4 a^2-b^2}}\right )}{d \sqrt {4 a^2-b^2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2666
Rubi steps
\begin {align*} \int \frac {1}{a+b \cos (c+d x) \sin (c+d x)} \, dx &=\int \frac {1}{a+\frac {1}{2} b \sin (2 c+2 d x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (2 c+2 d x)\right )\right )}{d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-4 a^2+b^2-x^2} \, dx,x,b+2 a \tan \left (\frac {1}{2} (2 c+2 d x)\right )\right )}{d}\\ &=\frac {2 \tan ^{-1}\left (\frac {b+2 a \tan (c+d x)}{\sqrt {4 a^2-b^2}}\right )}{\sqrt {4 a^2-b^2} d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {2 a \tan (c+d x)+b}{\sqrt {4 a^2-b^2}}\right )}{d \sqrt {4 a^2-b^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 290, normalized size = 6.04 \[ \left [-\frac {\sqrt {-4 \, a^{2} + b^{2}} \log \left (-\frac {2 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (8 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a^{2} - b^{2} + {\left (2 \, b \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - b\right )} \sqrt {-4 \, a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{4} - b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - a^{2}}\right )}{2 \, {\left (4 \, a^{2} - b^{2}\right )} d}, -\frac {\arctan \left (-\frac {{\left (4 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right )} \sqrt {4 \, a^{2} - b^{2}}}{2 \, {\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, a^{2} + b^{2}}\right )}{\sqrt {4 \, a^{2} - b^{2}} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 61, normalized size = 1.27 \[ \frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {2 \, a \tan \left (d x + c\right ) + b}{\sqrt {4 \, a^{2} - b^{2}}}\right )\right )}}{\sqrt {4 \, a^{2} - b^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 45, normalized size = 0.94 \[ \frac {2 \arctan \left (\frac {b +2 a \tan \left (d x +c \right )}{\sqrt {4 a^{2}-b^{2}}}\right )}{d \sqrt {4 a^{2}-b^{2}}} \]
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 = 3.11, size = 44, normalized size = 0.92 \[ \frac {2\,\mathrm {atan}\left (\frac {b+2\,a\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {4\,a^2-b^2}}\right )}{d\,\sqrt {4\,a^2-b^2}} \]
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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