Optimal. Leaf size=105 \[ \frac{2 a \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}}+\frac{b c x}{d^2}-\frac{2 b \sqrt{c-d} \sqrt{c+d} \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{d^2}-\frac{b \sin (x)}{d} \]
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Rubi [A] time = 0.259909, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {4401, 2659, 205, 2695, 2735} \[ \frac{2 a \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}}+\frac{b c x}{d^2}-\frac{2 b \sqrt{c-d} \sqrt{c+d} \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{d^2}-\frac{b \sin (x)}{d} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2659
Rule 205
Rule 2695
Rule 2735
Rubi steps
\begin{align*} \int \frac{a+b \sin ^2(x)}{c+d \cos (x)} \, dx &=\int \left (\frac{a}{c+d \cos (x)}+\frac{b \sin ^2(x)}{c+d \cos (x)}\right ) \, dx\\ &=a \int \frac{1}{c+d \cos (x)} \, dx+b \int \frac{\sin ^2(x)}{c+d \cos (x)} \, dx\\ &=-\frac{b \sin (x)}{d}+(2 a) \operatorname{Subst}\left (\int \frac{1}{c+d+(c-d) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{b \int \frac{-d-c \cos (x)}{c+d \cos (x)} \, dx}{d}\\ &=\frac{b c x}{d^2}+\frac{2 a \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}}-\frac{b \sin (x)}{d}+\frac{\left (b \left (-c^2+d^2\right )\right ) \int \frac{1}{c+d \cos (x)} \, dx}{d^2}\\ &=\frac{b c x}{d^2}+\frac{2 a \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}}-\frac{b \sin (x)}{d}+\frac{\left (2 b \left (-c^2+d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+d+(c-d) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{d^2}\\ &=\frac{b c x}{d^2}+\frac{2 a \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{\sqrt{c-d} \sqrt{c+d}}-\frac{2 b \sqrt{c-d} \sqrt{c+d} \tan ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{x}{2}\right )}{\sqrt{c+d}}\right )}{d^2}-\frac{b \sin (x)}{d}\\ \end{align*}
Mathematica [A] time = 0.153592, size = 73, normalized size = 0.7 \[ \frac{-\frac{2 \left (a d^2+b \left (d^2-c^2\right )\right ) \tanh ^{-1}\left (\frac{(c-d) \tan \left (\frac{x}{2}\right )}{\sqrt{d^2-c^2}}\right )}{\sqrt{d^2-c^2}}+b c x-b d \sin (x)}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 148, normalized size = 1.4 \begin{align*} 2\,{\frac{a}{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}\arctan \left ({\frac{ \left ( c-d \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) }-2\,{\frac{b{c}^{2}}{{d}^{2}\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}\arctan \left ({\frac{ \left ( c-d \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) }+2\,{\frac{b}{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}\arctan \left ({\frac{ \left ( c-d \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) }-2\,{\frac{b\tan \left ( x/2 \right ) }{d \left ( 1+ \left ( \tan \left ( x/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{cb\arctan \left ( \tan \left ( x/2 \right ) \right ) }{{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.72175, size = 558, normalized size = 5.31 \begin{align*} \left [\frac{{\left (b c^{2} -{\left (a + b\right )} d^{2}\right )} \sqrt{-c^{2} + d^{2}} \log \left (\frac{2 \, c d \cos \left (x\right ) +{\left (2 \, c^{2} - d^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt{-c^{2} + d^{2}}{\left (c \cos \left (x\right ) + d\right )} \sin \left (x\right ) - c^{2} + 2 \, d^{2}}{d^{2} \cos \left (x\right )^{2} + 2 \, c d \cos \left (x\right ) + c^{2}}\right ) + 2 \,{\left (b c^{3} - b c d^{2}\right )} x - 2 \,{\left (b c^{2} d - b d^{3}\right )} \sin \left (x\right )}{2 \,{\left (c^{2} d^{2} - d^{4}\right )}}, -\frac{{\left (b c^{2} -{\left (a + b\right )} d^{2}\right )} \sqrt{c^{2} - d^{2}} \arctan \left (-\frac{c \cos \left (x\right ) + d}{\sqrt{c^{2} - d^{2}} \sin \left (x\right )}\right ) -{\left (b c^{3} - b c d^{2}\right )} x +{\left (b c^{2} d - b d^{3}\right )} \sin \left (x\right )}{c^{2} d^{2} - d^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14017, size = 149, normalized size = 1.42 \begin{align*} \frac{b c x}{d^{2}} - \frac{2 \, b \tan \left (\frac{1}{2} \, x\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} d} + \frac{2 \,{\left (b c^{2} - a d^{2} - b d^{2}\right )}{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac{c \tan \left (\frac{1}{2} \, x\right ) - d \tan \left (\frac{1}{2} \, x\right )}{\sqrt{c^{2} - d^{2}}}\right )\right )}}{\sqrt{c^{2} - d^{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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