Optimal. Leaf size=42 \[ \frac{2 \tan (c+d x)}{3 a d}-\frac{\sec (c+d x)}{3 d (a \sin (c+d x)+a)} \]
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Rubi [A] time = 0.0522441, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2672, 3767, 8} \[ \frac{2 \tan (c+d x)}{3 a d}-\frac{\sec (c+d x)}{3 d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\sec (c+d x)}{3 d (a+a \sin (c+d x))}+\frac{2 \int \sec ^2(c+d x) \, dx}{3 a}\\ &=-\frac{\sec (c+d x)}{3 d (a+a \sin (c+d x))}-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a d}\\ &=-\frac{\sec (c+d x)}{3 d (a+a \sin (c+d x))}+\frac{2 \tan (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.0547872, size = 45, normalized size = 1.07 \[ \frac{2 \tan (c+d x)-\cos (2 (c+d x)) \sec (c+d x)}{3 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 70, normalized size = 1.7 \begin{align*} 2\,{\frac{1}{da} \left ( -1/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}-1/3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}+1/2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}-3/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.948407, size = 174, normalized size = 4.14 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}}{3 \,{\left (a + \frac{2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6413, size = 131, normalized size = 3.12 \begin{align*} -\frac{2 \, \cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 1}{3 \,{\left (a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec ^{2}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13329, size = 90, normalized size = 2.14 \begin{align*} -\frac{\frac{3}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} + \frac{9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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