Optimal. Leaf size=37 \[ \frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.093388, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2839, 2606, 30, 2607} \[ \frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2606
Rule 30
Rule 2607
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \sec ^3(c+d x) \tan (c+d x) \, dx}{a}-\frac{\int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 0.131107, size = 104, normalized size = 2.81 \[ \frac{-2 \sin (c+d x)+\frac{1}{2} \sin (2 (c+d x))+\cos (c+d x)+\cos (2 (c+d x))-3}{6 a d (\sin (c+d x)+1) \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 70, normalized size = 1.9 \begin{align*} 4\,{\frac{1}{da} \left ( -1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}+1/6\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}-1/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+1/8\, \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 = 1.02291, size = 149, normalized size = 4.03 \begin{align*} \frac{2 \,{\left (\frac{2 \, \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}} + 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 = 1.0251, size = 126, normalized size = 3.41 \begin{align*} -\frac{\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 2}{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{\sin{\left (c + d x \right )} \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 = 1.23009, size = 77, normalized size = 2.08 \begin{align*} -\frac{\frac{3}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} - \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}{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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