Optimal. Leaf size=71 \[ \frac{2 \tan (c+d x)}{5 a^2 d}-\frac{\sec (c+d x)}{5 d \left (a^2 \sin (c+d x)+a^2\right )}-\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.09312, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2672, 3767, 8} \[ \frac{2 \tan (c+d x)}{5 a^2 d}-\frac{\sec (c+d x)}{5 d \left (a^2 \sin (c+d x)+a^2\right )}-\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2} \]
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))^2} \, dx &=-\frac{\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}+\frac{3 \int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{5 a}\\ &=-\frac{\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac{2 \int \sec ^2(c+d x) \, dx}{5 a^2}\\ &=-\frac{\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 a^2 d}\\ &=-\frac{\sec (c+d x)}{5 d (a+a \sin (c+d x))^2}-\frac{\sec (c+d x)}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac{2 \tan (c+d x)}{5 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0756553, size = 53, normalized size = 0.75 \[ -\frac{\sec (c+d x) (-5 \sin (c+d x)+\sin (3 (c+d x))+4 \cos (2 (c+d x)))}{10 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 98, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{d{a}^{2}} \left ( -1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}-2/5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}-3/2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}+5/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}-{\frac{7}{8\,\tan \left ( 1/2\,dx+c/2 \right ) +8}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.98925, size = 275, normalized size = 3.87 \begin{align*} -\frac{2 \,{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{5 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 2\right )}}{5 \,{\left (a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92073, size = 200, normalized size = 2.82 \begin{align*} \frac{4 \, \cos \left (d x + c\right )^{2} +{\left (2 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) - 2}{5 \,{\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} 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 ^{2}{\left (c + d x \right )} + 2 \sin{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15695, size = 126, normalized size = 1.77 \begin{align*} -\frac{\frac{5}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} + \frac{35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 90 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 70 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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