Optimal. Leaf size=79 \[ -\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.119843, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2839, 2622, 302, 207, 3767} \[ -\frac{\tan ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}+\frac{\sec (c+d x)}{a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2622
Rule 302
Rule 207
Rule 3767
Rubi steps
\begin{align*} \int \frac{\csc (c+d x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \sec ^4(c+d x) \, dx}{a}+\frac{\int \csc (c+d x) \sec ^4(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a d}+\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=-\frac{\tan (c+d x)}{a d}-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{\sec (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}-\frac{\tan ^3(c+d x)}{3 a d}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{a d}+\frac{\sec ^3(c+d x)}{3 a d}-\frac{\tan (c+d x)}{a d}-\frac{\tan ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.606308, size = 149, normalized size = 1.89 \[ \frac{6 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-6 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\sin \left (\frac{1}{2} (c+d x)\right ) \left (-\frac{11}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-\frac{2}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{3}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )+\frac{1}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}}{6 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 103, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{2}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{1}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{5}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01971, size = 184, normalized size = 2.33 \begin{align*} \frac{\frac{2 \,{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 4\right )}}{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}}} + \frac{3 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12055, size = 333, normalized size = 4.22 \begin{align*} \frac{4 \, \cos \left (d x + c\right )^{2} - 3 \,{\left (\cos \left (d x + c\right ) \sin \left (d x + c\right ) + \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (d x + c\right ) \sin \left (d x + c\right ) + \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \, \sin \left (d x + c\right ) + 4}{6 \,{\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{\csc{\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 = 1.29881, size = 112, normalized size = 1.42 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{3}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} + \frac{15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 13}{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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