Optimal. Leaf size=36 \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0742867, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2838, 2622, 321, 207, 3767, 8} \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2622
Rule 321
Rule 207
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \sec ^2(c+d x) \, dx+a \int \csc (c+d x) \sec ^2(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac{a \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{a \sec (c+d x)}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \sec (c+d x)}{d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0322275, size = 56, normalized size = 1.56 \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 47, normalized size = 1.3 \begin{align*}{\frac{a\tan \left ( dx+c \right ) }{d}}+{\frac{a}{d\cos \left ( dx+c \right ) }}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06426, size = 65, normalized size = 1.81 \begin{align*} \frac{a{\left (\frac{2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.08253, size = 302, normalized size = 8.39 \begin{align*} \frac{2 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \, a \sin \left (d x + c\right ) + 2 \, a}{2 \,{\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\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 = 1.26279, size = 46, normalized size = 1.28 \begin{align*} \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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