Optimal. Leaf size=39 \[ \frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x \]
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Rubi [A] time = 0.10432, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2708, 2746, 12, 2735, 2648} \[ \frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x)}{d (1-\sin (c+d x))}-a x \]
Antiderivative was successfully verified.
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Rule 2708
Rule 2746
Rule 12
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int (a+a \sin (c+d x)) \tan ^2(c+d x) \, dx &=a^2 \int \frac{\sin ^2(c+d x)}{a-a \sin (c+d x)} \, dx\\ &=\frac{a \cos (c+d x)}{d}+a \int \frac{a \sin (c+d x)}{a-a \sin (c+d x)} \, dx\\ &=\frac{a \cos (c+d x)}{d}+a^2 \int \frac{\sin (c+d x)}{a-a \sin (c+d x)} \, dx\\ &=-a x+\frac{a \cos (c+d x)}{d}+a^2 \int \frac{1}{a-a \sin (c+d x)} \, dx\\ &=-a x+\frac{a \cos (c+d x)}{d}+\frac{a^2 \cos (c+d x)}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0333655, size = 47, normalized size = 1.21 \[ \frac{a \cos (c+d x)}{d}-\frac{a \tan ^{-1}(\tan (c+d x))}{d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 59, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}+ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +a \left ( \tan \left ( dx+c \right ) -dx-c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57888, size = 53, normalized size = 1.36 \begin{align*} -\frac{{\left (d x + c - \tan \left (d x + c\right )\right )} a - a{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.04934, size = 194, normalized size = 4.97 \begin{align*} -\frac{a d x - a \cos \left (d x + c\right )^{2} +{\left (a d x - 2 \, a\right )} \cos \left (d x + c\right ) -{\left (a d x - a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \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 [B] time = 1.25164, size = 109, normalized size = 2.79 \begin{align*} -\frac{{\left (d x + c\right )} a + \frac{2 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \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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