Optimal. Leaf size=61 \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.0667725, antiderivative size = 61, 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 = {2836, 12, 77, 206} \[ \frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{8 d (a \sin (c+d x)+a)}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 77
Rule 206
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x)) \tan (c+d x) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x}{a (a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \frac{x}{(a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 \operatorname{Subst}\left (\int \left (\frac{1}{4 a (a-x)^3}-\frac{1}{8 a^2 (a+x)^2}-\frac{1}{8 a^2 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{8 d (a+a \sin (c+d x))}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{8 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0291179, size = 74, normalized size = 1.21 \[ \frac{a \sec ^4(c+d x)}{4 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{a \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 92, normalized size = 1.5 \begin{align*}{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a\sin \left ( dx+c \right ) }{8\,d}}-{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{a}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05883, size = 113, normalized size = 1.85 \begin{align*} -\frac{a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49932, size = 343, normalized size = 5.62 \begin{align*} \frac{2 \, a \cos \left (d x + c\right )^{2} -{\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, a \sin \left (d x + c\right ) - 6 \, a}{16 \,{\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\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 = 1.31221, size = 123, normalized size = 2.02 \begin{align*} -\frac{2 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 2 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (a \sin \left (d x + c\right ) + 3 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac{3 \, a \sin \left (d x + c\right )^{2} - 6 \, a \sin \left (d x + c\right ) - a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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