Optimal. Leaf size=73 \[ \frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{13 a^4 x}{2} \]
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Rubi [A] time = 0.078554, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3791, 2637, 2635, 8, 3770, 3767} \[ \frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{13 a^4 x}{2} \]
Antiderivative was successfully verified.
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Rule 3791
Rule 2637
Rule 2635
Rule 8
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (6 a^4+4 a^4 \cos (c+d x)+a^4 \cos ^2(c+d x)+4 a^4 \sec (c+d x)+a^4 \sec ^2(c+d x)\right ) \, dx\\ &=6 a^4 x+a^4 \int \cos ^2(c+d x) \, dx+a^4 \int \sec ^2(c+d x) \, dx+\left (4 a^4\right ) \int \cos (c+d x) \, dx+\left (4 a^4\right ) \int \sec (c+d x) \, dx\\ &=6 a^4 x+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a^4 \int 1 \, dx-\frac{a^4 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{13 a^4 x}{2}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^4 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 1.77213, size = 241, normalized size = 3.3 \[ \frac{1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac{1}{2} (c+d x)\right ) \left (\frac{16 \sin (c) \cos (d x)}{d}+\frac{\sin (2 c) \cos (2 d x)}{d}+\frac{16 \cos (c) \sin (d x)}{d}+\frac{\cos (2 c) \sin (2 d x)}{d}+\frac{4 \sin \left (\frac{d x}{2}\right )}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{4 \sin \left (\frac{d x}{2}\right )}{d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{16 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{16 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}+26 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 86, normalized size = 1.2 \begin{align*}{\frac{{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{13\,{a}^{4}x}{2}}+{\frac{13\,{a}^{4}c}{2\,d}}+4\,{\frac{{a}^{4}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{4}\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06882, size = 115, normalized size = 1.58 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 24 \,{\left (d x + c\right )} a^{4} + 8 \, a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, a^{4} \sin \left (d x + c\right ) + 4 \, a^{4} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7898, size = 270, normalized size = 3.7 \begin{align*} \frac{13 \, a^{4} d x \cos \left (d x + c\right ) + 4 \, a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} \cos \left (d x + c\right ) + 2 \, a^{4}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\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 = 1.41339, size = 174, normalized size = 2.38 \begin{align*} \frac{13 \,{\left (d x + c\right )} a^{4} + 8 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{4 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} + \frac{2 \,{\left (7 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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