Optimal. Leaf size=96 \[ \frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.125709, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2757, 3770, 3767, 8, 3768} \[ \frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^4 \sec ^5(c+d x) \, dx &=\int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx\\ &=a^4 \int \sec (c+d x) \, dx+a^4 \int \sec ^5(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^2(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^4(c+d x) \, dx+\left (6 a^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^4 \sec (c+d x) \tan (c+d x)}{d}+\frac{a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{1}{4} \left (3 a^4\right ) \int \sec ^3(c+d x) \, dx+\left (3 a^4\right ) \int \sec (c+d x) \, dx-\frac{\left (4 a^4\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{\left (4 a^4\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{27 a^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{1}{8} \left (3 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{27 a^4 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{4 a^4 \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 6.30807, size = 797, normalized size = 8.3 \[ -\frac{35 (\cos (c+d x) a+a)^4 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d}+\frac{35 (\cos (c+d x) a+a)^4 \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{128 d}+\frac{5 (\cos (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{5 (\cos (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{(\cos (c+d x) a+a)^4 \left (97 \cos \left (\frac{c}{2}\right )-65 \sin \left (\frac{c}{2}\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{768 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{(\cos (c+d x) a+a)^4 \left (-97 \cos \left (\frac{c}{2}\right )-65 \sin \left (\frac{c}{2}\right )\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{768 d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}+\frac{(\cos (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}+\frac{(\cos (c+d x) a+a)^4 \sin \left (\frac{d x}{2}\right ) \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3}+\frac{(\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{256 d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4}-\frac{(\cos (c+d x) a+a)^4 \sec ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{256 d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 102, normalized size = 1.1 \begin{align*}{\frac{35\,{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{20\,{a}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{27\,{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{4\,{a}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{{a}^{4} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15794, size = 246, normalized size = 2.56 \begin{align*} \frac{64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4} - 3 \, a^{4}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, a^{4}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 192 \, a^{4} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95208, size = 292, normalized size = 3.04 \begin{align*} \frac{105 \, a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (160 \, a^{4} \cos \left (d x + c\right )^{3} + 81 \, a^{4} \cos \left (d x + c\right )^{2} + 32 \, a^{4} \cos \left (d x + c\right ) + 6 \, a^{4}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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 = 1.52136, size = 165, normalized size = 1.72 \begin{align*} \frac{105 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (105 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 385 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 511 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 279 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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