Optimal. Leaf size=127 \[ -\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16} \]
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Rubi [A] time = 0.127833, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3872, 2838, 2592, 302, 206, 2635, 8} \[ -\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2838
Rule 2592
Rule 302
Rule 206
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \sin ^6(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^5(c+d x) \tan (c+d x) \, dx\\ &=a \int \sin ^6(c+d x) \, dx+a \int \sin ^5(c+d x) \tan (c+d x) \, dx\\ &=-\frac{a \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac{1}{6} (5 a) \int \sin ^4(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \frac{x^6}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac{a \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac{1}{8} (5 a) \int \sin ^2(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \left (-1-x^2-x^4+\frac{1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a \sin (c+d x)}{d}-\frac{5 a \cos (c+d x) \sin (c+d x)}{16 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \cos (c+d x) \sin ^5(c+d x)}{6 d}+\frac{1}{16} (5 a) \int 1 \, dx+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{5 a x}{16}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin (c+d x)}{d}-\frac{5 a \cos (c+d x) \sin (c+d x)}{16 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{5 a \cos (c+d x) \sin ^3(c+d x)}{24 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \cos (c+d x) \sin ^5(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.184909, size = 86, normalized size = 0.68 \[ \frac{a \left (-192 \sin ^5(c+d x)-320 \sin ^3(c+d x)-960 \sin (c+d x)+5 (-45 \sin (2 (c+d x))+9 \sin (4 (c+d x))-\sin (6 (c+d x))+60 c+60 d x)+960 \tanh ^{-1}(\sin (c+d x))\right )}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 130, normalized size = 1. \begin{align*} -{\frac{a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{6\,d}}-{\frac{5\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{5\,a\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{16\,d}}+{\frac{5\,ax}{16}}+{\frac{5\,ac}{16\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{a\sin \left ( dx+c \right ) }{d}}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \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 = 0.998566, size = 143, normalized size = 1.13 \begin{align*} -\frac{32 \,{\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a - 5 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87778, size = 290, normalized size = 2.28 \begin{align*} \frac{75 \, a d x + 120 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 120 \, a \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (40 \, a \cos \left (d x + c\right )^{5} + 48 \, a \cos \left (d x + c\right )^{4} - 130 \, a \cos \left (d x + c\right )^{3} - 176 \, a \cos \left (d x + c\right )^{2} + 165 \, a \cos \left (d x + c\right ) + 368 \, a\right )} \sin \left (d x + c\right )}{240 \, 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 [A] time = 1.48382, size = 197, normalized size = 1.55 \begin{align*} \frac{75 \,{\left (d x + c\right )} a + 240 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 240 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (165 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1095 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 3138 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5118 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1945 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, a \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 )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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