Optimal. Leaf size=102 \[ \frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^4 x}{2} \]
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Rubi [A] time = 0.11388, antiderivative size = 102, 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 = {3791, 2637, 2635, 8, 2633} \[ \frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^4 x}{2} \]
Antiderivative was successfully verified.
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Rule 3791
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (a^4 \cos (c+d x)+4 a^4 \cos ^2(c+d x)+6 a^4 \cos ^3(c+d x)+4 a^4 \cos ^4(c+d x)+a^4 \cos ^5(c+d x)\right ) \, dx\\ &=a^4 \int \cos (c+d x) \, dx+a^4 \int \cos ^5(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^2(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^4(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^4 \sin (c+d x)}{d}+\frac{2 a^4 \cos (c+d x) \sin (c+d x)}{d}+\frac{a^4 \cos ^3(c+d x) \sin (c+d x)}{d}+\left (2 a^4\right ) \int 1 \, dx+\left (3 a^4\right ) \int \cos ^2(c+d x) \, dx-\frac{a^4 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{\left (6 a^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=2 a^4 x+\frac{8 a^4 \sin (c+d x)}{d}+\frac{7 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^4 \cos ^3(c+d x) \sin (c+d x)}{d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin ^5(c+d x)}{5 d}+\frac{1}{2} \left (3 a^4\right ) \int 1 \, dx\\ &=\frac{7 a^4 x}{2}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{7 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^4 \cos ^3(c+d x) \sin (c+d x)}{d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.146386, size = 63, normalized size = 0.62 \[ \frac{a^4 (1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x))+840 d x)}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 133, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+4\,{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +2\,{a}^{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) +4\,{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{4}\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15616, size = 173, normalized size = 1.7 \begin{align*} \frac{8 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} - 240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, a^{4} \sin \left (d x + c\right )}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6863, size = 190, normalized size = 1.86 \begin{align*} \frac{105 \, a^{4} d x +{\left (6 \, a^{4} \cos \left (d x + c\right )^{4} + 30 \, a^{4} \cos \left (d x + c\right )^{3} + 68 \, a^{4} \cos \left (d x + c\right )^{2} + 105 \, a^{4} \cos \left (d x + c\right ) + 166 \, a^{4}\right )} \sin \left (d x + c\right )}{30 \, 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 = 1.38698, size = 151, normalized size = 1.48 \begin{align*} \frac{105 \,{\left (d x + c\right )} a^{4} + \frac{2 \,{\left (105 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 490 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 896 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 790 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 375 \, 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 )}^{5}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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