Optimal. Leaf size=101 \[ -\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^4 x}{2} \]
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Rubi [A] time = 0.16332, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2872, 2650, 2648, 2638, 2635, 8} \[ -\frac{4 a^4 \cos (c+d x)}{d}-\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{17 a^4 x}{2} \]
Antiderivative was successfully verified.
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Rule 2872
Rule 2650
Rule 2648
Rule 2638
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx &=a^4 \int \left (8+\frac{4}{(-1+\sin (c+d x))^2}+\frac{12}{-1+\sin (c+d x)}+4 \sin (c+d x)+\sin ^2(c+d x)\right ) \, dx\\ &=8 a^4 x+a^4 \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \frac{1}{(-1+\sin (c+d x))^2} \, dx+\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (12 a^4\right ) \int \frac{1}{-1+\sin (c+d x)} \, dx\\ &=8 a^4 x-\frac{4 a^4 \cos (c+d x)}{d}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{12 a^4 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a^4 \int 1 \, dx-\frac{1}{3} \left (4 a^4\right ) \int \frac{1}{-1+\sin (c+d x)} \, dx\\ &=\frac{17 a^4 x}{2}-\frac{4 a^4 \cos (c+d x)}{d}+\frac{4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac{a^4 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.9119, size = 158, normalized size = 1.56 \[ -\frac{a^4 \left (-3 (204 c+204 d x+161) \cos \left (\frac{1}{2} (c+d x)\right )+(204 c+204 d x+647) \cos \left (\frac{3}{2} (c+d x)\right )-39 \cos \left (\frac{5}{2} (c+d x)\right )+3 \cos \left (\frac{7}{2} (c+d x)\right )+6 \sin \left (\frac{1}{2} (c+d x)\right ) ((68 c+68 d x-59) \cos (c+d x)-14 \cos (2 (c+d x))-\cos (3 (c+d x))+136 c+136 d x+146)\right )}{48 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.101, size = 268, normalized size = 2.7 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\,\cos \left ( dx+c \right ) }}-{\frac{4\,\cos \left ( dx+c \right ) }{3} \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{2}}+{\frac{5\,c}{2}} \right ) +4\,{a}^{4} \left ( 1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{\cos \left ( dx+c \right ) }}- \left ( 8/3+ \left ( \sin \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +6\,{a}^{4} \left ( 1/3\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}-\tan \left ( dx+c \right ) +dx+c \right ) +4\,{a}^{4} \left ( 1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-1/3\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}-1/3\, \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62033, size = 213, normalized size = 2.11 \begin{align*} \frac{2 \, a^{4} \tan \left (d x + c\right )^{3} +{\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac{3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{4} + 12 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} - 8 \, a^{4}{\left (\frac{6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac{8 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{4}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45128, size = 477, normalized size = 4.72 \begin{align*} \frac{3 \, a^{4} \cos \left (d x + c\right )^{4} - 18 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x - 8 \, a^{4} + 17 \,{\left (3 \, a^{4} d x + 5 \, a^{4}\right )} \cos \left (d x + c\right )^{2} -{\left (51 \, a^{4} d x - 98 \, a^{4}\right )} \cos \left (d x + c\right ) -{\left (3 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x + 21 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} -{\left (51 \, a^{4} d x - 106 \, a^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \,{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\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 = 1.31767, size = 182, normalized size = 1.8 \begin{align*} \frac{51 \,{\left (d x + c\right )} a^{4} + \frac{6 \,{\left (a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, a^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac{16 \,{\left (6 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7 \, a^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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