Optimal. Leaf size=117 \[ \frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \cos (c+d x)}{d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 a x}{2} \]
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Rubi [A] time = 0.134516, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2838, 2591, 288, 302, 203, 2590, 270} \[ \frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \cos (c+d x)}{d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{5 a x}{2} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2591
Rule 288
Rule 302
Rule 203
Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \sin ^2(c+d x) (a+a \sin (c+d x)) \tan ^4(c+d x) \, dx &=a \int \sin ^2(c+d x) \tan ^4(c+d x) \, dx+a \int \sin ^3(c+d x) \tan ^4(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^4} \, dx,x,\cos (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}-\frac{a \operatorname{Subst}\left (\int \left (3+\frac{1}{x^4}-\frac{3}{x^2}-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{3 a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{(5 a) \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{3 a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{5 a x}{2}-\frac{3 a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}-\frac{3 a \sec (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}-\frac{5 a \tan (c+d x)}{2 d}+\frac{5 a \tan ^3(c+d x)}{6 d}-\frac{a \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.411456, size = 84, normalized size = 0.72 \[ \frac{a \left (-3 \sin (2 (c+d x))-33 \cos (c+d x)+\cos (3 (c+d x))-28 \tan (c+d x)+4 \sec ^3(c+d x)-36 \sec (c+d x)+4 \tan (c+d x) \sec ^2(c+d x)+30 c+30 d x\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 164, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{3\,\cos \left ( dx+c \right ) }}-{\frac{5\,\cos \left ( dx+c \right ) }{3} \left ({\frac{16}{5}}+ \left ( \sin \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) +a \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51775, size = 130, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{3} - \frac{9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a +{\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}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4633, size = 285, normalized size = 2.44 \begin{align*} \frac{a \cos \left (d x + c\right )^{4} - 15 \, a d x \cos \left (d x + c\right ) + 29 \, a \cos \left (d x + c\right )^{2} +{\left (2 \, a \cos \left (d x + c\right )^{4} + 15 \, a d x \cos \left (d x + c\right ) - 15 \, a \cos \left (d x + c\right )^{2} - 4 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{6 \,{\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\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 = 1.28295, size = 246, normalized size = 2.1 \begin{align*} \frac{15 \,{\left (d x + c\right )} a - \frac{3 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1} + \frac{33 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 102 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 200 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 330 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 402 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 410 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 264 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 150 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 61 \, a}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \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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