Optimal. Leaf size=151 \[ \frac{\cos (c+d x)}{a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}+\frac{3 x}{a^3} \]
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Rubi [A] time = 0.34146, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2875, 2873, 2607, 30, 2606, 194, 3473, 8, 2590, 270} \[ \frac{\cos (c+d x)}{a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}+\frac{3 x}{a^3} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2607
Rule 30
Rule 2606
Rule 194
Rule 3473
Rule 8
Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \sec ^2(c+d x) (a-a \sin (c+d x))^3 \tan ^6(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (a^3 \sec ^2(c+d x) \tan ^6(c+d x)-3 a^3 \sec (c+d x) \tan ^7(c+d x)+3 a^3 \tan ^8(c+d x)-a^3 \sin (c+d x) \tan ^8(c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^3}-\frac{\int \sin (c+d x) \tan ^8(c+d x) \, dx}{a^3}-\frac{3 \int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^3}+\frac{3 \int \tan ^8(c+d x) \, dx}{a^3}\\ &=\frac{3 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \int \tan ^6(c+d x) \, dx}{a^3}+\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^4}{x^8} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{3 \int \tan ^4(c+d x) \, dx}{a^3}+\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^8}-\frac{4}{x^6}+\frac{6}{x^4}-\frac{4}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{\cos (c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}-\frac{3 \int \tan ^2(c+d x) \, dx}{a^3}\\ &=\frac{\cos (c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}+\frac{3 \int 1 \, dx}{a^3}\\ &=\frac{3 x}{a^3}+\frac{\cos (c+d x)}{a^3 d}+\frac{7 \sec (c+d x)}{a^3 d}-\frac{5 \sec ^3(c+d x)}{a^3 d}+\frac{13 \sec ^5(c+d x)}{5 a^3 d}-\frac{4 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{\tan ^3(c+d x)}{a^3 d}-\frac{3 \tan ^5(c+d x)}{5 a^3 d}+\frac{4 \tan ^7(c+d x)}{7 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.650508, size = 224, normalized size = 1.48 \[ \frac{8008 \sin (c+d x)+11760 c \sin (2 (c+d x))+11760 d x \sin (2 (c+d x))-20762 \sin (2 (c+d x))+6588 \sin (3 (c+d x))-840 c \sin (4 (c+d x))-840 d x \sin (4 (c+d x))+1483 \sin (4 (c+d x))-140 \sin (5 (c+d x))+14 (840 c+840 d x-1483) \cos (c+d x)+5152 \cos (2 (c+d x))-5040 c \cos (3 (c+d x))-5040 d x \cos (3 (c+d x))+8898 \cos (3 (c+d x))-2288 \cos (4 (c+d x))+8400}{2240 a^3 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.12, size = 211, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+2\,{\frac{1}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}+6\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{8}{7\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-7}}+4\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{14}{5\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-3\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+{\frac{1}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{17}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{49}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56908, size = 568, normalized size = 3.76 \begin{align*} \frac{2 \,{\left (\frac{\frac{951 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{2010 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1980 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{574 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{966 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{1890 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{1540 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{630 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{105 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 176}{a^{3} + \frac{6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{15 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{20 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{14 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{20 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{6 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac{105 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{35 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.14304, size = 431, normalized size = 2.85 \begin{align*} \frac{315 \, d x \cos \left (d x + c\right )^{3} + 286 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 447 \, \cos \left (d x + c\right )^{2} +{\left (105 \, d x \cos \left (d x + c\right )^{3} + 35 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 438 \, \cos \left (d x + c\right )^{2} - 20\right )} \sin \left (d x + c\right ) - 15}{35 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) +{\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \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 = 1.1958, size = 239, normalized size = 1.58 \begin{align*} \frac{\frac{840 \,{\left (d x + c\right )}}{a^{3}} - \frac{35 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 17\right )}}{{\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 )} a^{3}} + \frac{1715 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 11480 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 31815 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 45920 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 35161 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 13832 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2221}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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