Optimal. Leaf size=178 \[ -\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{x}{a^3} \]
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Rubi [A] time = 0.367597, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2875, 2873, 2606, 270, 2607, 30, 194, 3473, 8} \[ -\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{3 \sec (c+d x)}{a^3 d}+\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2606
Rule 270
Rule 2607
Rule 30
Rule 194
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \sec ^3(c+d x) (a-a \sin (c+d x))^3 \tan ^7(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (a^3 \sec ^3(c+d x) \tan ^7(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^8(c+d x)+3 a^3 \sec (c+d x) \tan ^9(c+d x)-a^3 \tan ^{10}(c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \sec ^3(c+d x) \tan ^7(c+d x) \, dx}{a^3}-\frac{\int \tan ^{10}(c+d x) \, dx}{a^3}-\frac{3 \int \sec ^2(c+d x) \tan ^8(c+d x) \, dx}{a^3}+\frac{3 \int \sec (c+d x) \tan ^9(c+d x) \, dx}{a^3}\\ &=-\frac{\tan ^9(c+d x)}{9 a^3 d}+\frac{\int \tan ^8(c+d x) \, dx}{a^3}+\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^8 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac{3 \operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{\int \tan ^6(c+d x) \, dx}{a^3}+\frac{\operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac{3 \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{3 \sec (c+d x)}{a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\int \tan ^4(c+d x) \, dx}{a^3}\\ &=\frac{3 \sec (c+d x)}{a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \tan ^9(c+d x)}{9 a^3 d}-\frac{\int \tan ^2(c+d x) \, dx}{a^3}\\ &=\frac{3 \sec (c+d x)}{a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\int 1 \, dx}{a^3}\\ &=\frac{x}{a^3}+\frac{3 \sec (c+d x)}{a^3 d}-\frac{13 \sec ^3(c+d x)}{3 a^3 d}+\frac{21 \sec ^5(c+d x)}{5 a^3 d}-\frac{15 \sec ^7(c+d x)}{7 a^3 d}+\frac{4 \sec ^9(c+d x)}{9 a^3 d}-\frac{\tan (c+d x)}{a^3 d}+\frac{\tan ^3(c+d x)}{3 a^3 d}-\frac{\tan ^5(c+d x)}{5 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \tan ^9(c+d x)}{9 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.51058, size = 273, normalized size = 1.53 \[ \frac{93312 \sin (c+d x)+272160 (c+d x) \sin (2 (c+d x))-506277 \sin (2 (c+d x))+125248 \sin (3 (c+d x))+120960 (c+d x) \sin (4 (c+d x))-225012 \sin (4 (c+d x))+67776 \sin (5 (c+d x))-10080 (c+d x) \sin (6 (c+d x))+18751 \sin (6 (c+d x))+362880 (c+d x) \cos (c+d x)-675036 \cos (c+d x)+173952 \cos (2 (c+d x))+20160 (c+d x) \cos (3 (c+d x))-37502 \cos (3 (c+d x))+54912 \cos (4 (c+d x))-60480 (c+d x) \cos (5 (c+d x))+112506 \cos (5 (c+d x))-21376 \cos (6 (c+d x))+169344}{322560 d (a \sin (c+d x)+a)^3 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.16, size = 272, normalized size = 1.5 \begin{align*} -{\frac{1}{24\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{7}{32\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}+{\frac{8}{9\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-9}}-4\,{\frac{1}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{8}}}+{\frac{40}{7\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-7}}-{\frac{4}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-6}}-{\frac{21}{10\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-{\frac{3}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}+{\frac{3}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{13}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{57}{32\,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 = 1.7886, size = 657, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8831, size = 478, normalized size = 2.69 \begin{align*} \frac{945 \, d x \cos \left (d x + c\right )^{5} + 668 \, \cos \left (d x + c\right )^{6} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1431 \, \cos \left (d x + c\right )^{4} + 465 \, \cos \left (d x + c\right )^{2} +{\left (315 \, d x \cos \left (d x + c\right )^{5} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1059 \, \cos \left (d x + c\right )^{4} + 305 \, \cos \left (d x + c\right )^{2} - 35\right )} \sin \left (d x + c\right ) - 70}{315 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} +{\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\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 = 1.29692, size = 244, normalized size = 1.37 \begin{align*} \frac{\frac{10080 \,{\left (d x + c\right )}}{a^{3}} + \frac{105 \,{\left (21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 23\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} + \frac{17955 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 160020 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 624960 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1387260 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1884582 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1556268 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 774792 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 215748 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 25967}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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