Optimal. Leaf size=149 \[ -\frac{2 \cos (c+d x)}{a^2 d}+\frac{9 \tan ^5(c+d x)}{10 a^2 d}-\frac{3 \tan ^3(c+d x)}{2 a^2 d}+\frac{9 \tan (c+d x)}{2 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac{9 x}{2 a^2} \]
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Rubi [A] time = 0.306488, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2875, 2710, 3473, 8, 2590, 270, 2591, 288, 302, 203} \[ -\frac{2 \cos (c+d x)}{a^2 d}+\frac{9 \tan ^5(c+d x)}{10 a^2 d}-\frac{3 \tan ^3(c+d x)}{2 a^2 d}+\frac{9 \tan (c+d x)}{2 a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac{9 x}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2710
Rule 3473
Rule 8
Rule 2590
Rule 270
Rule 2591
Rule 288
Rule 302
Rule 203
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int (a-a \sin (c+d x))^2 \tan ^6(c+d x) \, dx}{a^4}\\ &=\frac{\int \left (a^2 \tan ^6(c+d x)-2 a^2 \sin (c+d x) \tan ^6(c+d x)+a^2 \sin ^2(c+d x) \tan ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \tan ^6(c+d x) \, dx}{a^2}+\frac{\int \sin ^2(c+d x) \tan ^6(c+d x) \, dx}{a^2}-\frac{2 \int \sin (c+d x) \tan ^6(c+d x) \, dx}{a^2}\\ &=\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{\int \tan ^4(c+d x) \, dx}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{x^8}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac{2 \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^6} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}+\frac{\int \tan ^2(c+d x) \, dx}{a^2}+\frac{2 \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^6}-\frac{3}{x^4}+\frac{3}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac{7 \operatorname{Subst}\left (\int \frac{x^6}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}\\ &=-\frac{2 \cos (c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{\tan (c+d x)}{a^2 d}-\frac{\tan ^3(c+d x)}{3 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac{\int 1 \, dx}{a^2}+\frac{7 \operatorname{Subst}\left (\int \left (1-x^2+x^4-\frac{1}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{2 a^2 d}\\ &=-\frac{x}{a^2}-\frac{2 \cos (c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{9 \tan (c+d x)}{2 a^2 d}-\frac{3 \tan ^3(c+d x)}{2 a^2 d}+\frac{9 \tan ^5(c+d x)}{10 a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}\\ &=-\frac{9 x}{2 a^2}-\frac{2 \cos (c+d x)}{a^2 d}-\frac{6 \sec (c+d x)}{a^2 d}+\frac{2 \sec ^3(c+d x)}{a^2 d}-\frac{2 \sec ^5(c+d x)}{5 a^2 d}+\frac{9 \tan (c+d x)}{2 a^2 d}-\frac{3 \tan ^3(c+d x)}{2 a^2 d}+\frac{9 \tan ^5(c+d x)}{10 a^2 d}-\frac{\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.573373, size = 191, normalized size = 1.28 \[ -\frac{250 \sin (c+d x)+720 c \sin (2 (c+d x))+720 d x \sin (2 (c+d x))-824 \sin (2 (c+d x))+351 \sin (3 (c+d x))+5 \sin (5 (c+d x))+10 (90 c+90 d x-103) \cos (c+d x)+544 \cos (2 (c+d x))-180 c \cos (3 (c+d x))-180 d x \cos (3 (c+d x))+206 \cos (3 (c+d x))-20 \cos (4 (c+d x))+500}{160 a^2 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 267, normalized size = 1.8 \begin{align*} -{\frac{1}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{1}{d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-4\,{\frac{1}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-9\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}}-{\frac{4}{5\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}+2\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+{\frac{1}{d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{7}{2\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{31}{4\,d{a}^{2}} \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 = 1.5556, size = 568, normalized size = 3.81 \begin{align*} -\frac{\frac{\frac{211 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{268 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{212 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{84 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{174 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{300 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{300 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{180 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{45 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 64}{a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{6 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{7 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac{45 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15781, size = 350, normalized size = 2.35 \begin{align*} -\frac{45 \, d x \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{4} - 90 \, d x \cos \left (d x + c\right ) - 78 \, \cos \left (d x + c\right )^{2} -{\left (5 \, \cos \left (d x + c\right )^{4} + 90 \, d x \cos \left (d x + c\right ) + 84 \, \cos \left (d x + c\right )^{2} - 6\right )} \sin \left (d x + c\right ) + 4}{10 \,{\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} 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 = 1.29314, size = 216, normalized size = 1.45 \begin{align*} -\frac{\frac{90 \,{\left (d x + c\right )}}{a^{2}} + \frac{20 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, \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 ) + 4\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac{5}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} + \frac{155 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 690 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 750 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 181}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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