Optimal. Leaf size=85 \[ -\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{\sec ^5(c+d x)}{a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.271521, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2875, 2873, 2606, 270, 2607, 30, 194} \[ -\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{\sec ^5(c+d x)}{a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2606
Rule 270
Rule 2607
Rule 30
Rule 194
Rubi steps
\begin{align*} \int \frac{\sin (c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \sec ^3(c+d x) (a-a \sin (c+d x))^2 \tan ^5(c+d x) \, dx}{a^4}\\ &=\frac{\int \left (a^2 \sec ^3(c+d x) \tan ^5(c+d x)-2 a^2 \sec ^2(c+d x) \tan ^6(c+d x)+a^2 \sec (c+d x) \tan ^7(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sec ^3(c+d x) \tan ^5(c+d x) \, dx}{a^2}+\frac{\int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^2}-\frac{2 \int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{\operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac{\sec (c+d x)}{a^2 d}+\frac{4 \sec ^3(c+d x)}{3 a^2 d}-\frac{\sec ^5(c+d x)}{a^2 d}+\frac{2 \sec ^7(c+d x)}{7 a^2 d}-\frac{2 \tan ^7(c+d x)}{7 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.263848, size = 126, normalized size = 1.48 \[ -\frac{\sec ^3(c+d x) (28 \sin (c+d x)-104 \sin (2 (c+d x))+66 \sin (3 (c+d x))-52 \sin (4 (c+d x))+6 \sin (5 (c+d x))-182 \cos (c+d x)+104 \cos (2 (c+d x))-39 \cos (3 (c+d x))-18 \cos (4 (c+d x))+13 \cos (5 (c+d x))+42)}{336 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 145, normalized size = 1.7 \begin{align*} 64\,{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{768\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{3}}}-{\frac{1}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+{\frac{1}{256\,\tan \left ( 1/2\,dx+c/2 \right ) -256}}+{\frac{1}{112\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}-1/32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-6}+1/32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}-{\frac{5}{768\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{3}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-{\frac{1}{256\,\tan \left ( 1/2\,dx+c/2 \right ) +256}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11376, size = 400, normalized size = 4.71 \begin{align*} -\frac{16 \,{\left (\frac{4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{14 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}}{21 \,{\left (a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, 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{14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{14 \, 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{3 \, 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}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6397, size = 263, normalized size = 3.09 \begin{align*} -\frac{9 \, \cos \left (d x + c\right )^{4} - 22 \, \cos \left (d x + c\right )^{2} - 2 \,{\left (3 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 5}{21 \,{\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\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 = 1.2793, size = 197, normalized size = 2.32 \begin{align*} \frac{\frac{7 \,{\left (6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{42 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1015 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1750 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1344 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 511 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 79}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{168 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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