Optimal. Leaf size=245 \[ \frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \]
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Rubi [A] time = 0.40397, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2672, 3767, 8} \[ \frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}+\frac{9 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{17 a}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}+\frac{24 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{85 a^2}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}+\frac{168 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{1105 a^3}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}+\frac{1008 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{12155 a^4}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{112 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{2431 a^5}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac{64 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{2431 a^6}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{192 \int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{12155 a^7}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{128 \int \sec ^2(c+d x) \, dx}{12155 a^8}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}-\frac{128 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{12155 a^8 d}\\ &=-\frac{\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac{3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac{24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac{168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac{16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac{112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac{64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{128 \tan (c+d x)}{12155 a^8 d}\\ \end{align*}
Mathematica [A] time = 0.342428, size = 113, normalized size = 0.46 \[ \frac{\sec (c+d x) (4862 \sin (c+d x)-6188 \sin (3 (c+d x))+1700 \sin (5 (c+d x))-119 \sin (7 (c+d x))+\sin (9 (c+d x))-7072 \cos (2 (c+d x))+3808 \cos (4 (c+d x))-544 \cos (6 (c+d x))+16 \cos (8 (c+d x)))}{24310 a^8 d (\sin (c+d x)+1)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 280, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ( -{\frac{1}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512}}-{\frac{128}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{17}}}+64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-16}-{\frac{1376}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{15}}}+784\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-14}-{\frac{21400}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{13}}}+2692\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-12}-{\frac{38954}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{11}}}+{\frac{19109}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{10}}}-{\frac{6847}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}+{\frac{10241}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{8}}}-{\frac{12799}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+{\frac{13313}{16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}-{\frac{57083}{160\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}+{\frac{7937}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-{\frac{4351}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}+{\frac{1793}{256\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-{\frac{511}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15014, size = 999, normalized size = 4.08 \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 = 1.87013, size = 624, normalized size = 2.55 \begin{align*} \frac{1024 \, \cos \left (d x + c\right )^{8} - 10752 \, \cos \left (d x + c\right )^{6} + 29568 \, \cos \left (d x + c\right )^{4} - 27456 \, \cos \left (d x + c\right )^{2} +{\left (128 \, \cos \left (d x + c\right )^{8} - 4032 \, \cos \left (d x + c\right )^{6} + 18480 \, \cos \left (d x + c\right )^{4} - 24024 \, \cos \left (d x + c\right )^{2} + 6435\right )} \sin \left (d x + c\right ) + 5720}{12155 \,{\left (a^{8} d \cos \left (d x + c\right )^{9} - 32 \, a^{8} d \cos \left (d x + c\right )^{7} + 160 \, a^{8} d \cos \left (d x + c\right )^{5} - 256 \, a^{8} d \cos \left (d x + c\right )^{3} + 128 \, a^{8} d \cos \left (d x + c\right ) - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{7} - 10 \, a^{8} d \cos \left (d x + c\right )^{5} + 24 \, a^{8} d \cos \left (d x + c\right )^{3} - 16 \, a^{8} 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 = 1.21859, size = 336, normalized size = 1.37 \begin{align*} -\frac{\frac{12155}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} + \frac{6211205 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} + 55791450 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 303072770 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 1091397450 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 5901218466 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 9405145178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 11877161010 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 9710430158 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 6263238566 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3172666718 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 365303990 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 77883902 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10498214 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 982907}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{17}}}{3111680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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