Optimal. Leaf size=255 \[ \frac{\left (3 a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}-\frac{4 a \left (a^2-2 b^2\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left (-12 a^2 b^2+5 a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}-\frac{2 a \left (-8 a^2 b^2+3 a^4+6 b^4\right ) \sec (c+d x)}{b^7 d}+\frac{\left (a^2-b^2\right )^4}{a b^8 d (a+b \sec (c+d x))}+\frac{\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^5(c+d x)}{5 b^3 d}+\frac{\sec ^6(c+d x)}{6 b^2 d} \]
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Rubi [A] time = 0.205109, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3885, 894} \[ \frac{\left (3 a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}-\frac{4 a \left (a^2-2 b^2\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left (-12 a^2 b^2+5 a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}-\frac{2 a \left (-8 a^2 b^2+3 a^4+6 b^4\right ) \sec (c+d x)}{b^7 d}+\frac{\left (a^2-b^2\right )^4}{a b^8 d (a+b \sec (c+d x))}+\frac{\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 a \sec ^5(c+d x)}{5 b^3 d}+\frac{\sec ^6(c+d x)}{6 b^2 d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\tan ^9(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^4}{x (a+x)^2} \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a \left (3 a^4-8 a^2 b^2+6 b^4\right )+\frac{b^8}{a^2 x}+\left (5 a^4-12 a^2 b^2+6 b^4\right ) x-4 a \left (a^2-2 b^2\right ) x^2+\left (3 a^2-4 b^2\right ) x^3-2 a x^4+x^5-\frac{\left (a^2-b^2\right )^4}{a (a+x)^2}+\frac{\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right )}{a^2 (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\left (a^2-b^2\right )^3 \left (7 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^8 d}-\frac{2 a \left (3 a^4-8 a^2 b^2+6 b^4\right ) \sec (c+d x)}{b^7 d}+\frac{\left (5 a^4-12 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}-\frac{4 a \left (a^2-2 b^2\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac{\left (3 a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}-\frac{2 a \sec ^5(c+d x)}{5 b^3 d}+\frac{\sec ^6(c+d x)}{6 b^2 d}+\frac{\left (a^2-b^2\right )^4}{a b^8 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 6.27456, size = 528, normalized size = 2.07 \[ \frac{\left (3 a^2-4 b^2\right ) \sec ^6(c+d x) (a \cos (c+d x)+b)^2}{4 b^4 d (a+b \sec (c+d x))^2}+\frac{4 a \left (2 b^2-a^2\right ) \sec ^5(c+d x) (a \cos (c+d x)+b)^2}{3 b^5 d (a+b \sec (c+d x))^2}+\frac{\left (-12 a^2 b^2+5 a^4+6 b^4\right ) \sec ^4(c+d x) (a \cos (c+d x)+b)^2}{2 b^6 d (a+b \sec (c+d x))^2}-\frac{2 a \left (-8 a^2 b^2+3 a^4+6 b^4\right ) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{b^7 d (a+b \sec (c+d x))^2}-\frac{(b-a)^4 (a+b)^4 \sec ^2(c+d x) (a \cos (c+d x)+b)}{a^2 b^7 d (a+b \sec (c+d x))^2}+\frac{\left (20 a^4 b^2-18 a^2 b^4-7 a^6+4 b^6\right ) \sec ^2(c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)^2}{b^8 d (a+b \sec (c+d x))^2}+\frac{\left (-20 a^6 b^2+18 a^4 b^4-4 a^2 b^6+7 a^8-b^8\right ) \sec ^2(c+d x) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{a^2 b^8 d (a+b \sec (c+d x))^2}+\frac{\sec ^8(c+d x) (a \cos (c+d x)+b)^2}{6 b^2 d (a+b \sec (c+d x))^2}-\frac{2 a \sec ^7(c+d x) (a \cos (c+d x)+b)^2}{5 b^3 d (a+b \sec (c+d x))^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.072, size = 498, normalized size = 2. \begin{align*} -{\frac{{a}^{6}}{d{b}^{7} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+4\,{\frac{{a}^{4}}{d{b}^{5} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-6\,{\frac{{a}^{2}}{d{b}^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{b}{d{a}^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-7\,{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ){a}^{6}}{d{b}^{8}}}+20\,{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ){a}^{4}}{d{b}^{6}}}-18\,{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{4}}}-{\frac{2\,a}{5\,d{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{4\,{a}^{3}}{3\,d{b}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{8\,a}{3\,d{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-6\,{\frac{{a}^{5}}{d{b}^{7}\cos \left ( dx+c \right ) }}+16\,{\frac{{a}^{3}}{d{b}^{5}\cos \left ( dx+c \right ) }}-12\,{\frac{a}{d{b}^{3}\cos \left ( dx+c \right ) }}+7\,{\frac{{a}^{6}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{8}}}-20\,{\frac{{a}^{4}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{6}}}+18\,{\frac{{a}^{2}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{4}}}+{\frac{3\,{a}^{2}}{4\,d{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{5\,{a}^{4}}{2\,d{b}^{6} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-6\,{\frac{{a}^{2}}{d{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}-{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{2}}}+4\,{\frac{1}{db \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{1}{d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+3\,{\frac{1}{d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}+{\frac{1}{6\,d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02159, size = 433, normalized size = 1.7 \begin{align*} -\frac{\frac{14 \, a^{3} b^{5} \cos \left (d x + c\right ) - 10 \, a^{2} b^{6} + 60 \,{\left (7 \, a^{8} - 20 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{6} + 30 \,{\left (7 \, a^{7} b - 20 \, a^{5} b^{3} + 18 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{5} - 10 \,{\left (7 \, a^{6} b^{2} - 20 \, a^{4} b^{4} + 18 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (7 \, a^{5} b^{3} - 20 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (7 \, a^{4} b^{4} - 20 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2}}{a^{3} b^{7} \cos \left (d x + c\right )^{7} + a^{2} b^{8} \cos \left (d x + c\right )^{6}} + \frac{60 \,{\left (7 \, a^{6} - 20 \, a^{4} b^{2} + 18 \, a^{2} b^{4} - 4 \, b^{6}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{8}} - \frac{60 \,{\left (7 \, a^{8} - 20 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 4 \, a^{2} b^{6} - b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} b^{8}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55956, size = 946, normalized size = 3.71 \begin{align*} -\frac{14 \, a^{3} b^{6} \cos \left (d x + c\right ) - 10 \, a^{2} b^{7} + 60 \,{\left (7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} + 30 \,{\left (7 \, a^{7} b^{2} - 20 \, a^{5} b^{4} + 18 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{5} - 10 \,{\left (7 \, a^{6} b^{3} - 20 \, a^{4} b^{5} + 18 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (7 \, a^{5} b^{4} - 20 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (7 \, a^{4} b^{5} - 20 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} - 60 \,{\left ({\left (7 \, a^{9} - 20 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 4 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right )^{7} +{\left (7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7} - b^{9}\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + 60 \,{\left ({\left (7 \, a^{9} - 20 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 4 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{7} +{\left (7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{6}\right )} \log \left (-\cos \left (d x + c\right )\right )}{60 \,{\left (a^{3} b^{8} d \cos \left (d x + c\right )^{7} + a^{2} b^{9} d \cos \left (d x + c\right )^{6}\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 [B] time = 10.865, size = 2290, normalized size = 8.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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