Optimal. Leaf size=250 \[ \frac{\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left (-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac{\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d} \]
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Rubi [A] time = 0.196866, antiderivative size = 250, 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} \[ \frac{\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left (-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac{\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}-\frac{\log (\cos (c+d x))}{a d}+\frac{\sec ^7(c+d x)}{7 b d} \]
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)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^4}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^6 \left (1+\frac{-4 a^4 b^2+6 a^2 b^4-4 b^6}{a^6}\right )+\frac{b^8}{a x}-a \left (a^4-4 a^2 b^2+6 b^4\right ) x+\left (a^4-4 a^2 b^2+6 b^4\right ) x^2-a \left (a^2-4 b^2\right ) x^3+\left (a^2-4 b^2\right ) x^4-a x^5+x^6-\frac{\left (a^2-b^2\right )^4}{a (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=-\frac{\log (\cos (c+d x))}{a d}-\frac{\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}+\frac{\left (a^6-4 a^4 b^2+6 a^2 b^4-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac{a \left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac{\left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac{a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac{\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac{a \sec ^6(c+d x)}{6 b^2 d}+\frac{\sec ^7(c+d x)}{7 b d}\\ \end{align*}
Mathematica [B] time = 6.22729, size = 520, normalized size = 2.08 \[ \frac{\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^4(c+d x) (a \cos (c+d x)+b)}{3 b^5 d (a+b \sec (c+d x))}-\frac{a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x) (a \cos (c+d x)+b)}{2 b^6 d (a+b \sec (c+d x))}-\frac{\left (2 b^2-a^2\right ) \left (-2 a^2 b^2+a^4+2 b^4\right ) \sec ^2(c+d x) (a \cos (c+d x)+b)}{b^7 d (a+b \sec (c+d x))}+\frac{\left (-4 a^5 b^2+6 a^3 b^4+a^7-4 a b^6\right ) \sec (c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)}{b^8 d (a+b \sec (c+d x))}+\frac{\left (4 a^6 b^2-6 a^4 b^4+4 a^2 b^6-a^8-b^8\right ) \sec (c+d x) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a b^8 d (a+b \sec (c+d x))}-\frac{a \sec ^7(c+d x) (a \cos (c+d x)+b)}{6 b^2 d (a+b \sec (c+d x))}-\frac{(2 b-a) (a+2 b) \sec ^6(c+d x) (a \cos (c+d x)+b)}{5 b^3 d (a+b \sec (c+d x))}+\frac{a (2 b-a) (a+2 b) \sec ^5(c+d x) (a \cos (c+d x)+b)}{4 b^4 d (a+b \sec (c+d x))}+\frac{\sec ^8(c+d x) (a \cos (c+d x)+b)}{7 b d (a+b \sec (c+d x))} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.064, size = 460, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{db \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4}{5\,db \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{ad}}+{\frac{1}{7\,db \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}-4\,{\frac{1}{db\cos \left ( dx+c \right ) }}+{\frac{{a}^{4}}{3\,d{b}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4\,{a}^{2}}{3\,d{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{7}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{8}}}-4\,{\frac{{a}^{5}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{6}}}+6\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{4}}}-4\,{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}-{\frac{{a}^{7}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{8}}}+{\frac{{a}^{6}}{d{b}^{7}\cos \left ( dx+c \right ) }}-4\,{\frac{{a}^{4}}{d{b}^{5}\cos \left ( dx+c \right ) }}+6\,{\frac{{a}^{2}}{d{b}^{3}\cos \left ( dx+c \right ) }}-{\frac{{a}^{3}}{4\,d{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a}{d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{a}^{5}}{2\,d{b}^{6} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{a}^{3}}{d{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{a}{d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{{a}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{6}}}-6\,{\frac{{a}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{4}}}+4\,{\frac{a\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{b}^{2}}}-{\frac{a}{6\,d{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{a}^{2}}{5\,d{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00017, size = 362, normalized size = 1.45 \begin{align*} \frac{\frac{420 \,{\left (a^{7} - 4 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 4 \, a b^{6}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{8}} - \frac{420 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{8}} - \frac{70 \, a b^{5} \cos \left (d x + c\right ) - 420 \,{\left (a^{6} - 4 \, a^{4} b^{2} + 6 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - 60 \, b^{6} + 210 \,{\left (a^{5} b - 4 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 140 \,{\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 6 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 105 \,{\left (a^{3} b^{3} - 4 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 84 \,{\left (a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{2}}{b^{7} \cos \left (d x + c\right )^{7}}}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13327, size = 674, normalized size = 2.7 \begin{align*} -\frac{70 \, a^{2} b^{6} \cos \left (d x + c\right ) + 420 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{7} \log \left (a \cos \left (d x + c\right ) + b\right ) - 420 \,{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 60 \, a b^{7} - 420 \,{\left (a^{7} b - 4 \, a^{5} b^{3} + 6 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} + 210 \,{\left (a^{6} b^{2} - 4 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{5} - 140 \,{\left (a^{5} b^{3} - 4 \, a^{3} b^{5} + 6 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 105 \,{\left (a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - 84 \,{\left (a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{2}}{420 \, a b^{8} d \cos \left (d x + c\right )^{7}} \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 = 11.2655, size = 2267, normalized size = 9.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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