Optimal. Leaf size=185 \[ \frac{a^2 \sec ^8(c+d x)}{8 d}-\frac{2 a^2 \sec ^6(c+d x)}{3 d}+\frac{3 a^2 \sec ^4(c+d x)}{2 d}-\frac{2 a^2 \sec ^2(c+d x)}{d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^9(c+d x)}{9 d}-\frac{8 a b \sec ^7(c+d x)}{7 d}+\frac{12 a b \sec ^5(c+d x)}{5 d}-\frac{8 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \tan ^{10}(c+d x)}{10 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.131187, antiderivative size = 217, normalized size of antiderivative = 1.17, 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, 948} \[ \frac{\left (a^2-4 b^2\right ) \sec ^8(c+d x)}{8 d}-\frac{\left (2 a^2-3 b^2\right ) \sec ^6(c+d x)}{3 d}+\frac{\left (3 a^2-2 b^2\right ) \sec ^4(c+d x)}{2 d}-\frac{\left (4 a^2-b^2\right ) \sec ^2(c+d x)}{2 d}-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec ^9(c+d x)}{9 d}-\frac{8 a b \sec ^7(c+d x)}{7 d}+\frac{12 a b \sec ^5(c+d x)}{5 d}-\frac{8 a b \sec ^3(c+d x)}{3 d}+\frac{2 a b \sec (c+d x)}{d}+\frac{b^2 \sec ^{10}(c+d x)}{10 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3885
Rule 948
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 \tan ^9(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^2 \left (b^2-x^2\right )^4}{x} \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a b^8+\frac{a^2 b^8}{x}-b^6 \left (4 a^2-b^2\right ) x-8 a b^6 x^2+2 b^4 \left (3 a^2-2 b^2\right ) x^3+12 a b^4 x^4-2 b^2 \left (2 a^2-3 b^2\right ) x^5-8 a b^2 x^6+\left (a^2-4 b^2\right ) x^7+2 a x^8+x^9\right ) \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=-\frac{a^2 \log (\cos (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{\left (4 a^2-b^2\right ) \sec ^2(c+d x)}{2 d}-\frac{8 a b \sec ^3(c+d x)}{3 d}+\frac{\left (3 a^2-2 b^2\right ) \sec ^4(c+d x)}{2 d}+\frac{12 a b \sec ^5(c+d x)}{5 d}-\frac{\left (2 a^2-3 b^2\right ) \sec ^6(c+d x)}{3 d}-\frac{8 a b \sec ^7(c+d x)}{7 d}+\frac{\left (a^2-4 b^2\right ) \sec ^8(c+d x)}{8 d}+\frac{2 a b \sec ^9(c+d x)}{9 d}+\frac{b^2 \sec ^{10}(c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 0.423763, size = 173, normalized size = 0.94 \[ \frac{315 \left (a^2-4 b^2\right ) \sec ^8(c+d x)-840 \left (2 a^2-3 b^2\right ) \sec ^6(c+d x)+1260 \left (3 a^2-2 b^2\right ) \sec ^4(c+d x)-1260 \left (4 a^2-b^2\right ) \sec ^2(c+d x)-2520 a^2 \log (\cos (c+d x))+560 a b \sec ^9(c+d x)-2880 a b \sec ^7(c+d x)+6048 a b \sec ^5(c+d x)-6720 a b \sec ^3(c+d x)+5040 a b \sec (c+d x)+252 b^2 \sec ^{10}(c+d x)}{2520 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 317, normalized size = 1.7 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{8}{a}^{2}}{8\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{9\,d \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}-{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{63\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{105\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{63\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,ab \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{9\,d\cos \left ( dx+c \right ) }}+{\frac{256\,a\cos \left ( dx+c \right ) b}{315\,d}}+{\frac{2\,a\cos \left ( dx+c \right ) b \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{9\,d}}+{\frac{16\,a\cos \left ( dx+c \right ) b \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{63\,d}}+{\frac{32\,a\cos \left ( dx+c \right ) b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{105\,d}}+{\frac{128\,a\cos \left ( dx+c \right ) b \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{315\,d}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{10\,d \left ( \cos \left ( dx+c \right ) \right ) ^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0182, size = 235, normalized size = 1.27 \begin{align*} -\frac{2520 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac{5040 \, a b \cos \left (d x + c\right )^{9} - 6720 \, a b \cos \left (d x + c\right )^{7} - 1260 \,{\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{8} + 6048 \, a b \cos \left (d x + c\right )^{5} + 1260 \,{\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 2880 \, a b \cos \left (d x + c\right )^{3} - 840 \,{\left (2 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 560 \, a b \cos \left (d x + c\right ) + 315 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 252 \, b^{2}}{\cos \left (d x + c\right )^{10}}}{2520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.93027, size = 483, normalized size = 2.61 \begin{align*} -\frac{2520 \, a^{2} \cos \left (d x + c\right )^{10} \log \left (-\cos \left (d x + c\right )\right ) - 5040 \, a b \cos \left (d x + c\right )^{9} + 6720 \, a b \cos \left (d x + c\right )^{7} + 1260 \,{\left (4 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{8} - 6048 \, a b \cos \left (d x + c\right )^{5} - 1260 \,{\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{6} + 2880 \, a b \cos \left (d x + c\right )^{3} + 840 \,{\left (2 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 560 \, a b \cos \left (d x + c\right ) - 315 \,{\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 252 \, b^{2}}{2520 \, d \cos \left (d x + c\right )^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 90.7571, size = 314, normalized size = 1.7 \begin{align*} \begin{cases} \frac{a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{2} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac{a^{2} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac{a^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{2 a b \tan ^{8}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{9 d} - \frac{16 a b \tan ^{6}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{63 d} + \frac{32 a b \tan ^{4}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{105 d} - \frac{128 a b \tan ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{315 d} + \frac{256 a b \sec{\left (c + d x \right )}}{315 d} + \frac{b^{2} \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac{b^{2} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac{b^{2} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac{b^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac{b^{2} \sec ^{2}{\left (c + d x \right )}}{10 d} & \text{for}\: d \neq 0 \\x \left (a + b \sec{\left (c \right )}\right )^{2} \tan ^{9}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 13.7938, size = 660, normalized size = 3.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]