Optimal. Leaf size=139 \[ -\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.140545, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30} \[ -\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{2 a^2 \csc (c+d x)}{d}+a^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^8(c+d x)+2 a^2 \cot ^7(c+d x) \csc (c+d x)+a^2 \cot ^6(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^8(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^7(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^2 \cot ^7(c+d x)}{7 d}-a^2 \int \cot ^6(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+a^2 \int \cot ^4(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{2 a^2 \csc (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-a^2 \int \cot ^2(c+d x) \, dx\\ &=\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{2 a^2 \csc (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}+a^2 \int 1 \, dx\\ &=a^2 x+\frac{a^2 \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}+\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}+\frac{2 a^2 \csc (c+d x)}{d}-\frac{2 a^2 \csc ^3(c+d x)}{d}+\frac{6 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [B] time = 1.07294, size = 312, normalized size = 2.24 \[ \frac{a^2 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^7\left (\frac{1}{2} (c+d x)\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) (-16002 \sin (c+d x)+9144 \sin (2 (c+d x))+3429 \sin (3 (c+d x))-4572 \sin (4 (c+d x))+1143 \sin (5 (c+d x))-11760 \sin (2 c+d x)+8864 \sin (c+2 d x)+3360 \sin (3 c+2 d x)+2064 \sin (2 c+3 d x)+2520 \sin (4 c+3 d x)-4432 \sin (3 c+4 d x)-1680 \sin (5 c+4 d x)+1528 \sin (4 c+5 d x)-5880 d x \cos (2 c+d x)-3360 d x \cos (c+2 d x)+3360 d x \cos (3 c+2 d x)-1260 d x \cos (2 c+3 d x)+1260 d x \cos (4 c+3 d x)+1680 d x \cos (3 c+4 d x)-1680 d x \cos (5 c+4 d x)-420 d x \cos (4 c+5 d x)+420 d x \cos (6 c+5 d x)+4032 \sin (c)-9632 \sin (d x)+5880 d x \cos (d x))}{860160 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.078, size = 188, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5}}-{\frac{ \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3}}+\cot \left ( dx+c \right ) +dx+c \right ) +2\,{a}^{2} \left ( -1/7\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+1/35\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-1/35\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+1/7\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{\sin \left ( dx+c \right ) }}+1/7\, \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.76686, size = 158, normalized size = 1.14 \begin{align*} \frac{{\left (105 \, d x + 105 \, c + \frac{105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{2} + \frac{6 \,{\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a^{2}}{\sin \left (d x + c\right )^{7}} - \frac{15 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.07674, size = 435, normalized size = 3.13 \begin{align*} \frac{191 \, a^{2} \cos \left (d x + c\right )^{5} - 172 \, a^{2} \cos \left (d x + c\right )^{4} - 253 \, a^{2} \cos \left (d x + c\right )^{3} + 258 \, a^{2} \cos \left (d x + c\right )^{2} + 87 \, a^{2} \cos \left (d x + c\right ) - 96 \, a^{2} + 105 \,{\left (a^{2} d x \cos \left (d x + c\right )^{4} - 2 \, a^{2} d x \cos \left (d x + c\right )^{3} + 2 \, a^{2} d x \cos \left (d x + c\right ) - a^{2} d x\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.49927, size = 151, normalized size = 1.09 \begin{align*} \frac{35 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3360 \,{\left (d x + c\right )} a^{2} - 735 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{4410 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 770 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 147 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{3360 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]