Optimal. Leaf size=107 \[ -\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.127637, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{2 a^2 \cot ^5(c+d x)}{5 d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}-a^2 x \]
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 ^6(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x)+2 a^2 \cot ^5(c+d x) \csc (c+d x)+a^2 \cot ^4(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^5(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^2 \cot ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}+a^2 \int \cot ^2(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^2 \cot (c+d x)}{d}+\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 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{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{4 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.776342, size = 194, normalized size = 1.81 \[ \frac{a^2 \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \csc ^5\left (\frac{1}{2} (c+d x)\right ) \sec \left (\frac{1}{2} (c+d x)\right ) (445 \sin (c+d x)-356 \sin (2 (c+d x))+89 \sin (3 (c+d x))+240 \sin (2 c+d x)-296 \sin (c+2 d x)-120 \sin (3 c+2 d x)+104 \sin (2 c+3 d x)+150 d x \cos (2 c+d x)+120 d x \cos (c+2 d x)-120 d x \cos (3 c+2 d x)-30 d x \cos (2 c+3 d x)+30 d x \cos (4 c+3 d x)-80 \sin (c)+280 \sin (d x)-150 d x \cos (d x))}{3840 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.067, size = 155, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\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 ( -1/5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}+1/15\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-1/5\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{\sin \left ( dx+c \right ) }}-1/5\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \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 ) ^{5}}{5\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.78702, size = 131, normalized size = 1.22 \begin{align*} -\frac{{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} + \frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} a^{2}}{\sin \left (d x + c\right )^{5}} + \frac{3 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.852108, size = 298, normalized size = 2.79 \begin{align*} -\frac{26 \, a^{2} \cos \left (d x + c\right )^{3} - 22 \, a^{2} \cos \left (d x + c\right )^{2} - 17 \, a^{2} \cos \left (d x + c\right ) + 16 \, a^{2} + 15 \,{\left (a^{2} d x \cos \left (d x + c\right )^{2} - 2 \, a^{2} d x \cos \left (d x + c\right ) + a^{2} d x\right )} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{2} - 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 = 1.48166, size = 108, normalized size = 1.01 \begin{align*} -\frac{120 \,{\left (d x + c\right )} a^{2} - 15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{165 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 25 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]