Optimal. Leaf size=149 \[ \frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.097923, antiderivative size = 149, 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 = {3879, 88} \[ \frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\frac{a^{10} \operatorname{Subst}\left (\int \frac{x^6}{(a-a x)^5 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^{10} \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^7 (-1+x)^5}-\frac{5}{4 a^7 (-1+x)^4}-\frac{39}{16 a^7 (-1+x)^3}-\frac{9}{4 a^7 (-1+x)^2}-\frac{57}{64 a^7 (-1+x)}+\frac{1}{32 a^7 (1+x)^2}-\frac{7}{64 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (1+\cos (c+d x))}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (1+\cos (c+d x))}{64 d}\\ \end{align*}
Mathematica [A] time = 0.334718, size = 130, normalized size = 0.87 \[ \frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (-3 \csc ^8\left (\frac{1}{2} (c+d x)\right )+40 \csc ^6\left (\frac{1}{2} (c+d x)\right )-234 \csc ^4\left (\frac{1}{2} (c+d x)\right )+864 \csc ^2\left (\frac{1}{2} (c+d x)\right )+12 \left (\sec ^2\left (\frac{1}{2} (c+d x)\right )+114 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+14 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{6144 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.096, size = 141, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{32\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{7\,{a}^{3}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{64\,d}}-{\frac{{a}^{3}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{3}}{6\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{11\,{a}^{3}}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{13\,{a}^{3}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{57\,{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{64\,d}}-{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.6136, size = 192, normalized size = 1.29 \begin{align*} \frac{21 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 171 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (213 \, a^{3} \cos \left (d x + c\right )^{4} - 303 \, a^{3} \cos \left (d x + c\right )^{3} - 95 \, a^{3} \cos \left (d x + c\right )^{2} + 333 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.22619, size = 706, normalized size = 4.74 \begin{align*} -\frac{426 \, a^{3} \cos \left (d x + c\right )^{4} - 606 \, a^{3} \cos \left (d x + c\right )^{3} - 190 \, a^{3} \cos \left (d x + c\right )^{2} + 666 \, a^{3} \cos \left (d x + c\right ) - 272 \, a^{3} - 21 \,{\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 171 \,{\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{192 \,{\left (d \cos \left (d x + c\right )^{5} - 3 \, 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 )^{2} - 3 \, d \cos \left (d x + c\right ) + d\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 = 1.44596, size = 288, normalized size = 1.93 \begin{align*} \frac{684 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 768 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{12 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{{\left (3 \, a^{3} + \frac{28 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{132 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{504 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1425 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]