Optimal. Leaf size=95 \[ \frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0643238, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 88} \[ \frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^4}{(a-a x)^3 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^5 (-1+x)^3}-\frac{3}{4 a^5 (-1+x)^2}-\frac{11}{16 a^5 (-1+x)}+\frac{1}{8 a^5 (1+x)^2}-\frac{5}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (1+\cos (c+d x))}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (1+\cos (c+d x))}{16 d}\\ \end{align*}
Mathematica [A] time = 0.538715, size = 127, normalized size = 1.34 \[ \frac{a \left (-16 \cot ^4(c+d x)+32 \cot ^2(c+d x)-\csc ^4\left (\frac{1}{2} (c+d x)\right )+10 \csc ^2\left (\frac{1}{2} (c+d x)\right )+\sec ^4\left (\frac{1}{2} (c+d x)\right )-10 \sec ^2\left (\frac{1}{2} (c+d x)\right )+24 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+64 \log (\tan (c+d x))-24 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64 \log (\cos (c+d x))\right )}{64 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.073, size = 93, normalized size = 1. \begin{align*} -{\frac{a}{8\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{5\,a\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{a}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a}{2\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{11\,a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{a\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 = 1.19467, size = 116, normalized size = 1.22 \begin{align*} \frac{5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (5 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 1}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.906858, size = 402, normalized size = 4.23 \begin{align*} -\frac{10 \, a \cos \left (d x + c\right )^{2} + 6 \, a \cos \left (d x + c\right ) - 5 \,{\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 11 \,{\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 12 \, a}{16 \,{\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - 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 = 1.49646, size = 201, normalized size = 2.12 \begin{align*} \frac{22 \, a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (a + \frac{10 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{33 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{2 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]