Optimal. Leaf size=61 \[ \frac {1}{2 a d (\cos (c+d x)+1)}+\frac {\log (1-\cos (c+d x))}{4 a d}+\frac {3 \log (\cos (c+d x)+1)}{4 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{2 a d (\cos (c+d x)+1)}+\frac {\log (1-\cos (c+d x))}{4 a d}+\frac {3 \log (\cos (c+d x)+1)}{4 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {x^2}{(a-a x) (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (-\frac {1}{4 a^3 (-1+x)}+\frac {1}{2 a^3 (1+x)^2}-\frac {3}{4 a^3 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {1}{2 a d (1+\cos (c+d x))}+\frac {\log (1-\cos (c+d x))}{4 a d}+\frac {3 \log (1+\cos (c+d x))}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 67, normalized size = 1.10 \[ \frac {\sec (c+d x) \left (2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+1\right )}{2 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 60, normalized size = 0.98 \[ \frac {3 \, {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2}{4 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.24, size = 86, normalized size = 1.41 \[ \frac {\frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} - \frac {4 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac {\cos \left (d x + c\right ) - 1}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.61, size = 54, normalized size = 0.89 \[ \frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{4 d a}+\frac {1}{2 a d \left (1+\cos \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\cos \left (d x +c \right )\right )}{4 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 47, normalized size = 0.77 \[ \frac {\frac {3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a} + \frac {2}{a \cos \left (d x + c\right ) + a}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 49, normalized size = 0.80 \[ \frac {\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________