3.61 \(\int \frac{\cot (c+d x)}{a+a \sec (c+d x)} \, dx\)

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]

1/(2*a*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(4*a*d) + (3*Log[1 + Cos[c + d*x]])/(4*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0556855, antiderivative size = 61, 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} \[ \frac{1}{2 a d (\cos (c+d x)+1)}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (\cos (c+d x)+1)}{4 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]/(a + a*Sec[c + d*x]),x]

[Out]

1/(2*a*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(4*a*d) + (3*Log[1 + Cos[c + d*x]])/(4*a*d)

Rule 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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 = 0.120026, size = 67, normalized size = 1.1 \[ \frac{\sec (c+d x) \left (2 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+1\right )}{2 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x]),x]

[Out]

((1 + 2*Cos[(c + d*x)/2]^2*(3*Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))*Sec[c + d*x])/(2*a*d*(1 + Sec[c
+ d*x]))

________________________________________________________________________________________

Maple [A]  time = 0.067, size = 54, normalized size = 0.9 \begin{align*}{\frac{1}{2\,da \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{3\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{4\,da}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{4\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)/(a+a*sec(d*x+c)),x)

[Out]

1/2/a/d/(cos(d*x+c)+1)+3/4*ln(cos(d*x+c)+1)/a/d+1/4/d/a*ln(-1+cos(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.25335, size = 63, normalized size = 1.03 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(3*log(cos(d*x + c) + 1)/a + log(cos(d*x + c) - 1)/a + 2/(a*cos(d*x + c) + a))/d

________________________________________________________________________________________

Fricas [A]  time = 1.16061, size = 182, normalized size = 2.98 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(3*(cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + (cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 2)/(
a*d*cos(d*x + c) + a*d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sec(d*x+c)),x)

[Out]

Integral(cot(c + d*x)/(sec(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.41428, size = 116, normalized size = 1.9 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

1/4*(log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a - 4*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) +
1))/a - (cos(d*x + c) - 1)/(a*(cos(d*x + c) + 1)))/d