3.6 \(\int \cot (c+d x) (a+a \sec (c+d x)) \, dx\)

Optimal. Leaf size=16 \[ \frac{a \log (1-\cos (c+d x))}{d} \]

[Out]

(a*Log[1 - Cos[c + d*x]])/d

________________________________________________________________________________________

Rubi [A]  time = 0.0203078, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3879, 31} \[ \frac{a \log (1-\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Log[1 - Cos[c + d*x]])/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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \cot (c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a-a x} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a \log (1-\cos (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.024625, size = 29, normalized size = 1.81 \[ \frac{2 a \left (\log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*(Log[Cos[(c + d*x)/2]] + Log[Tan[(c + d*x)/2]]))/d

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 29, normalized size = 1.8 \begin{align*}{\frac{a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{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]

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

[Out]

1/d*a*ln(-1+sec(d*x+c))-1/d*a*ln(sec(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.14439, size = 19, normalized size = 1.19 \begin{align*} \frac{a \log \left (\cos \left (d x + c\right ) - 1\right )}{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]

a*log(cos(d*x + c) - 1)/d

________________________________________________________________________________________

Fricas [A]  time = 0.847773, size = 46, normalized size = 2.88 \begin{align*} \frac{a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{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="fricas")

[Out]

a*log(-1/2*cos(d*x + c) + 1/2)/d

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.41107, size = 78, normalized size = 4.88 \begin{align*} \frac{a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{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]

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