3.76 \(\int \frac{\cot (c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{5}{4 a^2 d (\cos (c+d x)+1)}-\frac{1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{\log (1-\cos (c+d x))}{8 a^2 d}+\frac{7 \log (\cos (c+d x)+1)}{8 a^2 d} \]

[Out]

-1/(4*a^2*d*(1 + Cos[c + d*x])^2) + 5/(4*a^2*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(8*a^2*d) + (7*Log[
1 + Cos[c + d*x]])/(8*a^2*d)

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Rubi [A]  time = 0.060841, antiderivative size = 81, 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} \[ \frac{5}{4 a^2 d (\cos (c+d x)+1)}-\frac{1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac{\log (1-\cos (c+d x))}{8 a^2 d}+\frac{7 \log (\cos (c+d x)+1)}{8 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]/(a + a*Sec[c + d*x])^2,x]

[Out]

-1/(4*a^2*d*(1 + Cos[c + d*x])^2) + 5/(4*a^2*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(8*a^2*d) + (7*Log[
1 + Cos[c + d*x]])/(8*a^2*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))^2} \, dx &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{x^3}{(a-a x) (a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (-\frac{1}{8 a^4 (-1+x)}-\frac{1}{2 a^4 (1+x)^3}+\frac{5}{4 a^4 (1+x)^2}-\frac{7}{8 a^4 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{4 a^2 d (1+\cos (c+d x))^2}+\frac{5}{4 a^2 d (1+\cos (c+d x))}+\frac{\log (1-\cos (c+d x))}{8 a^2 d}+\frac{7 \log (1+\cos (c+d x))}{8 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.186381, size = 83, normalized size = 1.02 \[ \frac{\sec ^2(c+d x) \left (10 \cos ^2\left (\frac{1}{2} (c+d x)\right )+4 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+7 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-1\right )}{4 a^2 d (\sec (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]/(a + a*Sec[c + d*x])^2,x]

[Out]

((-1 + 10*Cos[(c + d*x)/2]^2 + 4*Cos[(c + d*x)/2]^4*(7*Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))*Sec[c +
d*x]^2)/(4*a^2*d*(1 + Sec[c + d*x])^2)

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Maple [A]  time = 0.076, size = 72, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{5}{4\,d{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{7\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{8\,d{a}^{2}}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{8\,d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)/(a+a*sec(d*x+c))^2,x)

[Out]

-1/4/d/a^2/(cos(d*x+c)+1)^2+5/4/d/a^2/(cos(d*x+c)+1)+7/8*ln(cos(d*x+c)+1)/a^2/d+1/8/d/a^2*ln(-1+cos(d*x+c))

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Maxima [A]  time = 1.19077, size = 100, normalized size = 1.23 \begin{align*} \frac{\frac{2 \,{\left (5 \, \cos \left (d x + c\right ) + 4\right )}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac{7 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac{\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{8 \, 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))^2,x, algorithm="maxima")

[Out]

1/8*(2*(5*cos(d*x + c) + 4)/(a^2*cos(d*x + c)^2 + 2*a^2*cos(d*x + c) + a^2) + 7*log(cos(d*x + c) + 1)/a^2 + lo
g(cos(d*x + c) - 1)/a^2)/d

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Fricas [A]  time = 1.19867, size = 297, normalized size = 3.67 \begin{align*} \frac{7 \,{\left (\cos \left (d x + c\right )^{2} + 2 \, \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 )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 10 \, \cos \left (d x + c\right ) + 8}{8 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} 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))^2,x, algorithm="fricas")

[Out]

1/8*(7*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + (cos(d*x + c)^2 + 2*cos(d*x + c) +
1)*log(-1/2*cos(d*x + c) + 1/2) + 10*cos(d*x + c) + 8)/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)/(a+a*sec(d*x+c))**2,x)

[Out]

Integral(cot(c + d*x)/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2

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Giac [A]  time = 1.32623, size = 158, normalized size = 1.95 \begin{align*} \frac{\frac{2 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} - \frac{16 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac{\frac{8 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{4}}}{16 \, 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))^2,x, algorithm="giac")

[Out]

1/16*(2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2 - 16*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) +
1) + 1))/a^2 - (8*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/
a^4)/d