3.24 \(\int \cot (c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=35 \[ \frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0404139, antiderivative size = 35, 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, 72} \[ \frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*Log[1 - Cos[c + d*x]])/d - (a^2*Log[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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0378019, size = 29, normalized size = 0.83 \[ -\frac{a^2 \left (\log (\cos (c+d x))-4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 34, normalized size = 1. \begin{align*} 2\,{\frac{{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}\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))^2,x)

[Out]

2/d*a^2*ln(-1+sec(d*x+c))-1/d*a^2*ln(sec(d*x+c))

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Maxima [A]  time = 1.11702, size = 42, normalized size = 1.2 \begin{align*} \frac{2 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - a^{2} \log \left (\cos \left (d x + c\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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.03161, size = 89, normalized size = 2.54 \begin{align*} -\frac{a^{2} \log \left (-\cos \left (d x + c\right )\right ) - 2 \, a^{2} \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))^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.46376, size = 86, normalized size = 2.46 \begin{align*} \frac{2 \, a^{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} \log \left ({\left | \frac{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 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))^2,x, algorithm="giac")

[Out]

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