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

Optimal. Leaf size=43 \[ \frac{(a+b) \log (1-\cos (c+d x))}{2 d}+\frac{(a-b) \log (\cos (c+d x)+1)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.080858, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3883, 2668, 633, 31} \[ \frac{(a+b) \log (1-\cos (c+d x))}{2 d}+\frac{(a-b) \log (\cos (c+d x)+1)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3883

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))/cot[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[(b + a*Sin[c + d*x])/Cos[
c + d*x], x] /; FreeQ[{a, b, c, d}, x]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

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

Mathematica [A]  time = 0.0357285, size = 60, normalized size = 1.4 \[ \frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{b \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 35, normalized size = 0.8 \begin{align*}{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{b\ln \left ( \csc \left ( dx+c \right ) -\cot \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+b*sec(d*x+c)),x)

[Out]

1/d*a*ln(sin(d*x+c))+1/d*b*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 0.973253, size = 46, normalized size = 1.07 \begin{align*} \frac{{\left (a - b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) +{\left (a + b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.969679, size = 113, normalized size = 2.63 \begin{align*} \frac{{\left (a - b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a + b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a - b)*log(1/2*cos(d*x + c) + 1/2) + (a + b)*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*} \int \left (a + b \sec{\left (c + d x \right )}\right ) \cot{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.28501, size = 82, normalized size = 1.91 \begin{align*} \frac{{\left (a + b\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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