3.42 \(\int \cot (c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=48 \[ \frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0460482, antiderivative size = 48, 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} \[ \frac{a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]

[Out]

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

Mathematica [A]  time = 0.0876936, size = 36, normalized size = 0.75 \[ \frac{a^3 \left (\sec (c+d x)+8 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3 \log (\cos (c+d x))\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*(-3*Log[Cos[c + d*x]] + 8*Log[Sin[(c + d*x)/2]] + Sec[c + d*x]))/d

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Maple [A]  time = 0.046, size = 47, normalized size = 1. \begin{align*}{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}\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))^3,x)

[Out]

a^3*sec(d*x+c)/d+4/d*a^3*ln(-1+sec(d*x+c))-1/d*a^3*ln(sec(d*x+c))

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Maxima [A]  time = 1.50157, size = 58, normalized size = 1.21 \begin{align*} \frac{4 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 3 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac{a^{3}}{\cos \left (d x + c\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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.19775, size = 155, normalized size = 3.23 \begin{align*} -\frac{3 \, a^{3} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a^{3} \cos \left (d x + c\right ) \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - a^{3}}{d \cos \left (d x + c\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))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int 3 \cot{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 3 \cot{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot{\left (c + d x \right )} \sec ^{3}{\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))**3,x)

[Out]

a**3*(Integral(3*cot(c + d*x)*sec(c + d*x), x) + Integral(3*cot(c + d*x)*sec(c + d*x)**2, x) + Integral(cot(c
+ d*x)*sec(c + d*x)**3, x) + Integral(cot(c + d*x), x))

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Giac [B]  time = 1.26674, size = 196, normalized size = 4.08 \begin{align*} \frac{4 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 3 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{5 \, a^{3} + \frac{3 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{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))^3,x, algorithm="giac")

[Out]

(4*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
+ 1)) - 3*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (5*a^3 + 3*a^3*(cos(d*x + c) - 1)/(cos(d*
x + c) + 1))/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/d