3.8 \(\int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx\)

Optimal. Leaf size=95 \[ \frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d} \]

[Out]

-a/(8*d*(1 - Cos[c + d*x])^2) + (3*a)/(4*d*(1 - Cos[c + d*x])) + a/(8*d*(1 + Cos[c + d*x])) + (11*a*Log[1 - Co
s[c + d*x]])/(16*d) + (5*a*Log[1 + Cos[c + d*x]])/(16*d)

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Rubi [A]  time = 0.0643238, antiderivative size = 95, 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} \[ \frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (\cos (c+d x)+1)}-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x]),x]

[Out]

-a/(8*d*(1 - Cos[c + d*x])^2) + (3*a)/(4*d*(1 - Cos[c + d*x])) + a/(8*d*(1 + Cos[c + d*x])) + (11*a*Log[1 - Co
s[c + d*x]])/(16*d) + (5*a*Log[1 + Cos[c + d*x]])/(16*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 ^5(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^4}{(a-a x)^3 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^5 (-1+x)^3}-\frac{3}{4 a^5 (-1+x)^2}-\frac{11}{16 a^5 (-1+x)}+\frac{1}{8 a^5 (1+x)^2}-\frac{5}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a}{8 d (1-\cos (c+d x))^2}+\frac{3 a}{4 d (1-\cos (c+d x))}+\frac{a}{8 d (1+\cos (c+d x))}+\frac{11 a \log (1-\cos (c+d x))}{16 d}+\frac{5 a \log (1+\cos (c+d x))}{16 d}\\ \end{align*}

Mathematica [A]  time = 0.538715, size = 127, normalized size = 1.34 \[ \frac{a \left (-16 \cot ^4(c+d x)+32 \cot ^2(c+d x)-\csc ^4\left (\frac{1}{2} (c+d x)\right )+10 \csc ^2\left (\frac{1}{2} (c+d x)\right )+\sec ^4\left (\frac{1}{2} (c+d x)\right )-10 \sec ^2\left (\frac{1}{2} (c+d x)\right )+24 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+64 \log (\tan (c+d x))-24 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64 \log (\cos (c+d x))\right )}{64 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x]),x]

[Out]

(a*(32*Cot[c + d*x]^2 - 16*Cot[c + d*x]^4 + 10*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 - 24*Log[Cos[(c + d*x)/
2]] + 64*Log[Cos[c + d*x]] + 24*Log[Sin[(c + d*x)/2]] + 64*Log[Tan[c + d*x]] - 10*Sec[(c + d*x)/2]^2 + Sec[(c
+ d*x)/2]^4))/(64*d)

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Maple [A]  time = 0.073, size = 93, normalized size = 1. \begin{align*} -{\frac{a}{8\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{5\,a\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{a}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a}{2\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{11\,a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{16\,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)^5*(a+a*sec(d*x+c)),x)

[Out]

-1/8/d*a/(1+sec(d*x+c))+5/16/d*a*ln(1+sec(d*x+c))-1/8/d*a/(-1+sec(d*x+c))^2+1/2/d*a/(-1+sec(d*x+c))+11/16/d*a*
ln(-1+sec(d*x+c))-1/d*a*ln(sec(d*x+c))

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Maxima [A]  time = 1.19467, size = 116, normalized size = 1.22 \begin{align*} \frac{5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (5 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 1}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/16*(5*a*log(cos(d*x + c) + 1) + 11*a*log(cos(d*x + c) - 1) - 2*(5*a*cos(d*x + c)^2 + 3*a*cos(d*x + c) - 6*a)
/(cos(d*x + c)^3 - cos(d*x + c)^2 - cos(d*x + c) + 1))/d

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Fricas [A]  time = 0.906858, size = 402, normalized size = 4.23 \begin{align*} -\frac{10 \, a \cos \left (d x + c\right )^{2} + 6 \, a \cos \left (d x + c\right ) - 5 \,{\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 11 \,{\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 12 \, a}{16 \,{\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

-1/16*(10*a*cos(d*x + c)^2 + 6*a*cos(d*x + c) - 5*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*l
og(1/2*cos(d*x + c) + 1/2) - 11*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(-1/2*cos(d*x +
c) + 1/2) - 12*a)/(d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5*(a+a*sec(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.49646, size = 201, normalized size = 2.12 \begin{align*} \frac{22 \, a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (a + \frac{10 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{33 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{2 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{32 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

1/32*(22*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 32*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) + 1)) - (a + 10*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 33*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)
*(cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2 - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/d