3.26 \(\int \cot ^5(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=85 \[ \frac{5 a^2}{4 d (1-\cos (c+d x))}-\frac{a^2}{4 d (1-\cos (c+d x))^2}+\frac{7 a^2 \log (1-\cos (c+d x))}{8 d}+\frac{a^2 \log (\cos (c+d x)+1)}{8 d} \]

[Out]

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

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Rubi [A]  time = 0.0669472, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac{5 a^2}{4 d (1-\cos (c+d x))}-\frac{a^2}{4 d (1-\cos (c+d x))^2}+\frac{7 a^2 \log (1-\cos (c+d x))}{8 d}+\frac{a^2 \log (\cos (c+d x)+1)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]

[Out]

-a^2/(4*d*(1 - Cos[c + d*x])^2) + (5*a^2)/(4*d*(1 - Cos[c + d*x])) + (7*a^2*Log[1 - Cos[c + d*x]])/(8*d) + (a^
2*Log[1 + Cos[c + d*x]])/(8*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))^2 \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^3}{(a-a x)^3 (a+a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{2 a^4 (-1+x)^3}-\frac{5}{4 a^4 (-1+x)^2}-\frac{7}{8 a^4 (-1+x)}-\frac{1}{8 a^4 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2}{4 d (1-\cos (c+d x))^2}+\frac{5 a^2}{4 d (1-\cos (c+d x))}+\frac{7 a^2 \log (1-\cos (c+d x))}{8 d}+\frac{a^2 \log (1+\cos (c+d x))}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.267052, size = 86, normalized size = 1.01 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (\csc ^4\left (\frac{1}{2} (c+d x)\right )-10 \csc ^2\left (\frac{1}{2} (c+d x)\right )-4 \left (7 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{64 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*(1 + Cos[c + d*x])^2*(-10*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 - 4*(Log[Cos[(c + d*x)/2]] + 7*Log[Sin
[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4)/(64*d)

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Maple [A]  time = 0.08, size = 87, normalized size = 1. \begin{align*}{\frac{{a}^{2}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{a}^{2}}{4\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2}}{4\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{7\,{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{8\,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)^5*(a+a*sec(d*x+c))^2,x)

[Out]

1/8/d*a^2*ln(1+sec(d*x+c))-1/4/d*a^2/(-1+sec(d*x+c))^2+3/4/d*a^2/(-1+sec(d*x+c))+7/8/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 = 1.10416, size = 97, normalized size = 1.14 \begin{align*} \frac{a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) + 7 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (5 \, a^{2} \cos \left (d x + c\right ) - 4 \, a^{2}\right )}}{\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}}{8 \, 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.13477, size = 320, normalized size = 3.76 \begin{align*} -\frac{10 \, a^{2} \cos \left (d x + c\right ) - 8 \, a^{2} -{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 7 \,{\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{8 \,{\left (d \cos \left (d x + c\right )^{2} - 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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.48089, size = 186, normalized size = 2.19 \begin{align*} \frac{14 \, a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 16 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (a^{2} + \frac{8 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{21 \, a^{2}{\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}}}{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))^2,x, algorithm="giac")

[Out]

1/16*(14*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 16*a^2*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) + 21*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c
) + 1)^2)*(cos(d*x + c) + 1)^2/(cos(d*x + c) - 1)^2)/d