3.63 \(\int \frac{\cot ^5(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=145 \[ \frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d} \]

[Out]

-1/(32*a*d*(1 - Cos[c + d*x])^2) + 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(24*a*d*(1 + Cos[c + d*x])^3) - 9/(32*a*d*
(1 + Cos[c + d*x])^2) + 15/(16*a*d*(1 + Cos[c + d*x])) + (11*Log[1 - Cos[c + d*x]])/(32*a*d) + (21*Log[1 + Cos
[c + d*x]])/(32*a*d)

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Rubi [A]  time = 0.0980931, antiderivative size = 145, 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} \[ \frac{1}{4 a d (1-\cos (c+d x))}+\frac{15}{16 a d (\cos (c+d x)+1)}-\frac{1}{32 a d (1-\cos (c+d x))^2}-\frac{9}{32 a d (\cos (c+d x)+1)^2}+\frac{1}{24 a d (\cos (c+d x)+1)^3}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (\cos (c+d x)+1)}{32 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(32*a*d*(1 - Cos[c + d*x])^2) + 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(24*a*d*(1 + Cos[c + d*x])^3) - 9/(32*a*d*
(1 + Cos[c + d*x])^2) + 15/(16*a*d*(1 + Cos[c + d*x])) + (11*Log[1 - Cos[c + d*x]])/(32*a*d) + (21*Log[1 + Cos
[c + d*x]])/(32*a*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 \frac{\cot ^5(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^6}{(a-a x)^3 (a+a x)^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{16 a^7 (-1+x)^3}-\frac{1}{4 a^7 (-1+x)^2}-\frac{11}{32 a^7 (-1+x)}+\frac{1}{8 a^7 (1+x)^4}-\frac{9}{16 a^7 (1+x)^3}+\frac{15}{16 a^7 (1+x)^2}-\frac{21}{32 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{32 a d (1-\cos (c+d x))^2}+\frac{1}{4 a d (1-\cos (c+d x))}+\frac{1}{24 a d (1+\cos (c+d x))^3}-\frac{9}{32 a d (1+\cos (c+d x))^2}+\frac{15}{16 a d (1+\cos (c+d x))}+\frac{11 \log (1-\cos (c+d x))}{32 a d}+\frac{21 \log (1+\cos (c+d x))}{32 a d}\\ \end{align*}

Mathematica [A]  time = 0.538169, size = 135, normalized size = 0.93 \[ -\frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (3 \csc ^4\left (\frac{1}{2} (c+d x)\right )-48 \csc ^2\left (\frac{1}{2} (c+d x)\right )-2 \sec ^6\left (\frac{1}{2} (c+d x)\right )+27 \sec ^4\left (\frac{1}{2} (c+d x)\right )-180 \sec ^2\left (\frac{1}{2} (c+d x)\right )-264 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-504 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{192 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[(c + d*x)/2]^2*(-48*Csc[(c + d*x)/2]^2 + 3*Csc[(c + d*x)/2]^4 - 504*Log[Cos[(c + d*x)/2]] - 264*Log[Sin[
(c + d*x)/2]] - 180*Sec[(c + d*x)/2]^2 + 27*Sec[(c + d*x)/2]^4 - 2*Sec[(c + d*x)/2]^6)*Sec[c + d*x])/(192*a*d*
(1 + Sec[c + d*x]))

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Maple [A]  time = 0.068, size = 126, normalized size = 0.9 \begin{align*}{\frac{1}{24\,da \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{9}{32\,da \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{15}{16\,da \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{21\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{32\,da}}-{\frac{1}{32\,da \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{4\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{11\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{32\,da}} \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]

1/24/a/d/(cos(d*x+c)+1)^3-9/32/a/d/(cos(d*x+c)+1)^2+15/16/a/d/(cos(d*x+c)+1)+21/32*ln(cos(d*x+c)+1)/a/d-1/32/a
/d/(-1+cos(d*x+c))^2-1/4/a/d/(-1+cos(d*x+c))+11/32/d/a*ln(-1+cos(d*x+c))

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Maxima [A]  time = 1.17699, size = 176, normalized size = 1.21 \begin{align*} \frac{\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{4} - 39 \, \cos \left (d x + c\right )^{3} - 79 \, \cos \left (d x + c\right )^{2} + 29 \, \cos \left (d x + c\right ) + 44\right )}}{a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + a} + \frac{63 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac{33 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{96 \, 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]

1/96*(2*(33*cos(d*x + c)^4 - 39*cos(d*x + c)^3 - 79*cos(d*x + c)^2 + 29*cos(d*x + c) + 44)/(a*cos(d*x + c)^5 +
a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 - 2*a*cos(d*x + c)^2 + a*cos(d*x + c) + a) + 63*log(cos(d*x + c) + 1)/a
+ 33*log(cos(d*x + c) - 1)/a)/d

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Fricas [A]  time = 1.2057, size = 610, normalized size = 4.21 \begin{align*} \frac{66 \, \cos \left (d x + c\right )^{4} - 78 \, \cos \left (d x + c\right )^{3} - 158 \, \cos \left (d x + c\right )^{2} + 63 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 33 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 58 \, \cos \left (d x + c\right ) + 88}{96 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a 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]

1/96*(66*cos(d*x + c)^4 - 78*cos(d*x + c)^3 - 158*cos(d*x + c)^2 + 63*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos
(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 33*(cos(d*x + c)^5 + cos(d*x
+ c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 58*cos(d*x + c
) + 88)/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x +
c) + a*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{5}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \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]

Integral(cot(c + d*x)**5/(sec(c + d*x) + 1), x)/a

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Giac [A]  time = 1.4128, size = 285, normalized size = 1.97 \begin{align*} -\frac{\frac{3 \,{\left (\frac{14 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{66 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{132 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac{384 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} + \frac{\frac{132 \, 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}} + \frac{2 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}}}{384 \, 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]

-1/384*(3*(14*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 66*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1)*(cos(d
*x + c) + 1)^2/(a*(cos(d*x + c) - 1)^2) - 132*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a + 384*log(ab
s(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a + (132*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 21*a^2*(co
s(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 2*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/a^3)/d