3.94 \(\int \frac{\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=185 \[ \frac{5}{64 a^3 d (1-\cos (c+d x))}+\frac{303}{128 a^3 d (\cos (c+d x)+1)}-\frac{1}{128 a^3 d (1-\cos (c+d x))^2}-\frac{99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac{35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac{13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac{1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac{37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac{219 \log (\cos (c+d x)+1)}{256 a^3 d} \]

[Out]

-1/(128*a^3*d*(1 - Cos[c + d*x])^2) + 5/(64*a^3*d*(1 - Cos[c + d*x])) + 1/(40*a^3*d*(1 + Cos[c + d*x])^5) - 13
/(64*a^3*d*(1 + Cos[c + d*x])^4) + 35/(48*a^3*d*(1 + Cos[c + d*x])^3) - 99/(64*a^3*d*(1 + Cos[c + d*x])^2) + 3
03/(128*a^3*d*(1 + Cos[c + d*x])) + (37*Log[1 - Cos[c + d*x]])/(256*a^3*d) + (219*Log[1 + Cos[c + d*x]])/(256*
a^3*d)

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Rubi [A]  time = 0.123572, antiderivative size = 185, 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} \[ \frac{5}{64 a^3 d (1-\cos (c+d x))}+\frac{303}{128 a^3 d (\cos (c+d x)+1)}-\frac{1}{128 a^3 d (1-\cos (c+d x))^2}-\frac{99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac{35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac{13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac{1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac{37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac{219 \log (\cos (c+d x)+1)}{256 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/(128*a^3*d*(1 - Cos[c + d*x])^2) + 5/(64*a^3*d*(1 - Cos[c + d*x])) + 1/(40*a^3*d*(1 + Cos[c + d*x])^5) - 13
/(64*a^3*d*(1 + Cos[c + d*x])^4) + 35/(48*a^3*d*(1 + Cos[c + d*x])^3) - 99/(64*a^3*d*(1 + Cos[c + d*x])^2) + 3
03/(128*a^3*d*(1 + Cos[c + d*x])) + (37*Log[1 - Cos[c + d*x]])/(256*a^3*d) + (219*Log[1 + Cos[c + d*x]])/(256*
a^3*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))^3} \, dx &=-\frac{a^6 \operatorname{Subst}\left (\int \frac{x^8}{(a-a x)^3 (a+a x)^6} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^6 \operatorname{Subst}\left (\int \left (-\frac{1}{64 a^9 (-1+x)^3}-\frac{5}{64 a^9 (-1+x)^2}-\frac{37}{256 a^9 (-1+x)}+\frac{1}{8 a^9 (1+x)^6}-\frac{13}{16 a^9 (1+x)^5}+\frac{35}{16 a^9 (1+x)^4}-\frac{99}{32 a^9 (1+x)^3}+\frac{303}{128 a^9 (1+x)^2}-\frac{219}{256 a^9 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{128 a^3 d (1-\cos (c+d x))^2}+\frac{5}{64 a^3 d (1-\cos (c+d x))}+\frac{1}{40 a^3 d (1+\cos (c+d x))^5}-\frac{13}{64 a^3 d (1+\cos (c+d x))^4}+\frac{35}{48 a^3 d (1+\cos (c+d x))^3}-\frac{99}{64 a^3 d (1+\cos (c+d x))^2}+\frac{303}{128 a^3 d (1+\cos (c+d x))}+\frac{37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac{219 \log (1+\cos (c+d x))}{256 a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.18225, size = 169, normalized size = 0.91 \[ \frac{\sec ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (1400 \cos ^4\left (\frac{1}{2} (c+d x)\right )-195 \cos ^2\left (\frac{1}{2} (c+d x)\right )+60 \cos ^8\left (\frac{1}{2} (c+d x)\right ) \left (10 \cot ^2\left (\frac{1}{2} (c+d x)\right )+303\right )-30 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\cot ^4\left (\frac{1}{2} (c+d x)\right )+198\right )+120 \cos ^{10}\left (\frac{1}{2} (c+d x)\right ) \left (37 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+219 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+12\right )}{1920 a^3 d (\sec (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]

[Out]

((12 - 195*Cos[(c + d*x)/2]^2 + 1400*Cos[(c + d*x)/2]^4 + 60*Cos[(c + d*x)/2]^8*(303 + 10*Cot[(c + d*x)/2]^2)
- 30*Cos[(c + d*x)/2]^6*(198 + Cot[(c + d*x)/2]^4) + 120*Cos[(c + d*x)/2]^10*(219*Log[Cos[(c + d*x)/2]] + 37*L
og[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4*Sec[c + d*x]^3)/(1920*a^3*d*(1 + Sec[c + d*x])^3)

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Maple [A]  time = 0.082, size = 162, normalized size = 0.9 \begin{align*}{\frac{1}{40\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{5}}}-{\frac{13}{64\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{4}}}+{\frac{35}{48\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}-{\frac{99}{64\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{303}{128\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{219\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{256\,d{a}^{3}}}-{\frac{1}{128\,d{a}^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5}{64\,d{a}^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{37\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{256\,d{a}^{3}}} \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))^3,x)

[Out]

1/40/a^3/d/(cos(d*x+c)+1)^5-13/64/d/a^3/(cos(d*x+c)+1)^4+35/48/d/a^3/(cos(d*x+c)+1)^3-99/64/d/a^3/(cos(d*x+c)+
1)^2+303/128/d/a^3/(cos(d*x+c)+1)+219/256*ln(cos(d*x+c)+1)/a^3/d-1/128/d/a^3/(-1+cos(d*x+c))^2-5/64/d/a^3/(-1+
cos(d*x+c))+37/256/d/a^3*ln(-1+cos(d*x+c))

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Maxima [A]  time = 1.14742, size = 254, normalized size = 1.37 \begin{align*} \frac{\frac{2 \,{\left (4395 \, \cos \left (d x + c\right )^{6} + 5505 \, \cos \left (d x + c\right )^{5} - 6940 \, \cos \left (d x + c\right )^{4} - 12780 \, \cos \left (d x + c\right )^{3} - 367 \, \cos \left (d x + c\right )^{2} + 6939 \, \cos \left (d x + c\right ) + 2768\right )}}{a^{3} \cos \left (d x + c\right )^{7} + 3 \, a^{3} \cos \left (d x + c\right )^{6} + a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac{3285 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{555 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{3840 \, 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))^3,x, algorithm="maxima")

[Out]

1/3840*(2*(4395*cos(d*x + c)^6 + 5505*cos(d*x + c)^5 - 6940*cos(d*x + c)^4 - 12780*cos(d*x + c)^3 - 367*cos(d*
x + c)^2 + 6939*cos(d*x + c) + 2768)/(a^3*cos(d*x + c)^7 + 3*a^3*cos(d*x + c)^6 + a^3*cos(d*x + c)^5 - 5*a^3*c
os(d*x + c)^4 - 5*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2 + 3*a^3*cos(d*x + c) + a^3) + 3285*log(cos(d*x + c)
+ 1)/a^3 + 555*log(cos(d*x + c) - 1)/a^3)/d

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Fricas [A]  time = 1.29477, size = 879, normalized size = 4.75 \begin{align*} \frac{8790 \, \cos \left (d x + c\right )^{6} + 11010 \, \cos \left (d x + c\right )^{5} - 13880 \, \cos \left (d x + c\right )^{4} - 25560 \, \cos \left (d x + c\right )^{3} - 734 \, \cos \left (d x + c\right )^{2} + 3285 \,{\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 555 \,{\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 13878 \, \cos \left (d x + c\right ) + 5536}{3840 \,{\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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))^3,x, algorithm="fricas")

[Out]

1/3840*(8790*cos(d*x + c)^6 + 11010*cos(d*x + c)^5 - 13880*cos(d*x + c)^4 - 25560*cos(d*x + c)^3 - 734*cos(d*x
+ c)^2 + 3285*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos
(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 555*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d
*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) +
1/2) + 13878*cos(d*x + c) + 5536)/(a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a
^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*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 ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \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))**3,x)

[Out]

Integral(cot(c + d*x)**5/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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Giac [A]  time = 1.36609, size = 352, normalized size = 1.9 \begin{align*} -\frac{\frac{30 \,{\left (\frac{18 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{111 \,{\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^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac{2220 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac{15360 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac{\frac{9780 \, a^{12}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{2790 \, a^{12}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{740 \, a^{12}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{135 \, a^{12}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{12 \, a^{12}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{15}}}{15360 \, 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))^3,x, algorithm="giac")

[Out]

-1/15360*(30*(18*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 111*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1)*(c
os(d*x + c) + 1)^2/(a^3*(cos(d*x + c) - 1)^2) - 2220*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 + 1
5360*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^3 + (9780*a^12*(cos(d*x + c) - 1)/(cos(d*x + c) +
1) + 2790*a^12*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 740*a^12*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3
+ 135*a^12*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 12*a^12*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5)/a^15
)/d