3.27 \(\int \cot ^7(c+d x) (a+a \sec (c+d x))^2 \, dx\)

Optimal. Leaf size=127 \[ -\frac{23 a^2}{16 d (1-\cos (c+d x))}-\frac{a^2}{16 d (\cos (c+d x)+1)}+\frac{a^2}{2 d (1-\cos (c+d x))^2}-\frac{a^2}{12 d (1-\cos (c+d x))^3}-\frac{13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac{3 a^2 \log (\cos (c+d x)+1)}{16 d} \]

[Out]

-a^2/(12*d*(1 - Cos[c + d*x])^3) + a^2/(2*d*(1 - Cos[c + d*x])^2) - (23*a^2)/(16*d*(1 - Cos[c + d*x])) - a^2/(
16*d*(1 + Cos[c + d*x])) - (13*a^2*Log[1 - Cos[c + d*x]])/(16*d) - (3*a^2*Log[1 + Cos[c + d*x]])/(16*d)

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Rubi [A]  time = 0.0865423, antiderivative size = 127, 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} \[ -\frac{23 a^2}{16 d (1-\cos (c+d x))}-\frac{a^2}{16 d (\cos (c+d x)+1)}+\frac{a^2}{2 d (1-\cos (c+d x))^2}-\frac{a^2}{12 d (1-\cos (c+d x))^3}-\frac{13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac{3 a^2 \log (\cos (c+d x)+1)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]

[Out]

-a^2/(12*d*(1 - Cos[c + d*x])^3) + a^2/(2*d*(1 - Cos[c + d*x])^2) - (23*a^2)/(16*d*(1 - Cos[c + d*x])) - a^2/(
16*d*(1 + Cos[c + d*x])) - (13*a^2*Log[1 - Cos[c + d*x]])/(16*d) - (3*a^2*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 ^7(c+d x) (a+a \sec (c+d x))^2 \, dx &=-\frac{a^8 \operatorname{Subst}\left (\int \frac{x^5}{(a-a x)^4 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^8 \operatorname{Subst}\left (\int \left (\frac{1}{4 a^6 (-1+x)^4}+\frac{1}{a^6 (-1+x)^3}+\frac{23}{16 a^6 (-1+x)^2}+\frac{13}{16 a^6 (-1+x)}-\frac{1}{16 a^6 (1+x)^2}+\frac{3}{16 a^6 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2}{12 d (1-\cos (c+d x))^3}+\frac{a^2}{2 d (1-\cos (c+d x))^2}-\frac{23 a^2}{16 d (1-\cos (c+d x))}-\frac{a^2}{16 d (1+\cos (c+d x))}-\frac{13 a^2 \log (1-\cos (c+d x))}{16 d}-\frac{3 a^2 \log (1+\cos (c+d x))}{16 d}\\ \end{align*}

Mathematica [A]  time = 0.227062, size = 114, normalized size = 0.9 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (\csc ^6\left (\frac{1}{2} (c+d x)\right )-12 \csc ^4\left (\frac{1}{2} (c+d x)\right )+69 \csc ^2\left (\frac{1}{2} (c+d x)\right )+3 \left (\sec ^2\left (\frac{1}{2} (c+d x)\right )+52 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{384 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2,x]

[Out]

-(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(69*Csc[(c + d*x)/2]^2 - 12*Csc[(c + d*x)/2]^4 + Csc[(c + d*x)/2
]^6 + 3*(12*Log[Cos[(c + d*x)/2]] + 52*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2)))/(384*d)

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Maple [A]  time = 0.081, size = 122, normalized size = 1. \begin{align*}{\frac{{a}^{2}}{16\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{3\,{a}^{2}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{{a}^{2}}{12\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}+{\frac{{a}^{2}}{4\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{11\,{a}^{2}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}-{\frac{13\,{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{16\,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)^7*(a+a*sec(d*x+c))^2,x)

[Out]

1/16/d*a^2/(1+sec(d*x+c))-3/16/d*a^2*ln(1+sec(d*x+c))-1/12/d*a^2/(-1+sec(d*x+c))^3+1/4/d*a^2/(-1+sec(d*x+c))^2
-11/16/d*a^2/(-1+sec(d*x+c))-13/16/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 = 1.17354, size = 147, normalized size = 1.16 \begin{align*} -\frac{9 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) + 39 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (33 \, a^{2} \cos \left (d x + c\right )^{3} - 18 \, a^{2} \cos \left (d x + c\right )^{2} - 37 \, a^{2} \cos \left (d x + c\right ) + 26 \, a^{2}\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right ) - 1}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/48*(9*a^2*log(cos(d*x + c) + 1) + 39*a^2*log(cos(d*x + c) - 1) - 2*(33*a^2*cos(d*x + c)^3 - 18*a^2*cos(d*x
+ c)^2 - 37*a^2*cos(d*x + c) + 26*a^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^3 + 2*cos(d*x + c) - 1))/d

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Fricas [A]  time = 0.896504, size = 481, normalized size = 3.79 \begin{align*} \frac{66 \, a^{2} \cos \left (d x + c\right )^{3} - 36 \, a^{2} \cos \left (d x + c\right )^{2} - 74 \, a^{2} \cos \left (d x + c\right ) + 52 \, a^{2} - 9 \,{\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} + 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 ) - 39 \,{\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} + 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 )}{48 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 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)^7*(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/48*(66*a^2*cos(d*x + c)^3 - 36*a^2*cos(d*x + c)^2 - 74*a^2*cos(d*x + c) + 52*a^2 - 9*(a^2*cos(d*x + c)^4 - 2
*a^2*cos(d*x + c)^3 + 2*a^2*cos(d*x + c) - a^2)*log(1/2*cos(d*x + c) + 1/2) - 39*(a^2*cos(d*x + c)^4 - 2*a^2*c
os(d*x + c)^3 + 2*a^2*cos(d*x + c) - a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3
+ 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)**7*(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.6418, size = 251, normalized size = 1.98 \begin{align*} -\frac{78 \, a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 96 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{3 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{{\left (a^{2} + \frac{9 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{48 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{143 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-1/96*(78*a^2*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 96*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) + 1)) - 3*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - (a^2 + 9*a^2*(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) + 48*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 143*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(
cos(d*x + c) + 1)^3/(cos(d*x + c) - 1)^3)/d