3.45 \(\int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=107 \[ -\frac{17 a^3}{8 d (1-\cos (c+d x))}+\frac{7 a^3}{8 d (1-\cos (c+d x))^2}-\frac{a^3}{6 d (1-\cos (c+d x))^3}-\frac{15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{a^3 \log (\cos (c+d x)+1)}{16 d} \]

[Out]

-a^3/(6*d*(1 - Cos[c + d*x])^3) + (7*a^3)/(8*d*(1 - Cos[c + d*x])^2) - (17*a^3)/(8*d*(1 - Cos[c + d*x])) - (15
*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (a^3*Log[1 + Cos[c + d*x]])/(16*d)

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Rubi [A]  time = 0.0763585, antiderivative size = 107, 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} \[ -\frac{17 a^3}{8 d (1-\cos (c+d x))}+\frac{7 a^3}{8 d (1-\cos (c+d x))^2}-\frac{a^3}{6 d (1-\cos (c+d x))^3}-\frac{15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{a^3 \log (\cos (c+d x)+1)}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]

[Out]

-a^3/(6*d*(1 - Cos[c + d*x])^3) + (7*a^3)/(8*d*(1 - Cos[c + d*x])^2) - (17*a^3)/(8*d*(1 - Cos[c + d*x])) - (15
*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (a^3*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))^3 \, dx &=-\frac{a^8 \operatorname{Subst}\left (\int \frac{x^4}{(a-a x)^4 (a+a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^8 \operatorname{Subst}\left (\int \left (\frac{1}{2 a^5 (-1+x)^4}+\frac{7}{4 a^5 (-1+x)^3}+\frac{17}{8 a^5 (-1+x)^2}+\frac{15}{16 a^5 (-1+x)}+\frac{1}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^3}{6 d (1-\cos (c+d x))^3}+\frac{7 a^3}{8 d (1-\cos (c+d x))^2}-\frac{17 a^3}{8 d (1-\cos (c+d x))}-\frac{15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{a^3 \log (1+\cos (c+d x))}{16 d}\\ \end{align*}

Mathematica [A]  time = 0.669128, size = 102, normalized size = 0.95 \[ -\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (2 \csc ^6\left (\frac{1}{2} (c+d x)\right )-21 \csc ^4\left (\frac{1}{2} (c+d x)\right )+102 \csc ^2\left (\frac{1}{2} (c+d x)\right )+12 \left (15 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{768 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3,x]

[Out]

-(a^3*(1 + Cos[c + d*x])^3*(102*Csc[(c + d*x)/2]^2 - 21*Csc[(c + d*x)/2]^4 + 2*Csc[(c + d*x)/2]^6 + 12*(Log[Co
s[(c + d*x)/2]] + 15*Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^6)/(768*d)

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Maple [A]  time = 0.087, size = 104, normalized size = 1. \begin{align*} -{\frac{{a}^{3}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{{a}^{3}}{6\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}+{\frac{3\,{a}^{3}}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{7\,{a}^{3}}{8\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}-{\frac{15\,{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{{a}^{3}\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))^3,x)

[Out]

-1/16/d*a^3*ln(1+sec(d*x+c))-1/6/d*a^3/(-1+sec(d*x+c))^3+3/8/d*a^3/(-1+sec(d*x+c))^2-7/8/d*a^3/(-1+sec(d*x+c))
-15/16/d*a^3*ln(-1+sec(d*x+c))+1/d*a^3*ln(sec(d*x+c))

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Maxima [A]  time = 1.2166, size = 130, normalized size = 1.21 \begin{align*} -\frac{3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 45 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (51 \, a^{3} \cos \left (d x + c\right )^{2} - 81 \, a^{3} \cos \left (d x + c\right ) + 34 \, a^{3}\right )}}{\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 3 \, \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))^3,x, algorithm="maxima")

[Out]

-1/48*(3*a^3*log(cos(d*x + c) + 1) + 45*a^3*log(cos(d*x + c) - 1) - 2*(51*a^3*cos(d*x + c)^2 - 81*a^3*cos(d*x
+ c) + 34*a^3)/(cos(d*x + c)^3 - 3*cos(d*x + c)^2 + 3*cos(d*x + c) - 1))/d

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Fricas [A]  time = 1.19566, size = 451, normalized size = 4.21 \begin{align*} \frac{102 \, a^{3} \cos \left (d x + c\right )^{2} - 162 \, a^{3} \cos \left (d x + c\right ) + 68 \, a^{3} - 3 \,{\left (a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 45 \,{\left (a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}\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 )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, 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))^3,x, algorithm="fricas")

[Out]

1/48*(102*a^3*cos(d*x + c)^2 - 162*a^3*cos(d*x + c) + 68*a^3 - 3*(a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 +
3*a^3*cos(d*x + c) - a^3)*log(1/2*cos(d*x + c) + 1/2) - 45*(a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + 3*a^3*
cos(d*x + c) - a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 + 3*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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.61034, size = 223, normalized size = 2.08 \begin{align*} -\frac{90 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 96 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (2 \, a^{3} + \frac{15 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{66 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{165 \, a^{3}{\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))^3,x, algorithm="giac")

[Out]

-1/96*(90*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 96*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) + 1)) - (2*a^3 + 15*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 66*a^3*(cos(d*x + c) - 1)^2/(cos(d*x
+ c) + 1)^2 + 165*a^3*(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