3.46 \(\int \cot ^9(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=149 \[ \frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d} \]

[Out]

-a^3/(16*d*(1 - Cos[c + d*x])^4) + (5*a^3)/(12*d*(1 - Cos[c + d*x])^3) - (39*a^3)/(32*d*(1 - Cos[c + d*x])^2)
+ (9*a^3)/(4*d*(1 - Cos[c + d*x])) + a^3/(32*d*(1 + Cos[c + d*x])) + (57*a^3*Log[1 - Cos[c + d*x]])/(64*d) + (
7*a^3*Log[1 + Cos[c + d*x]])/(64*d)

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Rubi [A]  time = 0.097923, antiderivative size = 149, 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} \[ \frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (\cos (c+d x)+1)}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (\cos (c+d x)+1)}{64 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]

[Out]

-a^3/(16*d*(1 - Cos[c + d*x])^4) + (5*a^3)/(12*d*(1 - Cos[c + d*x])^3) - (39*a^3)/(32*d*(1 - Cos[c + d*x])^2)
+ (9*a^3)/(4*d*(1 - Cos[c + d*x])) + a^3/(32*d*(1 + Cos[c + d*x])) + (57*a^3*Log[1 - Cos[c + d*x]])/(64*d) + (
7*a^3*Log[1 + Cos[c + d*x]])/(64*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 ^9(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\frac{a^{10} \operatorname{Subst}\left (\int \frac{x^6}{(a-a x)^5 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^{10} \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^7 (-1+x)^5}-\frac{5}{4 a^7 (-1+x)^4}-\frac{39}{16 a^7 (-1+x)^3}-\frac{9}{4 a^7 (-1+x)^2}-\frac{57}{64 a^7 (-1+x)}+\frac{1}{32 a^7 (1+x)^2}-\frac{7}{64 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^3}{16 d (1-\cos (c+d x))^4}+\frac{5 a^3}{12 d (1-\cos (c+d x))^3}-\frac{39 a^3}{32 d (1-\cos (c+d x))^2}+\frac{9 a^3}{4 d (1-\cos (c+d x))}+\frac{a^3}{32 d (1+\cos (c+d x))}+\frac{57 a^3 \log (1-\cos (c+d x))}{64 d}+\frac{7 a^3 \log (1+\cos (c+d x))}{64 d}\\ \end{align*}

Mathematica [A]  time = 0.334718, size = 130, normalized size = 0.87 \[ \frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (-3 \csc ^8\left (\frac{1}{2} (c+d x)\right )+40 \csc ^6\left (\frac{1}{2} (c+d x)\right )-234 \csc ^4\left (\frac{1}{2} (c+d x)\right )+864 \csc ^2\left (\frac{1}{2} (c+d x)\right )+12 \left (\sec ^2\left (\frac{1}{2} (c+d x)\right )+114 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+14 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{6144 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(864*Csc[(c + d*x)/2]^2 - 234*Csc[(c + d*x)/2]^4 + 40*Csc[(c + d*
x)/2]^6 - 3*Csc[(c + d*x)/2]^8 + 12*(14*Log[Cos[(c + d*x)/2]] + 114*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2
)))/(6144*d)

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Maple [A]  time = 0.096, size = 141, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{32\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}+{\frac{7\,{a}^{3}\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{64\,d}}-{\frac{{a}^{3}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{3}}{6\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}-{\frac{11\,{a}^{3}}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{13\,{a}^{3}}{16\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}+{\frac{57\,{a}^{3}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{64\,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)^9*(a+a*sec(d*x+c))^3,x)

[Out]

-1/32/d*a^3/(1+sec(d*x+c))+7/64/d*a^3*ln(1+sec(d*x+c))-1/16/d*a^3/(-1+sec(d*x+c))^4+1/6/d*a^3/(-1+sec(d*x+c))^
3-11/32/d*a^3/(-1+sec(d*x+c))^2+13/16/d*a^3/(-1+sec(d*x+c))+57/64/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 = 1.6136, size = 192, normalized size = 1.29 \begin{align*} \frac{21 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 171 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (213 \, a^{3} \cos \left (d x + c\right )^{4} - 303 \, a^{3} \cos \left (d x + c\right )^{3} - 95 \, a^{3} \cos \left (d x + c\right )^{2} + 333 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/192*(21*a^3*log(cos(d*x + c) + 1) + 171*a^3*log(cos(d*x + c) - 1) - 2*(213*a^3*cos(d*x + c)^4 - 303*a^3*cos(
d*x + c)^3 - 95*a^3*cos(d*x + c)^2 + 333*a^3*cos(d*x + c) - 136*a^3)/(cos(d*x + c)^5 - 3*cos(d*x + c)^4 + 2*co
s(d*x + c)^3 + 2*cos(d*x + c)^2 - 3*cos(d*x + c) + 1))/d

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Fricas [B]  time = 1.22619, size = 706, normalized size = 4.74 \begin{align*} -\frac{426 \, a^{3} \cos \left (d x + c\right )^{4} - 606 \, a^{3} \cos \left (d x + c\right )^{3} - 190 \, a^{3} \cos \left (d x + c\right )^{2} + 666 \, a^{3} \cos \left (d x + c\right ) - 272 \, a^{3} - 21 \,{\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, 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 ) - 171 \,{\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} + 2 \, 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 )}{192 \,{\left (d \cos \left (d x + c\right )^{5} - 3 \, 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 )^{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)^9*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/192*(426*a^3*cos(d*x + c)^4 - 606*a^3*cos(d*x + c)^3 - 190*a^3*cos(d*x + c)^2 + 666*a^3*cos(d*x + c) - 272*
a^3 - 21*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 + 2*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) - 171*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x +
c)^3 + 2*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)^5 - 3*d
*cos(d*x + c)^4 + 2*d*cos(d*x + c)^3 + 2*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)**9*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.44596, size = 288, normalized size = 1.93 \begin{align*} \frac{684 \, a^{3} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 768 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{12 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{{\left (3 \, a^{3} + \frac{28 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{132 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{504 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{1425 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}}{768 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/768*(684*a^3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 768*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*
x + c) + 1) + 1)) - 12*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - (3*a^3 + 28*a^3*(cos(d*x + c) - 1)/(cos(d*x
+ c) + 1) + 132*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 504*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) +
1)^3 + 1425*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)*(cos(d*x + c) + 1)^4/(cos(d*x + c) - 1)^4)/d