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

Optimal. Leaf size=133 \[ -\frac{15 a}{16 d (1-\cos (c+d x))}-\frac{a}{4 d (\cos (c+d x)+1)}+\frac{9 a}{32 d (1-\cos (c+d x))^2}+\frac{a}{32 d (\cos (c+d x)+1)^2}-\frac{a}{24 d (1-\cos (c+d x))^3}-\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{11 a \log (\cos (c+d x)+1)}{32 d} \]

[Out]

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

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Rubi [A]  time = 0.0823685, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 88} \[ -\frac{15 a}{16 d (1-\cos (c+d x))}-\frac{a}{4 d (\cos (c+d x)+1)}+\frac{9 a}{32 d (1-\cos (c+d x))^2}+\frac{a}{32 d (\cos (c+d x)+1)^2}-\frac{a}{24 d (1-\cos (c+d x))^3}-\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{11 a \log (\cos (c+d x)+1)}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.378039, size = 165, normalized size = 1.24 \[ -\frac{a \left (64 \cot ^6(c+d x)-96 \cot ^4(c+d x)+192 \cot ^2(c+d x)+\csc ^6\left (\frac{1}{2} (c+d x)\right )-12 \csc ^4\left (\frac{1}{2} (c+d x)\right )+66 \csc ^2\left (\frac{1}{2} (c+d x)\right )-\sec ^6\left (\frac{1}{2} (c+d x)\right )+12 \sec ^4\left (\frac{1}{2} (c+d x)\right )-66 \sec ^2\left (\frac{1}{2} (c+d x)\right )+120 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+384 \log (\tan (c+d x))-120 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+384 \log (\cos (c+d x))\right )}{384 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(192*Cot[c + d*x]^2 - 96*Cot[c + d*x]^4 + 64*Cot[c + d*x]^6 + 66*Csc[(c + d*x)/2]^2 - 12*Csc[(c + d*x)/2]^
4 + Csc[(c + d*x)/2]^6 - 120*Log[Cos[(c + d*x)/2]] + 384*Log[Cos[c + d*x]] + 120*Log[Sin[(c + d*x)/2]] + 384*L
og[Tan[c + d*x]] - 66*Sec[(c + d*x)/2]^2 + 12*Sec[(c + d*x)/2]^4 - Sec[(c + d*x)/2]^6))/(384*d)

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Maple [A]  time = 0.087, size = 124, normalized size = 0.9 \begin{align*}{\frac{a}{32\,d \left ( 1+\sec \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a}{16\,d \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{11\,a\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{32\,d}}-{\frac{a}{24\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5\,a}{32\,d \left ( -1+\sec \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a}{2\,d \left ( -1+\sec \left ( dx+c \right ) \right ) }}-{\frac{21\,a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{32\,d}}+{\frac{a\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)),x)

[Out]

1/32/d*a/(1+sec(d*x+c))^2+3/16/d*a/(1+sec(d*x+c))-11/32/d*a*ln(1+sec(d*x+c))-1/24/d*a/(-1+sec(d*x+c))^3+5/32/d
*a/(-1+sec(d*x+c))^2-1/2/d*a/(-1+sec(d*x+c))-21/32/d*a*ln(-1+sec(d*x+c))+1/d*a*ln(sec(d*x+c))

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Maxima [A]  time = 1.07772, size = 170, normalized size = 1.28 \begin{align*} -\frac{33 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 63 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (33 \, a \cos \left (d x + c\right )^{4} + 39 \, a \cos \left (d x + c\right )^{3} - 79 \, a \cos \left (d x + c\right )^{2} - 29 \, a \cos \left (d x + c\right ) + 44 \, a\right )}}{\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}}{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)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.00248, size = 635, normalized size = 4.77 \begin{align*} \frac{66 \, a \cos \left (d x + c\right )^{4} + 78 \, a \cos \left (d x + c\right )^{3} - 158 \, a \cos \left (d x + c\right )^{2} - 58 \, a \cos \left (d x + c\right ) - 33 \,{\left (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\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 63 \,{\left (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\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 88 \, a}{96 \,{\left (d \cos \left (d x + c\right )^{5} - 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} + 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)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.62899, size = 266, normalized size = 2. \begin{align*} -\frac{252 \, a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \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 + \frac{21 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{132 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{462 \, a{\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}} - \frac{42 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{3 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{384 \, 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)),x, algorithm="giac")

[Out]

-1/384*(252*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 384*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x +
c) + 1) + 1)) - (2*a + 21*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 132*a*(cos(d*x + c) - 1)^2/(cos(d*x + c)
+ 1)^2 + 462*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)*(cos(d*x + c) + 1)^3/(cos(d*x + c) - 1)^3 - 42*a*(co
s(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/d