3.60 \(\int \frac{\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=106 \[ -\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d} \]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(16*a*d) - Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (5*Cot
[c + d*x]^3*Csc[c + d*x])/(24*a*d) + (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)

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Rubi [A]  time = 0.127459, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2706, 2607, 30, 2611, 3770} \[ -\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^8/(a + a*Sin[c + d*x]),x]

[Out]

(-5*ArcTanh[Cos[c + d*x]])/(16*a*d) - Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (5*Cot
[c + d*x]^3*Csc[c + d*x])/(24*a*d) + (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cot ^8(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^6(c+d x) \csc (c+d x) \, dx}{a}+\frac{\int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a}\\ &=\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac{5 \int \cot ^4(c+d x) \csc (c+d x) \, dx}{6 a}+\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}-\frac{5 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{8 a}\\ &=-\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac{5 \int \csc (c+d x) \, dx}{16 a}\\ &=-\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}-\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}+\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}\\ \end{align*}

Mathematica [B]  time = 0.914319, size = 284, normalized size = 2.68 \[ -\frac{\csc ^5(c+d x) \left (\csc \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (-1190 \sin (2 (c+d x))+392 \sin (4 (c+d x))-462 \sin (6 (c+d x))+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-3675 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2205 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-735 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+3675 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2205 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+735 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-105 \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{86016 a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^8/(a + a*Sin[c + d*x]),x]

[Out]

-(Csc[c + d*x]^5*(Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^2*(1680*Cos[c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[
5*(c + d*x)] + 48*Cos[7*(c + d*x)] + 3675*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 3675*Log[Sin[(c + d*x)/2]]*Sin[
c + d*x] - 1190*Sin[2*(c + d*x)] - 2205*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2205*Log[Sin[(c + d*x)/2]]*Si
n[3*(c + d*x)] + 392*Sin[4*(c + d*x)] + 735*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 735*Log[Sin[(c + d*x)/2]]
*Sin[5*(c + d*x)] - 462*Sin[6*(c + d*x)] - 105*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*Log[Sin[(c + d*x)/
2]]*Sin[7*(c + d*x)]))/(86016*a*d*(1 + Sin[c + d*x]))

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Maple [B]  time = 0.102, size = 284, normalized size = 2.7 \begin{align*}{\frac{1}{896\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{3}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{15}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{5}{128\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{896\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{5}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{3}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{5}{16\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}-{\frac{3}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{15}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^8/(a+a*sin(d*x+c)),x)

[Out]

1/896/d/a*tan(1/2*d*x+1/2*c)^7-1/384/d/a*tan(1/2*d*x+1/2*c)^6-1/128/d/a*tan(1/2*d*x+1/2*c)^5+3/128/d/a*tan(1/2
*d*x+1/2*c)^4+3/128/d/a*tan(1/2*d*x+1/2*c)^3-15/128/d/a*tan(1/2*d*x+1/2*c)^2-5/128/d/a*tan(1/2*d*x+1/2*c)-1/89
6/d/a/tan(1/2*d*x+1/2*c)^7+5/128/d/a/tan(1/2*d*x+1/2*c)+1/128/d/a/tan(1/2*d*x+1/2*c)^5-3/128/d/a/tan(1/2*d*x+1
/2*c)^4+5/16/d/a*ln(tan(1/2*d*x+1/2*c))+1/384/d/a/tan(1/2*d*x+1/2*c)^6-3/128/d/a/tan(1/2*d*x+1/2*c)^3+15/128/d
/a/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.12476, size = 425, normalized size = 4.01 \begin{align*} -\frac{\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{315 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{63 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{7 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} - \frac{840 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{{\left (\frac{7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{63 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{315 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 3\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a \sin \left (d x + c\right )^{7}}}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2688*((105*sin(d*x + c)/(cos(d*x + c) + 1) + 315*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 63*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 63*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 7*sin(d
*x + c)^6/(cos(d*x + c) + 1)^6 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a - 840*log(sin(d*x + c)/(cos(d*x + c)
+ 1))/a - (7*sin(d*x + c)/(cos(d*x + c) + 1) + 21*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 63*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 - 63*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 315*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 105*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6 - 3)*(cos(d*x + c) + 1)^7/(a*sin(d*x + c)^7))/d

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Fricas [B]  time = 1.57927, size = 545, normalized size = 5.14 \begin{align*} \frac{96 \, \cos \left (d x + c\right )^{7} - 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 105 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 14 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \,{\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/672*(96*cos(d*x + c)^7 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c)
+ 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) +
1/2)*sin(d*x + c) - 14*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c))/((a*d*cos(d*x +
c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))

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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)**8/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.97865, size = 329, normalized size = 3.1 \begin{align*} \frac{\frac{840 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{3 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 7 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 21 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 63 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 63 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 315 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 105 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{7}} - \frac{2178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 63 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2688*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a + (3*a^6*tan(1/2*d*x + 1/2*c)^7 - 7*a^6*tan(1/2*d*x + 1/2*c)^6 -
21*a^6*tan(1/2*d*x + 1/2*c)^5 + 63*a^6*tan(1/2*d*x + 1/2*c)^4 + 63*a^6*tan(1/2*d*x + 1/2*c)^3 - 315*a^6*tan(1/
2*d*x + 1/2*c)^2 - 105*a^6*tan(1/2*d*x + 1/2*c))/a^7 - (2178*tan(1/2*d*x + 1/2*c)^7 - 105*tan(1/2*d*x + 1/2*c)
^6 - 315*tan(1/2*d*x + 1/2*c)^5 + 63*tan(1/2*d*x + 1/2*c)^4 + 63*tan(1/2*d*x + 1/2*c)^3 - 21*tan(1/2*d*x + 1/2
*c)^2 - 7*tan(1/2*d*x + 1/2*c) + 3)/(a*tan(1/2*d*x + 1/2*c)^7))/d