3.715 \(\int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=138 \[ -\frac{15 \cos (c+d x)}{8 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{x}{a} \]

[Out]

-(x/a) + (15*ArcTanh[Cos[c + d*x]])/(8*a*d) - (15*Cos[c + d*x])/(8*a*d) - Cot[c + d*x]/(a*d) - (5*Cos[c + d*x]
*Cot[c + d*x]^2)/(8*a*d) + Cot[c + d*x]^3/(3*a*d) + (Cos[c + d*x]*Cot[c + d*x]^4)/(4*a*d) - Cot[c + d*x]^5/(5*
a*d)

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Rubi [A]  time = 0.151258, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2839, 3473, 8, 2592, 288, 321, 206} \[ -\frac{15 \cos (c+d x)}{8 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot (c+d x)}{a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]

[Out]

-(x/a) + (15*ArcTanh[Cos[c + d*x]])/(8*a*d) - (15*Cos[c + d*x])/(8*a*d) - Cot[c + d*x]/(a*d) - (5*Cos[c + d*x]
*Cot[c + d*x]^2)/(8*a*d) + Cot[c + d*x]^3/(3*a*d) + (Cos[c + d*x]*Cot[c + d*x]^4)/(4*a*d) - Cot[c + d*x]^5/(5*
a*d)

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cos (c+d x) \cot ^5(c+d x) \, dx}{a}+\frac{\int \cot ^6(c+d x) \, dx}{a}\\ &=-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\int \cot ^4(c+d x) \, dx}{a}+\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\int \cot ^2(c+d x) \, dx}{a}-\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 a d}\\ &=-\frac{\cot (c+d x)}{a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\int 1 \, dx}{a}+\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 a d}\\ &=-\frac{x}{a}-\frac{15 \cos (c+d x)}{8 a d}-\frac{\cot (c+d x)}{a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 a d}\\ &=-\frac{x}{a}+\frac{15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{15 \cos (c+d x)}{8 a d}-\frac{\cot (c+d x)}{a d}-\frac{5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}+\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac{\cot ^5(c+d x)}{5 a d}\\ \end{align*}

Mathematica [A]  time = 0.994794, size = 264, normalized size = 1.91 \[ -\frac{\csc ^5(c+d x) \left (1200 c \sin (c+d x)+1200 d x \sin (c+d x)+600 \sin (2 (c+d x))-600 c \sin (3 (c+d x))-600 d x \sin (3 (c+d x))-510 \sin (4 (c+d x))+120 c \sin (5 (c+d x))+120 d x \sin (5 (c+d x))+60 \sin (6 (c+d x))+400 \cos (c+d x)-200 \cos (3 (c+d x))+184 \cos (5 (c+d x))+2250 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-1125 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+225 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2250 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+1125 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-225 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1920 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]

[Out]

-(Csc[c + d*x]^5*(400*Cos[c + d*x] - 200*Cos[3*(c + d*x)] + 184*Cos[5*(c + d*x)] + 1200*c*Sin[c + d*x] + 1200*
d*x*Sin[c + d*x] - 2250*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 2250*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 600*Sin
[2*(c + d*x)] - 600*c*Sin[3*(c + d*x)] - 600*d*x*Sin[3*(c + d*x)] + 1125*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)
] - 1125*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 510*Sin[4*(c + d*x)] + 120*c*Sin[5*(c + d*x)] + 120*d*x*Sin[
5*(c + d*x)] - 225*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 225*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 60*Si
n[6*(c + d*x)]))/(1920*a*d)

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Maple [A]  time = 0.154, size = 249, normalized size = 1.8 \begin{align*}{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{7}{96\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{11}{16\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{1}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{7}{96\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{11}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{15}{8\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^6/(a+a*sin(d*x+c)),x)

[Out]

1/160/d/a*tan(1/2*d*x+1/2*c)^5-1/64/d/a*tan(1/2*d*x+1/2*c)^4-7/96/d/a*tan(1/2*d*x+1/2*c)^3+1/4/d/a*tan(1/2*d*x
+1/2*c)^2+11/16/d/a*tan(1/2*d*x+1/2*c)-2/a/d/(1+tan(1/2*d*x+1/2*c)^2)-2/a/d*arctan(tan(1/2*d*x+1/2*c))-1/160/d
/a/tan(1/2*d*x+1/2*c)^5+1/64/d/a/tan(1/2*d*x+1/2*c)^4+7/96/d/a/tan(1/2*d*x+1/2*c)^3-1/4/d/a/tan(1/2*d*x+1/2*c)
^2-11/16/d/a/tan(1/2*d*x+1/2*c)-15/8/d/a*ln(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.52293, size = 431, normalized size = 3.12 \begin{align*} \frac{\frac{\frac{660 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a} + \frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{64 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{225 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{590 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{2160 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{660 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 6}{\frac{a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac{1920 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{1800 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/960*((660*sin(d*x + c)/(cos(d*x + c) + 1) + 240*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 70*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 - 15*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 6*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a + (15*sin
(d*x + c)/(cos(d*x + c) + 1) + 64*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 225*sin(d*x + c)^3/(cos(d*x + c) + 1)^
3 - 590*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2160*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 660*sin(d*x + c)^6/(c
os(d*x + c) + 1)^6 - 6)/(a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) - 1920
*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - 1800*log(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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Fricas [A]  time = 1.16632, size = 593, normalized size = 4.3 \begin{align*} -\frac{368 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} - 225 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \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 ) + 225 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \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 ) + 30 \,{\left (8 \, d x \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{5} - 16 \, d x \cos \left (d x + c\right )^{2} - 25 \, \cos \left (d x + c\right )^{3} + 8 \, d x + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 240 \, \cos \left (d x + c\right )}{240 \,{\left (a d \cos \left (d x + c\right )^{4} - 2 \, 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(cos(d*x+c)^8*csc(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/240*(368*cos(d*x + c)^5 - 560*cos(d*x + c)^3 - 225*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x
+ c) + 1/2)*sin(d*x + c) + 225*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x +
c) + 30*(8*d*x*cos(d*x + c)^4 + 8*cos(d*x + c)^5 - 16*d*x*cos(d*x + c)^2 - 25*cos(d*x + c)^3 + 8*d*x + 15*cos(
d*x + c))*sin(d*x + c) + 240*cos(d*x + c))/((a*d*cos(d*x + c)^4 - 2*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(cos(d*x+c)**8*csc(d*x+c)**6/(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.38203, size = 293, normalized size = 2.12 \begin{align*} -\frac{\frac{960 \,{\left (d x + c\right )}}{a} + \frac{1800 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{1920}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a} - \frac{6 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 70 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 240 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 660 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{5}} - \frac{4110 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 660 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 70 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/960*(960*(d*x + c)/a + 1800*log(abs(tan(1/2*d*x + 1/2*c)))/a + 1920/((tan(1/2*d*x + 1/2*c)^2 + 1)*a) - (6*a
^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^4*tan(1/2*d*x + 1/2*c)^4 - 70*a^4*tan(1/2*d*x + 1/2*c)^3 + 240*a^4*tan(1/2*d*
x + 1/2*c)^2 + 660*a^4*tan(1/2*d*x + 1/2*c))/a^5 - (4110*tan(1/2*d*x + 1/2*c)^5 - 660*tan(1/2*d*x + 1/2*c)^4 -
240*tan(1/2*d*x + 1/2*c)^3 + 70*tan(1/2*d*x + 1/2*c)^2 + 15*tan(1/2*d*x + 1/2*c) - 6)/(a*tan(1/2*d*x + 1/2*c)
^5))/d