3.716 \(\int \frac{\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=142 \[ \frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac{x}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.178634, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2839, 2611, 3770, 3473, 8} \[ \frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot (c+d x)}{a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac{\cot ^5(c+d x) \csc (c+d x)}{6 a d}+\frac{5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac{5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

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

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]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^6(c+d x) \, dx}{a}+\frac{\int \cot ^6(c+d x) \csc (c+d x) \, dx}{a}\\ &=\frac{\cot ^5(c+d x)}{5 a d}-\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{\int \cot ^4(c+d x) \, dx}{a}\\ &=-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot ^5(c+d x)}{5 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{\int \cot ^2(c+d x) \, dx}{a}\\ &=\frac{\cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot ^5(c+d x)}{5 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{\int 1 \, dx}{a}\\ &=\frac{x}{a}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot (c+d x)}{a d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\cot ^5(c+d x)}{5 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 = 0.944289, size = 317, normalized size = 2.23 \[ -\frac{\csc ^6(c+d x) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (-1200 \sin (2 (c+d x))+768 \sin (4 (c+d x))-368 \sin (6 (c+d x))-1440 c \cos (4 (c+d x))+240 c \cos (6 (c+d x))+900 \cos (c+d x)+50 \cos (3 (c+d x))-1440 d x \cos (4 (c+d x))+330 \cos (5 (c+d x))+240 d x \cos (6 (c+d x))+750 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-450 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+75 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-750 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+225 \cos (2 (c+d x)) \left (16 (c+d x)-5 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+5 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+450 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-75 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2400 c-2400 d x\right )}{7680 a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]

[Out]

-(Csc[c + d*x]^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2400*c - 2400*d*x + 900*Cos[c + d*x] + 50*Cos[3*(c
+ d*x)] - 1440*c*Cos[4*(c + d*x)] - 1440*d*x*Cos[4*(c + d*x)] + 330*Cos[5*(c + d*x)] + 240*c*Cos[6*(c + d*x)]
+ 240*d*x*Cos[6*(c + d*x)] - 750*Log[Cos[(c + d*x)/2]] - 450*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 75*Cos[6
*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 225*Cos[2*(c + d*x)]*(16*(c + d*x) + 5*Log[Cos[(c + d*x)/2]] - 5*Log[Sin[(
c + d*x)/2]]) + 750*Log[Sin[(c + d*x)/2]] + 450*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 75*Cos[6*(c + d*x)]*L
og[Sin[(c + d*x)/2]] - 1200*Sin[2*(c + d*x)] + 768*Sin[4*(c + d*x)] - 368*Sin[6*(c + d*x)]))/(7680*a*d*(1 + Si
n[c + d*x]))

________________________________________________________________________________________

Maple [B]  time = 0.156, size = 264, normalized size = 1.9 \begin{align*}{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{1}{160\,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{7}{96\,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{11}{16\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}-{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{1}{160\,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{7}{96\,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{11}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{5}{16\,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)^7/(a+a*sin(d*x+c)),x)

[Out]

1/384/d/a*tan(1/2*d*x+1/2*c)^6-1/160/d/a*tan(1/2*d*x+1/2*c)^5-3/128/d/a*tan(1/2*d*x+1/2*c)^4+7/96/d/a*tan(1/2*
d*x+1/2*c)^3+15/128/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*arctan(tan(1/2*d*x+1/2*c))-1/3
84/d/a/tan(1/2*d*x+1/2*c)^6+1/160/d/a/tan(1/2*d*x+1/2*c)^5+3/128/d/a/tan(1/2*d*x+1/2*c)^4-7/96/d/a/tan(1/2*d*x
+1/2*c)^3-15/128/d/a/tan(1/2*d*x+1/2*c)^2+11/16/d/a/tan(1/2*d*x+1/2*c)-5/16/d/a*ln(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.53579, size = 402, normalized size = 2.83 \begin{align*} -\frac{\frac{\frac{1320 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{12 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a} - \frac{3840 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{600 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{{\left (\frac{12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{225 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1320 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a \sin \left (d x + c\right )^{6}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/1920*((1320*sin(d*x + c)/(cos(d*x + c) + 1) - 225*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 + 45*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 12*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6)/a - 3840*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 600*log(sin(d*x + c)/(c
os(d*x + c) + 1))/a - (12*sin(d*x + c)/(cos(d*x + c) + 1) + 45*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d
*x + c)^3/(cos(d*x + c) + 1)^3 - 225*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1320*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 - 5)*(cos(d*x + c) + 1)^6/(a*sin(d*x + c)^6))/d

________________________________________________________________________________________

Fricas [A]  time = 1.2034, size = 660, normalized size = 4.65 \begin{align*} \frac{480 \, d x \cos \left (d x + c\right )^{6} - 1440 \, d x \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{5} + 1440 \, d x \cos \left (d x + c\right )^{2} - 400 \, \cos \left (d x + c\right )^{3} - 480 \, d x + 75 \,{\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 ) - 75 \,{\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 ) - 32 \,{\left (23 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 150 \, \cos \left (d x + c\right )}{480 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/480*(480*d*x*cos(d*x + c)^6 - 1440*d*x*cos(d*x + c)^4 + 330*cos(d*x + c)^5 + 1440*d*x*cos(d*x + c)^2 - 400*c
os(d*x + c)^3 - 480*d*x + 75*(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) - 75*(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) - 32*(23*co
s(d*x + c)^5 - 35*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c) + 150*cos(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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.36038, size = 302, normalized size = 2.13 \begin{align*} \frac{\frac{1920 \,{\left (d x + c\right )}}{a} - \frac{600 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{5 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 12 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 140 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1320 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}} + \frac{1470 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1320 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 225 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 140 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 5}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1920*(1920*(d*x + c)/a - 600*log(abs(tan(1/2*d*x + 1/2*c)))/a + (5*a^5*tan(1/2*d*x + 1/2*c)^6 - 12*a^5*tan(1
/2*d*x + 1/2*c)^5 - 45*a^5*tan(1/2*d*x + 1/2*c)^4 + 140*a^5*tan(1/2*d*x + 1/2*c)^3 + 225*a^5*tan(1/2*d*x + 1/2
*c)^2 - 1320*a^5*tan(1/2*d*x + 1/2*c))/a^6 + (1470*tan(1/2*d*x + 1/2*c)^6 + 1320*tan(1/2*d*x + 1/2*c)^5 - 225*
tan(1/2*d*x + 1/2*c)^4 - 140*tan(1/2*d*x + 1/2*c)^3 + 45*tan(1/2*d*x + 1/2*c)^2 + 12*tan(1/2*d*x + 1/2*c) - 5)
/(a*tan(1/2*d*x + 1/2*c)^6))/d