3.632 \(\int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=102 \[ \frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {x}{a} \]

[Out]

-x/a-3/8*arctanh(cos(d*x+c))/a/d-cot(d*x+c)/a/d+1/3*cot(d*x+c)^3/a/d+3/8*cot(d*x+c)*csc(d*x+c)/a/d-1/4*cot(d*x
+c)^3*csc(d*x+c)/a/d

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Rubi [A]  time = 0.13, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 8

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

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 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 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 {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^4(c+d x) \, dx}{a}+\frac {\int \cot ^4(c+d x) \csc (c+d x) \, dx}{a}\\ &=\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac {3 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{4 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 {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac {3 \int \csc (c+d x) \, dx}{8 a}-\frac {\int 1 \, dx}{a}\\ &=-\frac {x}{a}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}\\ \end {align*}

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Mathematica [B]  time = 0.67, size = 232, normalized size = 2.27 \[ -\frac {\csc ^4(c+d x) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (32 \sin (2 (c+d x))-32 \sin (4 (c+d x))+24 c \cos (4 (c+d x))+18 \cos (c+d x)+30 \cos (3 (c+d x))+24 d x \cos (4 (c+d x))-27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 \cos (2 (c+d x)) \left (-3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 c+8 d x\right )-9 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+72 c+72 d x\right )}{192 a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

-1/192*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(72*c + 72*d*x + 18*Cos[c + d*x] + 30*Cos[3*(c
+ d*x)] + 24*c*Cos[4*(c + d*x)] + 24*d*x*Cos[4*(c + d*x)] + 27*Log[Cos[(c + d*x)/2]] + 9*Cos[4*(c + d*x)]*Log[
Cos[(c + d*x)/2]] - 12*Cos[2*(c + d*x)]*(8*c + 8*d*x + 3*Log[Cos[(c + d*x)/2]] - 3*Log[Sin[(c + d*x)/2]]) - 27
*Log[Sin[(c + d*x)/2]] - 9*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 32*Sin[2*(c + d*x)] - 32*Sin[4*(c + d*x)])
)/(a*d*(1 + Sin[c + d*x]))

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fricas [A]  time = 0.91, size = 171, normalized size = 1.68 \[ -\frac {48 \, d x \cos \left (d x + c\right )^{4} - 96 \, d x \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right )^{3} + 48 \, d x + 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (4 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 18 \, \cos \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/48*(48*d*x*cos(d*x + c)^4 - 96*d*x*cos(d*x + c)^2 + 30*cos(d*x + c)^3 + 48*d*x + 9*(cos(d*x + c)^4 - 2*cos(
d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 9*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c)
+ 1/2) - 16*(4*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c) - 18*cos(d*x + c))/(a*d*cos(d*x + c)^4 - 2*a*d*co
s(d*x + c)^2 + a*d)

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giac [A]  time = 0.20, size = 167, normalized size = 1.64 \[ -\frac {\frac {192 \, {\left (d x + c\right )}}{a} - \frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {150 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(192*(d*x + c)/a - 72*log(abs(tan(1/2*d*x + 1/2*c)))/a - (3*a^3*tan(1/2*d*x + 1/2*c)^4 - 8*a^3*tan(1/2*
d*x + 1/2*c)^3 - 24*a^3*tan(1/2*d*x + 1/2*c)^2 + 120*a^3*tan(1/2*d*x + 1/2*c))/a^4 + (150*tan(1/2*d*x + 1/2*c)
^4 + 120*tan(1/2*d*x + 1/2*c)^3 - 24*tan(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 3)/(a*tan(1/2*d*x + 1/2
*c)^4))/d

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maple [A]  time = 0.48, size = 188, normalized size = 1.84 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1}{64 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{24 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {5}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c)),x)

[Out]

1/64/a/d*tan(1/2*d*x+1/2*c)^4-1/24/a/d*tan(1/2*d*x+1/2*c)^3-1/8/a/d*tan(1/2*d*x+1/2*c)^2+5/8/a/d*tan(1/2*d*x+1
/2*c)-2/a/d*arctan(tan(1/2*d*x+1/2*c))-1/64/a/d/tan(1/2*d*x+1/2*c)^4+1/24/a/d/tan(1/2*d*x+1/2*c)^3+1/8/a/d/tan
(1/2*d*x+1/2*c)^2-5/8/a/d/tan(1/2*d*x+1/2*c)+3/8/a/d*ln(tan(1/2*d*x+1/2*c))

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maxima [B]  time = 0.48, size = 217, normalized size = 2.13 \[ \frac {\frac {\frac {120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a} - \frac {384 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {72 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a \sin \left (d x + c\right )^{4}}}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/192*((120*sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/a - 384*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 72
*log(sin(d*x + c)/(cos(d*x + c) + 1))/a + (8*sin(d*x + c)/(cos(d*x + c) + 1) + 24*sin(d*x + c)^2/(cos(d*x + c)
+ 1)^2 - 120*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3)*(cos(d*x + c) + 1)^4/(a*sin(d*x + c)^4))/d

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mupad [B]  time = 9.42, size = 317, normalized size = 3.11 \[ \frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+384\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+72\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^5*(a + a*sin(c + d*x))),x)

[Out]

(3*sin(c/2 + (d*x)/2)^8 - 3*cos(c/2 + (d*x)/2)^8 - 8*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^7 + 8*cos(c/2 + (d*
x)/2)^7*sin(c/2 + (d*x)/2) - 24*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^6 + 120*cos(c/2 + (d*x)/2)^3*sin(c/2 +
(d*x)/2)^5 - 120*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^3 + 24*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^2 + 3
84*atan((8*cos(c/2 + (d*x)/2) - 3*sin(c/2 + (d*x)/2))/(3*cos(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*cos(c/2 +
(d*x)/2)^4*sin(c/2 + (d*x)/2)^4 + 72*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^4*sin(c/2
+ (d*x)/2)^4)/(192*a*d*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**5/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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