3.652 \(\int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=114 \[ -\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}+\frac {13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]

[Out]

13/8*arctanh(cos(d*x+c))/a^3/d-4*cot(d*x+c)/a^3/d-5/3*cot(d*x+c)^3/a^3/d-1/5*cot(d*x+c)^5/a^3/d+13/8*cot(d*x+c
)*csc(d*x+c)/a^3/d+3/4*cot(d*x+c)*csc(d*x+c)^3/a^3/d

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Rubi [A]  time = 0.18, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2708, 2757, 3768, 3770, 3767} \[ -\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}+\frac {13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^3,x]

[Out]

(13*ArcTanh[Cos[c + d*x]])/(8*a^3*d) - (4*Cot[c + d*x])/(a^3*d) - (5*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^
5/(5*a^3*d) + (13*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)

Rule 2708

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Sin[
e + f*x]^p/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] &&
EqQ[p, 2*m]

Rule 2757

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \csc ^6(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \csc ^3(c+d x)+3 a^3 \csc ^4(c+d x)-3 a^3 \csc ^5(c+d x)+a^3 \csc ^6(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \csc ^3(c+d x) \, dx}{a^3}+\frac {\int \csc ^6(c+d x) \, dx}{a^3}+\frac {3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^5(c+d x) \, dx}{a^3}\\ &=\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {\int \csc (c+d x) \, dx}{2 a^3}-\frac {9 \int \csc ^3(c+d x) \, dx}{4 a^3}-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {9 \int \csc (c+d x) \, dx}{8 a^3}\\ &=\frac {13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}\\ \end {align*}

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Mathematica [A]  time = 1.80, size = 189, normalized size = 1.66 \[ \frac {\csc ^5(c+d x) \left (1500 \sin (2 (c+d x))-390 \sin (4 (c+d x))-1600 \cos (c+d x)+1520 \cos (3 (c+d x))-304 \cos (5 (c+d x))-1950 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+975 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-195 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1950 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-975 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+195 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{1920 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^5*(-1600*Cos[c + d*x] + 1520*Cos[3*(c + d*x)] - 304*Cos[5*(c + d*x)] + 1950*Log[Cos[(c + d*x)/2]
]*Sin[c + d*x] - 1950*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 1500*Sin[2*(c + d*x)] - 975*Log[Cos[(c + d*x)/2]]*S
in[3*(c + d*x)] + 975*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 390*Sin[4*(c + d*x)] + 195*Log[Cos[(c + d*x)/2]
]*Sin[5*(c + d*x)] - 195*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(1920*a^3*d)

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fricas [A]  time = 0.80, size = 179, normalized size = 1.57 \[ -\frac {608 \, \cos \left (d x + c\right )^{5} - 1520 \, \cos \left (d x + c\right )^{3} - 195 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 195 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (13 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^6/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/240*(608*cos(d*x + c)^5 - 1520*cos(d*x + c)^3 - 195*(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) + 195*(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*(13*cos(d*x + c)^3 - 19*cos(d*x + c))*sin(d*x + c) + 960*cos(d*x + c))/((a^3*d*cos(d*x + c)^4 - 2*a^3
*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))

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giac [A]  time = 0.28, size = 187, normalized size = 1.64 \[ -\frac {\frac {1560 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3562 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1380 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {6 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 170 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1380 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^6/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/960*(1560*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (3562*tan(1/2*d*x + 1/2*c)^5 - 1380*tan(1/2*d*x + 1/2*c)^4 +
480*tan(1/2*d*x + 1/2*c)^3 - 170*tan(1/2*d*x + 1/2*c)^2 + 45*tan(1/2*d*x + 1/2*c) - 6)/(a^3*tan(1/2*d*x + 1/2
*c)^5) - (6*a^12*tan(1/2*d*x + 1/2*c)^5 - 45*a^12*tan(1/2*d*x + 1/2*c)^4 + 170*a^12*tan(1/2*d*x + 1/2*c)^3 - 4
80*a^12*tan(1/2*d*x + 1/2*c)^2 + 1380*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d

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maple [A]  time = 0.68, size = 208, normalized size = 1.82 \[ \frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d \,a^{3}}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{3} d}+\frac {17 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d \,a^{3}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3} d}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{3}}-\frac {23}{16 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {13 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} d}-\frac {1}{160 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{2 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3}{64 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {17}{96 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^6/(a+a*sin(d*x+c))^3,x)

[Out]

1/160/d/a^3*tan(1/2*d*x+1/2*c)^5-3/64/a^3/d*tan(1/2*d*x+1/2*c)^4+17/96/d/a^3*tan(1/2*d*x+1/2*c)^3-1/2/a^3/d*ta
n(1/2*d*x+1/2*c)^2+23/16/d/a^3*tan(1/2*d*x+1/2*c)-23/16/d/a^3/tan(1/2*d*x+1/2*c)-13/8/a^3/d*ln(tan(1/2*d*x+1/2
*c))-1/160/d/a^3/tan(1/2*d*x+1/2*c)^5+1/2/a^3/d/tan(1/2*d*x+1/2*c)^2+3/64/a^3/d/tan(1/2*d*x+1/2*c)^4-17/96/d/a
^3/tan(1/2*d*x+1/2*c)^3

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maxima [B]  time = 0.32, size = 234, normalized size = 2.05 \[ \frac {\frac {\frac {1380 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {480 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {170 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1560 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {170 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {480 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1380 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^6/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/960*((1380*sin(d*x + c)/(cos(d*x + c) + 1) - 480*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 170*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 45*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 6*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 156
0*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 + (45*sin(d*x + c)/(cos(d*x + c) + 1) - 170*sin(d*x + c)^2/(cos(d*x
+ c) + 1)^2 + 480*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1380*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6)*(cos(d*
x + c) + 1)^5/(a^3*sin(d*x + c)^5))/d

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mupad [B]  time = 9.85, size = 291, normalized size = 2.55 \[ -\frac {6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+45\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-170\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-1380\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1380\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+170\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1560\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^6*(a + a*sin(c + d*x))^3),x)

[Out]

-(6*cos(c/2 + (d*x)/2)^10 - 6*sin(c/2 + (d*x)/2)^10 + 45*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^9 - 45*cos(c/2
+ (d*x)/2)^9*sin(c/2 + (d*x)/2) - 170*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^8 + 480*cos(c/2 + (d*x)/2)^3*sin
(c/2 + (d*x)/2)^7 - 1380*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^6 + 1380*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)
/2)^4 - 480*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^3 + 170*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^2 + 1560*l
og(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^5)/(960*a^3*d*cos(c/2 + (d*x
)/2)^5*sin(c/2 + (d*x)/2)^5)

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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)**6/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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