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*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)

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Rubi [A]  time = 0.133348, antiderivative size = 102, normalized size of antiderivative = 1., 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 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 ^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*}

Mathematica [B]  time = 0.659336, 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]

-(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[S
in[(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)]))/(192
*a*d*(1 + Sin[c + d*x]))

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Maple [A]  time = 0.138, size = 188, normalized size = 1.8 \begin{align*}{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{1}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{5}{8\,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}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{1}{24\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{5}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{3}{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)^6*csc(d*x+c)^5/(a+a*sin(d*x+c)),x)

[Out]

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

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Maxima [B]  time = 1.52618, size = 293, normalized size = 2.87 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.13399, size = 474, normalized size = 4.65 \begin{align*} -\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 )}} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 1.37655, size = 225, normalized size = 2.21 \begin{align*} -\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} \end{align*}

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