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

Optimal. Leaf size=101 \[ \frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{3 x}{8 a} \]

[Out]

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

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Rubi [A]  time = 0.123102, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2839, 2592, 302, 206, 2635, 8} \[ \frac{\cos ^3(c+d x)}{3 a d}+\frac{\cos (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{3 x}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(8*a) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) - (3*Cos[c + d*x]*Sin
[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[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 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.375653, size = 86, normalized size = 0.85 \[ \frac{120 \cos (c+d x)+8 \cos (3 (c+d x))-3 \left (8 \sin (2 (c+d x))+\sin (4 (c+d x))+4 \left (-8 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+8 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3 c+3 d x\right )\right )}{96 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(120*Cos[c + d*x] + 8*Cos[3*(c + d*x)] - 3*(4*(3*c + 3*d*x + 8*Log[Cos[(c + d*x)/2]] - 8*Log[Sin[(c + d*x)/2]]
) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(96*a*d)

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Maple [B]  time = 0.11, size = 296, normalized size = 2.9 \begin{align*}{\frac{5}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{3}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+8\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}+{\frac{3}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{20}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{5}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{8}{3\,da} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{3}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{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)/(a+a*sin(d*x+c)),x)

[Out]

5/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7+4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^6-
3/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5+8/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^4+
3/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3+20/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)
^2-5/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)+8/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^4-3/4/a/d*arctan(tan
(1/2*d*x+1/2*c))+1/d/a*ln(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.49523, size = 378, normalized size = 3.74 \begin{align*} -\frac{\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{80 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{96 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{9 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{48 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 32}{a + \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac{9 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{12 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*((15*sin(d*x + c)/(cos(d*x + c) + 1) - 80*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 9*sin(d*x + c)^3/(cos(d*
x + c) + 1)^3 - 96*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 9*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 48*sin(d*x +
c)^6/(cos(d*x + c) + 1)^6 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 32)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c)
+ 1)^2 + 6*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8
/(cos(d*x + c) + 1)^8) + 9*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - 12*log(sin(d*x + c)/(cos(d*x + c) + 1))
/a)/d

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Fricas [A]  time = 1.1497, size = 239, normalized size = 2.37 \begin{align*} \frac{8 \, \cos \left (d x + c\right )^{3} - 9 \, d x - 3 \,{\left (2 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 24 \, \cos \left (d x + c\right ) - 12 \, \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \, \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{24 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(8*cos(d*x + c)^3 - 9*d*x - 3*(2*cos(d*x + c)^3 + 3*cos(d*x + c))*sin(d*x + c) + 24*cos(d*x + c) - 12*log
(1/2*cos(d*x + c) + 1/2) + 12*log(-1/2*cos(d*x + c) + 1/2))/(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)/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.30466, size = 193, normalized size = 1.91 \begin{align*} -\frac{\frac{9 \,{\left (d x + c\right )}}{a} - \frac{24 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 96 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 80 \, \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 ) + 32\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(9*(d*x + c)/a - 24*log(abs(tan(1/2*d*x + 1/2*c)))/a - 2*(15*tan(1/2*d*x + 1/2*c)^7 + 48*tan(1/2*d*x + 1
/2*c)^6 - 9*tan(1/2*d*x + 1/2*c)^5 + 96*tan(1/2*d*x + 1/2*c)^4 + 9*tan(1/2*d*x + 1/2*c)^3 + 80*tan(1/2*d*x + 1
/2*c)^2 - 15*tan(1/2*d*x + 1/2*c) + 32)/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d