3.1244 \(\int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=157 \[ \frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a b x}{8}-\frac{b^2 \cos ^7(c+d x)}{7 d} \]

[Out]

(5*a*b*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c +
d*x]^5)/(5*d) - (b^2*Cos[c + d*x]^7)/(7*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Cos[c + d*x]^3*
Sin[c + d*x])/(12*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(3*d)

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Rubi [A]  time = 0.207032, antiderivative size = 157, 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 = {2911, 2635, 8, 14, 207} \[ \frac{a^2 \cos ^5(c+d x)}{5 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos (c+d x)}{d}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a b \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{5 a b \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac{5 a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a b x}{8}-\frac{b^2 \cos ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]

[Out]

(5*a*b*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c +
d*x]^5)/(5*d) - (b^2*Cos[c + d*x]^7)/(7*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Cos[c + d*x]^3*
Sin[c + d*x])/(12*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(3*d)

Rule 2911

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^2, x_Symbol] :> Dist[(2*a*b)/d, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e
+ f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 -
b^2, 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]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cos ^5(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^6(c+d x) \, dx+\int \cos ^5(c+d x) \cot (c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{a b \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac{1}{3} (5 a b) \int \cos ^4(c+d x) \, dx-\frac{\operatorname{Subst}\left (\int x^6 \left (b^2-\frac{a^2}{-1+x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac{a b \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac{1}{4} (5 a b) \int \cos ^2(c+d x) \, dx-\frac{\operatorname{Subst}\left (\int \left (-a^2-a^2 x^2-a^2 x^4+b^2 x^6-\frac{a^2}{-1+x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{b^2 \cos ^7(c+d x)}{7 d}+\frac{5 a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac{5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac{a b \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac{1}{8} (5 a b) \int 1 \, dx+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{5 a b x}{8}-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{b^2 \cos ^7(c+d x)}{7 d}+\frac{5 a b \cos (c+d x) \sin (c+d x)}{8 d}+\frac{5 a b \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac{a b \cos ^5(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.280247, size = 166, normalized size = 1.06 \[ \frac{105 \left (88 a^2-5 b^2\right ) \cos (c+d x)+35 \left (28 a^2-9 b^2\right ) \cos (3 (c+d x))+84 a^2 \cos (5 (c+d x))+6720 a^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-6720 a^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3150 a b \sin (2 (c+d x))+630 a b \sin (4 (c+d x))+70 a b \sin (6 (c+d x))+4200 a b c+4200 a b d x-105 b^2 \cos (5 (c+d x))-15 b^2 \cos (7 (c+d x))}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Cot[c + d*x]*(a + b*Sin[c + d*x])^2,x]

[Out]

(4200*a*b*c + 4200*a*b*d*x + 105*(88*a^2 - 5*b^2)*Cos[c + d*x] + 35*(28*a^2 - 9*b^2)*Cos[3*(c + d*x)] + 84*a^2
*Cos[5*(c + d*x)] - 105*b^2*Cos[5*(c + d*x)] - 15*b^2*Cos[7*(c + d*x)] - 6720*a^2*Log[Cos[(c + d*x)/2]] + 6720
*a^2*Log[Sin[(c + d*x)/2]] + 3150*a*b*Sin[2*(c + d*x)] + 630*a*b*Sin[4*(c + d*x)] + 70*a*b*Sin[6*(c + d*x)])/(
6720*d)

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Maple [A]  time = 0.086, size = 160, normalized size = 1. \begin{align*}{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2}\cos \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{5\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{12\,d}}+{\frac{5\,ab\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{5\,abx}{8}}+{\frac{5\,abc}{8\,d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)*(a+b*sin(d*x+c))^2,x)

[Out]

1/5*a^2*cos(d*x+c)^5/d+1/3*a^2*cos(d*x+c)^3/d+a^2*cos(d*x+c)/d+1/d*a^2*ln(csc(d*x+c)-cot(d*x+c))+1/3*a*b*cos(d
*x+c)^5*sin(d*x+c)/d+5/12*a*b*cos(d*x+c)^3*sin(d*x+c)/d+5/8*a*b*cos(d*x+c)*sin(d*x+c)/d+5/8*a*b*x+5/8/d*a*b*c-
1/7*b^2*cos(d*x+c)^7/d

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Maxima [A]  time = 0.996883, size = 165, normalized size = 1.05 \begin{align*} -\frac{480 \, b^{2} \cos \left (d x + c\right )^{7} - 112 \,{\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 35 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b}{3360 \, 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+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3360*(480*b^2*cos(d*x + c)^7 - 112*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(cos(d*x
+ c) + 1) + 15*log(cos(d*x + c) - 1))*a^2 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 4
8*sin(2*d*x + 2*c))*a*b)/d

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Fricas [A]  time = 1.83097, size = 386, normalized size = 2.46 \begin{align*} -\frac{120 \, b^{2} \cos \left (d x + c\right )^{7} - 168 \, a^{2} \cos \left (d x + c\right )^{5} - 280 \, a^{2} \cos \left (d x + c\right )^{3} - 525 \, a b d x - 840 \, a^{2} \cos \left (d x + c\right ) + 420 \, a^{2} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 420 \, a^{2} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 35 \,{\left (8 \, a b \cos \left (d x + c\right )^{5} + 10 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, 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+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/840*(120*b^2*cos(d*x + c)^7 - 168*a^2*cos(d*x + c)^5 - 280*a^2*cos(d*x + c)^3 - 525*a*b*d*x - 840*a^2*cos(d
*x + c) + 420*a^2*log(1/2*cos(d*x + c) + 1/2) - 420*a^2*log(-1/2*cos(d*x + c) + 1/2) - 35*(8*a*b*cos(d*x + c)^
5 + 10*a*b*cos(d*x + c)^3 + 15*a*b*cos(d*x + c))*sin(d*x + c))/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+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.22764, size = 393, normalized size = 2.5 \begin{align*} \frac{525 \,{\left (d x + c\right )} a b + 840 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \,{\left (1155 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 2520 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 840 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 980 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 10080 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 2975 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 20440 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 4200 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 24640 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 2975 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 16968 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2520 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 980 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6496 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1155 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1288 \, a^{2} + 120 \, b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{7}}}{840 \, 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+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/840*(525*(d*x + c)*a*b + 840*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(1155*a*b*tan(1/2*d*x + 1/2*c)^13 - 2520
*a^2*tan(1/2*d*x + 1/2*c)^12 + 840*b^2*tan(1/2*d*x + 1/2*c)^12 + 980*a*b*tan(1/2*d*x + 1/2*c)^11 - 10080*a^2*t
an(1/2*d*x + 1/2*c)^10 + 2975*a*b*tan(1/2*d*x + 1/2*c)^9 - 20440*a^2*tan(1/2*d*x + 1/2*c)^8 + 4200*b^2*tan(1/2
*d*x + 1/2*c)^8 - 24640*a^2*tan(1/2*d*x + 1/2*c)^6 - 2975*a*b*tan(1/2*d*x + 1/2*c)^5 - 16968*a^2*tan(1/2*d*x +
1/2*c)^4 + 2520*b^2*tan(1/2*d*x + 1/2*c)^4 - 980*a*b*tan(1/2*d*x + 1/2*c)^3 - 6496*a^2*tan(1/2*d*x + 1/2*c)^2
- 1155*a*b*tan(1/2*d*x + 1/2*c) - 1288*a^2 + 120*b^2)/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d