3.609 \(\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=185 \[ -\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{25 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{125 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{125 a^3 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{125 a^3 x}{128} \]

[Out]

(125*a^3*x)/128 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos
[c + d*x]^5)/(5*d) - (3*a^3*Cos[c + d*x]^7)/(7*d) + (125*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (125*a^3*Cos
[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^3*Cos[c + d*x]^7*Sin[c +
d*x])/(8*d)

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Rubi [A]  time = 0.235842, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2873, 2635, 8, 2592, 302, 206, 2565, 30, 2568} \[ -\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos (c+d x)}{d}-\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{25 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{125 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{125 a^3 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{125 a^3 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(125*a^3*x)/128 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos
[c + d*x]^5)/(5*d) - (3*a^3*Cos[c + d*x]^7)/(7*d) + (125*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (125*a^3*Cos
[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^3*Cos[c + d*x]^7*Sin[c +
d*x])/(8*d)

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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 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 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (3 a^3 \cos ^6(c+d x)+a^3 \cos ^5(c+d x) \cot (c+d x)+3 a^3 \cos ^6(c+d x) \sin (c+d x)+a^3 \cos ^6(c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^5(c+d x) \cot (c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin (c+d x) \, dx\\ &=\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{2 d}-\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} a^3 \int \cos ^6(c+d x) \, dx+\frac{1}{2} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \frac{x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{5 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac{25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{48} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{8} \left (15 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (-1-x^2-x^4+\frac{1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{64} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{16} \left (15 a^3\right ) \int 1 \, dx-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{15 a^3 x}{16}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{128} \left (5 a^3\right ) \int 1 \, dx\\ &=\frac{125 a^3 x}{128}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cos ^7(c+d x)}{7 d}+\frac{125 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{125 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{25 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.685172, size = 122, normalized size = 0.66 \[ \frac{a^3 \left (77280 \sin (2 (c+d x))+14280 \sin (4 (c+d x))+1120 \sin (6 (c+d x))-105 \sin (8 (c+d x))+122640 \cos (c+d x)+560 \cos (3 (c+d x))-3696 \cos (5 (c+d x))-720 \cos (7 (c+d x))+107520 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-107520 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+105000 c+105000 d x\right )}{107520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Cot[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(105000*c + 105000*d*x + 122640*Cos[c + d*x] + 560*Cos[3*(c + d*x)] - 3696*Cos[5*(c + d*x)] - 720*Cos[7*(
c + d*x)] - 107520*Log[Cos[(c + d*x)/2]] + 107520*Log[Sin[(c + d*x)/2]] + 77280*Sin[2*(c + d*x)] + 14280*Sin[4
*(c + d*x)] + 1120*Sin[6*(c + d*x)] - 105*Sin[8*(c + d*x)]))/(107520*d)

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Maple [A]  time = 0.081, size = 187, normalized size = 1. \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}\sin \left ( dx+c \right ) }{8\,d}}+{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{48\,d}}+{\frac{125\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{192\,d}}+{\frac{125\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{128\,d}}+{\frac{125\,{a}^{3}x}{128}}+{\frac{125\,{a}^{3}c}{128\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{3}\cos \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{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+a*sin(d*x+c))^3,x)

[Out]

-1/8*a^3*cos(d*x+c)^7*sin(d*x+c)/d+25/48*a^3*cos(d*x+c)^5*sin(d*x+c)/d+125/192*a^3*cos(d*x+c)^3*sin(d*x+c)/d+1
25/128*a^3*cos(d*x+c)*sin(d*x+c)/d+125/128*a^3*x+125/128/d*a^3*c-3/7*a^3*cos(d*x+c)^7/d+1/5*a^3*cos(d*x+c)^5/d
+1/3*a^3*cos(d*x+c)^3/d+a^3*cos(d*x+c)/d+1/d*a^3*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 1.12855, size = 231, normalized size = 1.25 \begin{align*} -\frac{46080 \, a^{3} \cos \left (d x + c\right )^{7} - 3584 \,{\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^{3} - 35 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 1680 \,{\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^{3}}{107520 \, 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))^3,x, algorithm="maxima")

[Out]

-1/107520*(46080*a^3*cos(d*x + c)^7 - 3584*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(co
s(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^3 - 35*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x +
8*c) - 24*sin(4*d*x + 4*c))*a^3 + 1680*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d
*x + 2*c))*a^3)/d

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Fricas [A]  time = 1.27421, size = 439, normalized size = 2.37 \begin{align*} -\frac{5760 \, a^{3} \cos \left (d x + c\right )^{7} - 2688 \, a^{3} \cos \left (d x + c\right )^{5} - 4480 \, a^{3} \cos \left (d x + c\right )^{3} - 13125 \, a^{3} d x - 13440 \, a^{3} \cos \left (d x + c\right ) + 6720 \, a^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 6720 \, a^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 35 \,{\left (48 \, a^{3} \cos \left (d x + c\right )^{7} - 200 \, a^{3} \cos \left (d x + c\right )^{5} - 250 \, a^{3} \cos \left (d x + c\right )^{3} - 375 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, 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))^3,x, algorithm="fricas")

[Out]

-1/13440*(5760*a^3*cos(d*x + c)^7 - 2688*a^3*cos(d*x + c)^5 - 4480*a^3*cos(d*x + c)^3 - 13125*a^3*d*x - 13440*
a^3*cos(d*x + c) + 6720*a^3*log(1/2*cos(d*x + c) + 1/2) - 6720*a^3*log(-1/2*cos(d*x + c) + 1/2) + 35*(48*a^3*c
os(d*x + c)^7 - 200*a^3*cos(d*x + c)^5 - 250*a^3*cos(d*x + c)^3 - 375*a^3*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+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.25884, size = 374, normalized size = 2.02 \begin{align*} \frac{13125 \,{\left (d x + c\right )} a^{3} + 13440 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \,{\left (27195 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 65135 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 161280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 63595 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 286720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 133175 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 519680 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 133175 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 544768 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 63595 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 254464 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 65135 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 118784 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 27195 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 14848 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, 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))^3,x, algorithm="giac")

[Out]

1/13440*(13125*(d*x + c)*a^3 + 13440*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(27195*a^3*tan(1/2*d*x + 1/2*c)^15
+ 65135*a^3*tan(1/2*d*x + 1/2*c)^13 - 161280*a^3*tan(1/2*d*x + 1/2*c)^12 + 63595*a^3*tan(1/2*d*x + 1/2*c)^11
- 286720*a^3*tan(1/2*d*x + 1/2*c)^10 + 133175*a^3*tan(1/2*d*x + 1/2*c)^9 - 519680*a^3*tan(1/2*d*x + 1/2*c)^8 -
133175*a^3*tan(1/2*d*x + 1/2*c)^7 - 544768*a^3*tan(1/2*d*x + 1/2*c)^6 - 63595*a^3*tan(1/2*d*x + 1/2*c)^5 - 25
4464*a^3*tan(1/2*d*x + 1/2*c)^4 - 65135*a^3*tan(1/2*d*x + 1/2*c)^3 - 118784*a^3*tan(1/2*d*x + 1/2*c)^2 - 27195
*a^3*tan(1/2*d*x + 1/2*c) - 14848*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d