3.397 \(\int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=131 \[ -\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}-\frac{3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{11 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^3 x}{8} \]

[Out]

(-3*a^3*x)/8 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d - (a^3*Cos[c
+ d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d + (11*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (3*a^3*Cos[c + d*x]*Sin[c
+ d*x]^3)/(4*d)

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Rubi [A]  time = 0.196466, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2872, 3770, 3767, 8, 2638, 2635, 2633} \[ -\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}-\frac{3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac{11 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^3 x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(-3*a^3*x)/8 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d - (a^3*Cos[c
+ d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d + (11*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (3*a^3*Cos[c + d*x]*Sin[c
+ d*x]^3)/(4*d)

Rule 2872

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 8

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 2633

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

Rubi steps

\begin{align*} \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (a^7+3 a^7 \csc (c+d x)+a^7 \csc ^2(c+d x)-5 a^7 \sin (c+d x)-5 a^7 \sin ^2(c+d x)+a^7 \sin ^3(c+d x)+3 a^7 \sin ^4(c+d x)+a^7 \sin ^5(c+d x)\right ) \, dx}{a^4}\\ &=a^3 x+a^3 \int \csc ^2(c+d x) \, dx+a^3 \int \sin ^3(c+d x) \, dx+a^3 \int \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx+\left (3 a^3\right ) \int \sin ^4(c+d x) \, dx-\left (5 a^3\right ) \int \sin (c+d x) \, dx-\left (5 a^3\right ) \int \sin ^2(c+d x) \, dx\\ &=a^3 x-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{5 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{4} \left (9 a^3\right ) \int \sin ^2(c+d x) \, dx-\frac{1}{2} \left (5 a^3\right ) \int 1 \, dx-\frac{a^3 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{a^3 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{3 a^3 x}{2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{d}-\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{11 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac{3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{8} \left (9 a^3\right ) \int 1 \, dx\\ &=-\frac{3 a^3 x}{8}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^3 \cos (c+d x)}{d}+\frac{a^3 \cos ^3(c+d x)}{d}-\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{11 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac{3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 1.96367, size = 148, normalized size = 1.13 \[ \frac{(a \sin (c+d x)+a)^3 \left (-60 (c+d x)+80 \sin (2 (c+d x))+15 \sin (4 (c+d x))+580 \cos (c+d x)+30 \cos (3 (c+d x))-2 \cos (5 (c+d x))+80 \tan \left (\frac{1}{2} (c+d x)\right )-80 \cot \left (\frac{1}{2} (c+d x)\right )+480 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-480 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{160 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

((a + a*Sin[c + d*x])^3*(-60*(c + d*x) + 580*Cos[c + d*x] + 30*Cos[3*(c + d*x)] - 2*Cos[5*(c + d*x)] - 80*Cot[
(c + d*x)/2] - 480*Log[Cos[(c + d*x)/2]] + 480*Log[Sin[(c + d*x)/2]] + 80*Sin[2*(c + d*x)] + 15*Sin[4*(c + d*x
)] + 80*Tan[(c + d*x)/2]))/(160*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [A]  time = 0.076, size = 152, normalized size = 1.2 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{3\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{3\,{a}^{3}x}{8}}-{\frac{3\,{a}^{3}c}{8\,d}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,{\frac{{a}^{3}\cos \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{d\sin \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.55699, size = 190, normalized size = 1.45 \begin{align*} -\frac{32 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \,{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 80 \,{\left (3 \, d x + 3 \, c + \frac{3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/160*(32*a^3*cos(d*x + c)^5 - 80*(2*cos(d*x + c)^3 + 6*cos(d*x + c) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*
x + c) - 1))*a^3 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^3 + 80*(3*d*x + 3*c + (3*tan(d
*x + c)^2 + 2)/(tan(d*x + c)^3 + tan(d*x + c)))*a^3)/d

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Fricas [A]  time = 1.17434, size = 394, normalized size = 3.01 \begin{align*} -\frac{30 \, a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + 60 \, a^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 60 \, a^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 15 \, a^{3} \cos \left (d x + c\right ) +{\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 40 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} d x - 120 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40 \, d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/40*(30*a^3*cos(d*x + c)^5 - 5*a^3*cos(d*x + c)^3 + 60*a^3*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 60*a^3
*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*a^3*cos(d*x + c) + (8*a^3*cos(d*x + c)^5 - 40*a^3*cos(d*x + c)
^3 + 15*a^3*d*x - 120*a^3*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))

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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)**4*csc(d*x+c)**2*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.52869, size = 305, normalized size = 2.33 \begin{align*} -\frac{15 \,{\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 20 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{20 \,{\left (6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \frac{2 \,{\left (55 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 200 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 10 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 10 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 55 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 152 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5}}}{40 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/40*(15*(d*x + c)*a^3 - 120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 20*a^3*tan(1/2*d*x + 1/2*c) + 20*(6*a^3*tan
(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 1/2*c) + 2*(55*a^3*tan(1/2*d*x + 1/2*c)^9 - 200*a^3*tan(1/2*d*x + 1/2*c
)^8 - 10*a^3*tan(1/2*d*x + 1/2*c)^7 - 720*a^3*tan(1/2*d*x + 1/2*c)^6 - 800*a^3*tan(1/2*d*x + 1/2*c)^4 + 10*a^3
*tan(1/2*d*x + 1/2*c)^3 - 560*a^3*tan(1/2*d*x + 1/2*c)^2 - 55*a^3*tan(1/2*d*x + 1/2*c) - 152*a^3)/(tan(1/2*d*x
+ 1/2*c)^2 + 1)^5)/d