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

Optimal. Leaf size=153 \[ -\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{4 a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+5 a^2 x \]

[Out]

5*a^2*x + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (4*a^2*Cot
[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[
c + d*x]^3)/(4*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/d

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Rubi [A]  time = 0.197078, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2872, 3767, 8, 3768, 3770, 2638, 2635, 2633} \[ -\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{4 a^2 \cot (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+5 a^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

5*a^2*x + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (4*a^2*Cot
[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[
c + d*x]^3)/(4*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/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 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 3768

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, 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 (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\int \left (6 a^8-6 a^8 \csc ^2(c+d x)-2 a^8 \csc ^3(c+d x)+2 a^8 \csc ^4(c+d x)+a^8 \csc ^5(c+d x)+2 a^8 \sin (c+d x)-2 a^8 \sin ^2(c+d x)-a^8 \sin ^3(c+d x)\right ) \, dx}{a^6}\\ &=6 a^2 x+a^2 \int \csc ^5(c+d x) \, dx-a^2 \int \sin ^3(c+d x) \, dx-\left (2 a^2\right ) \int \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^4(c+d x) \, dx+\left (2 a^2\right ) \int \sin (c+d x) \, dx-\left (2 a^2\right ) \int \sin ^2(c+d x) \, dx-\left (6 a^2\right ) \int \csc ^2(c+d x) \, dx\\ &=6 a^2 x-\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac{1}{4} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-a^2 \int 1 \, dx-a^2 \int \csc (c+d x) \, dx+\frac{a^2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac{\left (6 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=5 a^2 x+\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{4 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac{1}{8} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=5 a^2 x+\frac{5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{4 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \cot ^3(c+d x)}{3 d}+\frac{5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 1.20725, size = 227, normalized size = 1.48 \[ \frac{a^2 (\sin (c+d x)+1)^2 \left (960 (c+d x)+96 \sin (2 (c+d x))-240 \cos (c+d x)-16 \cos (3 (c+d x))-448 \tan \left (\frac{1}{2} (c+d x)\right )+448 \cot \left (\frac{1}{2} (c+d x)\right )-3 \csc ^4\left (\frac{1}{2} (c+d x)\right )+30 \csc ^2\left (\frac{1}{2} (c+d x)\right )+3 \sec ^4\left (\frac{1}{2} (c+d x)\right )-30 \sec ^2\left (\frac{1}{2} (c+d x)\right )-120 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+120 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+128 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-8 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )\right )}{192 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(1 + Sin[c + d*x])^2*(960*(c + d*x) - 240*Cos[c + d*x] - 16*Cos[3*(c + d*x)] + 448*Cot[(c + d*x)/2] + 30*
Csc[(c + d*x)/2]^2 - 3*Csc[(c + d*x)/2]^4 + 120*Log[Cos[(c + d*x)/2]] - 120*Log[Sin[(c + d*x)/2]] - 30*Sec[(c
+ d*x)/2]^2 + 3*Sec[(c + d*x)/2]^4 + 128*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 8*Csc[(c + d*x)/2]^4*Sin[c + d*x]
+ 96*Sin[2*(c + d*x)] - 448*Tan[(c + d*x)/2]))/(192*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)

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Maple [A]  time = 0.082, size = 247, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d}}-{\frac{5\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{5\,{a}^{2}\cos \left ( dx+c \right ) }{8\,d}}-{\frac{5\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{8\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{8\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{10\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{3\,d}}+5\,{\frac{{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+5\,{a}^{2}x+5\,{\frac{c{a}^{2}}{d}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}} \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))^2,x)

[Out]

-1/8/d*a^2/sin(d*x+c)^2*cos(d*x+c)^7-1/8*a^2*cos(d*x+c)^5/d-5/24*a^2*cos(d*x+c)^3/d-5/8*a^2*cos(d*x+c)/d-5/8/d
*a^2*ln(csc(d*x+c)-cot(d*x+c))-2/3/d*a^2/sin(d*x+c)^3*cos(d*x+c)^7+8/3/d*a^2/sin(d*x+c)*cos(d*x+c)^7+8/3*a^2*c
os(d*x+c)^5*sin(d*x+c)/d+10/3*a^2*cos(d*x+c)^3*sin(d*x+c)/d+5*a^2*cos(d*x+c)*sin(d*x+c)/d+5*a^2*x+5/d*c*a^2-1/
4/d*a^2/sin(d*x+c)^4*cos(d*x+c)^7

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Maxima [A]  time = 1.58052, size = 278, normalized size = 1.82 \begin{align*} -\frac{4 \,{\left (4 \, \cos \left (d x + c\right )^{3} - \frac{6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \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} - 16 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{2} + 3 \, a^{2}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \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 )}}{48 \, 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))^2,x, algorithm="maxima")

[Out]

-1/48*(4*(4*cos(d*x + c)^3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d*x + c) + 1)
+ 15*log(cos(d*x + c) - 1))*a^2 - 16*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(tan(d*x + c
)^5 + tan(d*x + c)^3))*a^2 + 3*a^2*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 +
1) - 16*cos(d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d

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Fricas [A]  time = 1.25567, size = 636, normalized size = 4.16 \begin{align*} -\frac{16 \, a^{2} \cos \left (d x + c\right )^{7} - 240 \, a^{2} d x \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{5} + 480 \, a^{2} d x \cos \left (d x + c\right )^{2} - 50 \, a^{2} \cos \left (d x + c\right )^{3} - 240 \, a^{2} d x + 30 \, a^{2} \cos \left (d x + c\right ) - 15 \,{\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 15 \,{\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 16 \,{\left (3 \, a^{2} \cos \left (d x + c\right )^{5} - 20 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + 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))^2,x, algorithm="fricas")

[Out]

-1/48*(16*a^2*cos(d*x + c)^7 - 240*a^2*d*x*cos(d*x + c)^4 + 16*a^2*cos(d*x + c)^5 + 480*a^2*d*x*cos(d*x + c)^2
- 50*a^2*cos(d*x + c)^3 - 240*a^2*d*x + 30*a^2*cos(d*x + c) - 15*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^2 +
a^2)*log(1/2*cos(d*x + c) + 1/2) + 15*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c)
+ 1/2) - 16*(3*a^2*cos(d*x + c)^5 - 20*a^2*cos(d*x + c)^3 + 15*a^2*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c
)^4 - 2*d*cos(d*x + c)^2 + 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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.34094, size = 350, normalized size = 2.29 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 16 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 960 \,{\left (d x + c\right )} a^{2} - 120 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 432 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{128 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 6 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, a^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}} + \frac{250 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 432 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a^{2}}{\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))^2,x, algorithm="giac")

[Out]

1/192*(3*a^2*tan(1/2*d*x + 1/2*c)^4 + 16*a^2*tan(1/2*d*x + 1/2*c)^3 - 24*a^2*tan(1/2*d*x + 1/2*c)^2 + 960*(d*x
+ c)*a^2 - 120*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 432*a^2*tan(1/2*d*x + 1/2*c) - 128*(3*a^2*tan(1/2*d*x + 1
/2*c)^5 + 6*a^2*tan(1/2*d*x + 1/2*c)^4 + 6*a^2*tan(1/2*d*x + 1/2*c)^2 - 3*a^2*tan(1/2*d*x + 1/2*c) + 4*a^2)/(t
an(1/2*d*x + 1/2*c)^2 + 1)^3 + (250*a^2*tan(1/2*d*x + 1/2*c)^4 + 432*a^2*tan(1/2*d*x + 1/2*c)^3 + 24*a^2*tan(1
/2*d*x + 1/2*c)^2 - 16*a^2*tan(1/2*d*x + 1/2*c) - 3*a^2)/tan(1/2*d*x + 1/2*c)^4)/d