3.584 \(\int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=122 \[ -\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

(5*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^7)/(7*d) + (5*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (5*
a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a*Cot[c + d*x]^5*Csc[c +
d*x]^3)/(8*d)

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Rubi [A]  time = 0.183989, antiderivative size = 122, 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 = {2838, 2611, 3768, 3770, 2607, 30} \[ -\frac{a \cot ^7(c+d x)}{7 d}+\frac{5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

(5*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^7)/(7*d) + (5*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (5*
a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a*Cot[c + d*x]^5*Csc[c +
d*x]^3)/(8*d)

Rule 2838

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

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

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 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.0585969, size = 215, normalized size = 1.76 \[ -\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \csc ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}+\frac{7 a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{1536 d}-\frac{15 a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{5 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}+\frac{a \sec ^8\left (\frac{1}{2} (c+d x)\right )}{2048 d}-\frac{7 a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{1536 d}+\frac{15 a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{5 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{512 d}-\frac{5 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{128 d}+\frac{5 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{128 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^7)/(7*d) + (5*a*Csc[(c + d*x)/2]^2)/(512*d) - (15*a*Csc[(c + d*x)/2]^4)/(1024*d) + (7*a*Csc[(
c + d*x)/2]^6)/(1536*d) - (a*Csc[(c + d*x)/2]^8)/(2048*d) + (5*a*Log[Cos[(c + d*x)/2]])/(128*d) - (5*a*Log[Sin
[(c + d*x)/2]])/(128*d) - (5*a*Sec[(c + d*x)/2]^2)/(512*d) + (15*a*Sec[(c + d*x)/2]^4)/(1024*d) - (7*a*Sec[(c
+ d*x)/2]^6)/(1536*d) + (a*Sec[(c + d*x)/2]^8)/(2048*d)

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Maple [A]  time = 0.066, size = 174, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{5\,a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{5\,\cos \left ( dx+c \right ) a}{128\,d}}-{\frac{5\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x)

[Out]

-1/7/d*a/sin(d*x+c)^7*cos(d*x+c)^7-1/8/d*a/sin(d*x+c)^8*cos(d*x+c)^7-1/48/d*a/sin(d*x+c)^6*cos(d*x+c)^7+1/192/
d*a/sin(d*x+c)^4*cos(d*x+c)^7-1/128/d*a/sin(d*x+c)^2*cos(d*x+c)^7-1/128*a*cos(d*x+c)^5/d-5/384*a*cos(d*x+c)^3/
d-5/128*a*cos(d*x+c)/d-5/128/d*a*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 1.18098, size = 170, normalized size = 1.39 \begin{align*} -\frac{7 \, a{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{768 \, a}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/5376*(7*a*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^8
- 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c)
- 1)) + 768*a/tan(d*x + c)^7)/d

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Fricas [B]  time = 1.16699, size = 625, normalized size = 5.12 \begin{align*} -\frac{768 \, a \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) + 210 \, a \cos \left (d x + c\right )^{7} + 1022 \, a \cos \left (d x + c\right )^{5} - 770 \, a \cos \left (d x + c\right )^{3} + 210 \, a \cos \left (d x + c\right ) - 105 \,{\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 105 \,{\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{5376 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, 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)^9*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/5376*(768*a*cos(d*x + c)^7*sin(d*x + c) + 210*a*cos(d*x + c)^7 + 1022*a*cos(d*x + c)^5 - 770*a*cos(d*x + c)
^3 + 210*a*cos(d*x + c) - 105*(a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(d*x + c)^2
+ a)*log(1/2*cos(d*x + c) + 1/2) + 105*(a*cos(d*x + c)^8 - 4*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 4*a*cos(
d*x + c)^2 + a)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*
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)**9*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.34801, size = 346, normalized size = 2.84 \begin{align*} \frac{21 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 48 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 112 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 336 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 168 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1008 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 336 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1680 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 1680 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{4566 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1680 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 336 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1008 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 168 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 336 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 112 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 21 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{43008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/43008*(21*a*tan(1/2*d*x + 1/2*c)^8 + 48*a*tan(1/2*d*x + 1/2*c)^7 - 112*a*tan(1/2*d*x + 1/2*c)^6 - 336*a*tan(
1/2*d*x + 1/2*c)^5 + 168*a*tan(1/2*d*x + 1/2*c)^4 + 1008*a*tan(1/2*d*x + 1/2*c)^3 + 336*a*tan(1/2*d*x + 1/2*c)
^2 - 1680*a*log(abs(tan(1/2*d*x + 1/2*c))) - 1680*a*tan(1/2*d*x + 1/2*c) + (4566*a*tan(1/2*d*x + 1/2*c)^8 + 16
80*a*tan(1/2*d*x + 1/2*c)^7 - 336*a*tan(1/2*d*x + 1/2*c)^6 - 1008*a*tan(1/2*d*x + 1/2*c)^5 - 168*a*tan(1/2*d*x
+ 1/2*c)^4 + 336*a*tan(1/2*d*x + 1/2*c)^3 + 112*a*tan(1/2*d*x + 1/2*c)^2 - 48*a*tan(1/2*d*x + 1/2*c) - 21*a)/
tan(1/2*d*x + 1/2*c)^8)/d