3.586 \(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=160 \[ -\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d} \]

[Out]

(3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (3*a*Cot[c + d*x]*
Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*
Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)

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Rubi [A]  time = 0.209296, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 10, 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, 14} \[ -\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot ^7(c+d x)}{7 d}+\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

(3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (3*a*Cot[c + d*x]*
Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*
Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*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 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]]

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{2} a \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{1}{16} (3 a) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{a \operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{32} a \int \csc ^5(c+d x) \, dx\\ &=-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{128} (3 a) \int \csc ^3(c+d x) \, dx\\ &=-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{1}{256} (3 a) \int \csc (c+d x) \, dx\\ &=\frac{3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{a \cot ^7(c+d x)}{7 d}-\frac{a \cot ^9(c+d x)}{9 d}+\frac{3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac{a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end{align*}

Mathematica [B]  time = 0.0917636, size = 341, normalized size = 2.13 \[ \frac{2 a \cot (c+d x)}{63 d}-\frac{a \csc ^{10}\left (\frac{1}{2} (c+d x)\right )}{10240 d}+\frac{3 a \csc ^8\left (\frac{1}{2} (c+d x)\right )}{4096 d}-\frac{3 a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{2048 d}-\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{3 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{1024 d}+\frac{a \sec ^{10}\left (\frac{1}{2} (c+d x)\right )}{10240 d}-\frac{3 a \sec ^8\left (\frac{1}{2} (c+d x)\right )}{4096 d}+\frac{3 a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{2048 d}+\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{3 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{1024 d}-\frac{3 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{256 d}+\frac{3 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{256 d}-\frac{a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac{19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac{5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{63 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

(2*a*Cot[c + d*x])/(63*d) + (3*a*Csc[(c + d*x)/2]^2)/(1024*d) - (a*Csc[(c + d*x)/2]^4)/(1024*d) - (3*a*Csc[(c
+ d*x)/2]^6)/(2048*d) + (3*a*Csc[(c + d*x)/2]^8)/(4096*d) - (a*Csc[(c + d*x)/2]^10)/(10240*d) + (a*Cot[c + d*x
]*Csc[c + d*x]^2)/(63*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^4)/(21*d) + (19*a*Cot[c + d*x]*Csc[c + d*x]^6)/(63*d
) - (a*Cot[c + d*x]*Csc[c + d*x]^8)/(9*d) + (3*a*Log[Cos[(c + d*x)/2]])/(256*d) - (3*a*Log[Sin[(c + d*x)/2]])/
(256*d) - (3*a*Sec[(c + d*x)/2]^2)/(1024*d) + (a*Sec[(c + d*x)/2]^4)/(1024*d) + (3*a*Sec[(c + d*x)/2]^6)/(2048
*d) - (3*a*Sec[(c + d*x)/2]^8)/(4096*d) + (a*Sec[(c + d*x)/2]^10)/(10240*d)

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Maple [A]  time = 0.066, size = 218, normalized size = 1.4 \begin{align*} -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{2\,a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{160\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{640\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1280\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{1280\,d}}-{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{256\,d}}-{\frac{3\,\cos \left ( dx+c \right ) a}{256\,d}}-{\frac{3\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{256\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x)

[Out]

-1/9/d*a/sin(d*x+c)^9*cos(d*x+c)^7-2/63/d*a/sin(d*x+c)^7*cos(d*x+c)^7-1/10/d*a/sin(d*x+c)^10*cos(d*x+c)^7-3/80
/d*a/sin(d*x+c)^8*cos(d*x+c)^7-1/160/d*a/sin(d*x+c)^6*cos(d*x+c)^7+1/640/d*a/sin(d*x+c)^4*cos(d*x+c)^7-3/1280/
d*a/sin(d*x+c)^2*cos(d*x+c)^7-3/1280*a*cos(d*x+c)^5/d-1/256*a*cos(d*x+c)^3/d-3/256*a*cos(d*x+c)/d-3/256/d*a*ln
(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 1.03815, size = 213, normalized size = 1.33 \begin{align*} -\frac{63 \, a{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \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{2560 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/161280*(63*a*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*
x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1) -
15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 2560*(9*tan(d*x + c)^2 + 7)*a/tan(d*x + c)^9)/d

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Fricas [B]  time = 1.1965, size = 798, normalized size = 4.99 \begin{align*} -\frac{1890 \, a \cos \left (d x + c\right )^{9} - 8820 \, a \cos \left (d x + c\right )^{7} - 16128 \, a \cos \left (d x + c\right )^{5} + 8820 \, a \cos \left (d x + c\right )^{3} - 1890 \, a \cos \left (d x + c\right ) - 945 \,{\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, 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 ) + 945 \,{\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, 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 ) + 2560 \,{\left (2 \, a \cos \left (d x + c\right )^{9} - 9 \, a \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \,{\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, 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)^11*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/161280*(1890*a*cos(d*x + c)^9 - 8820*a*cos(d*x + c)^7 - 16128*a*cos(d*x + c)^5 + 8820*a*cos(d*x + c)^3 - 18
90*a*cos(d*x + c) - 945*(a*cos(d*x + c)^10 - 5*a*cos(d*x + c)^8 + 10*a*cos(d*x + c)^6 - 10*a*cos(d*x + c)^4 +
5*a*cos(d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2) + 945*(a*cos(d*x + c)^10 - 5*a*cos(d*x + c)^8 + 10*a*cos(d
*x + c)^6 - 10*a*cos(d*x + c)^4 + 5*a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2) + 2560*(2*a*cos(d*x + c
)^9 - 9*a*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*c
os(d*x + c)^4 + 5*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)**11*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.36023, size = 383, normalized size = 2.39 \begin{align*} \frac{126 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 280 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 315 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1080 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 630 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2520 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 6720 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1260 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15120 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 15120 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{44286 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 15120 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1260 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 6720 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 2520 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 630 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1080 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 280 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 126 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}}}{1290240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1290240*(126*a*tan(1/2*d*x + 1/2*c)^10 + 280*a*tan(1/2*d*x + 1/2*c)^9 - 315*a*tan(1/2*d*x + 1/2*c)^8 - 1080*
a*tan(1/2*d*x + 1/2*c)^7 - 630*a*tan(1/2*d*x + 1/2*c)^6 + 2520*a*tan(1/2*d*x + 1/2*c)^4 + 6720*a*tan(1/2*d*x +
1/2*c)^3 + 1260*a*tan(1/2*d*x + 1/2*c)^2 - 15120*a*log(abs(tan(1/2*d*x + 1/2*c))) - 15120*a*tan(1/2*d*x + 1/2
*c) + (44286*a*tan(1/2*d*x + 1/2*c)^10 + 15120*a*tan(1/2*d*x + 1/2*c)^9 - 1260*a*tan(1/2*d*x + 1/2*c)^8 - 6720
*a*tan(1/2*d*x + 1/2*c)^7 - 2520*a*tan(1/2*d*x + 1/2*c)^6 + 630*a*tan(1/2*d*x + 1/2*c)^4 + 1080*a*tan(1/2*d*x
+ 1/2*c)^3 + 315*a*tan(1/2*d*x + 1/2*c)^2 - 280*a*tan(1/2*d*x + 1/2*c) - 126*a)/tan(1/2*d*x + 1/2*c)^10)/d