3.673 \(\int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=97 \[ -\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d} \]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^5)/(5*d) - (3*a*Csc[c + d*x]^7)/(7*d)
+ (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^11)/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 0.124318, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 14} \[ -\frac{a \cot ^{10}(c+d x)}{10 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^5)/(5*d) - (3*a*Csc[c + d*x]^7)/(7*d)
+ (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^11)/(11*d)

Rule 2834

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

Rule 2606

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^4 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int x^7 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \left (-x^4+3 x^6-3 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^{10}(c+d x)}{10 d}+\frac{a \csc ^5(c+d x)}{5 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{a \csc ^{11}(c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 0.138561, size = 86, normalized size = 0.89 \[ -\frac{a \csc ^4(c+d x) \left (840 \csc ^7(c+d x)+924 \csc ^6(c+d x)-3080 \csc ^5(c+d x)-3465 \csc ^4(c+d x)+3960 \csc ^3(c+d x)+4620 \csc ^2(c+d x)-1848 \csc (c+d x)-2310\right )}{9240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^4*(-2310 - 1848*Csc[c + d*x] + 4620*Csc[c + d*x]^2 + 3960*Csc[c + d*x]^3 - 3465*Csc[c + d*x]^
4 - 3080*Csc[c + d*x]^5 + 924*Csc[c + d*x]^6 + 840*Csc[c + d*x]^7))/(9240*d)

________________________________________________________________________________________

Maple [B]  time = 0.066, size = 194, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{10\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{40\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \right ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{11\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{33\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{231\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{1155\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{1155\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{231\,\sin \left ( dx+c \right ) }}+{\frac{\sin \left ( dx+c \right ) }{231} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/10/sin(d*x+c)^10*cos(d*x+c)^8-1/40/sin(d*x+c)^8*cos(d*x+c)^8)+a*(-1/11/sin(d*x+c)^11*cos(d*x+c)^8-1
/33/sin(d*x+c)^9*cos(d*x+c)^8-1/231/sin(d*x+c)^7*cos(d*x+c)^8+1/1155/sin(d*x+c)^5*cos(d*x+c)^8-1/1155/sin(d*x+
c)^3*cos(d*x+c)^8+1/231/sin(d*x+c)*cos(d*x+c)^8+1/231*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*si
n(d*x+c)))

________________________________________________________________________________________

Maxima [A]  time = 1.03288, size = 124, normalized size = 1.28 \begin{align*} \frac{2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/9240*(2310*a*sin(d*x + c)^7 + 1848*a*sin(d*x + c)^6 - 4620*a*sin(d*x + c)^5 - 3960*a*sin(d*x + c)^4 + 3465*a
*sin(d*x + c)^3 + 3080*a*sin(d*x + c)^2 - 924*a*sin(d*x + c) - 840*a)/(d*sin(d*x + c)^11)

________________________________________________________________________________________

Fricas [A]  time = 1.22169, size = 405, normalized size = 4.18 \begin{align*} \frac{1848 \, a \cos \left (d x + c\right )^{6} - 1584 \, a \cos \left (d x + c\right )^{4} + 704 \, a \cos \left (d x + c\right )^{2} + 231 \,{\left (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 )} \sin \left (d x + c\right ) - 128 \, a}{9240 \,{\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 )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/9240*(1848*a*cos(d*x + c)^6 - 1584*a*cos(d*x + c)^4 + 704*a*cos(d*x + c)^2 + 231*(10*a*cos(d*x + c)^6 - 10*a
*cos(d*x + c)^4 + 5*a*cos(d*x + c)^2 - a)*sin(d*x + c) - 128*a)/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*
d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)*sin(d*x + c))

________________________________________________________________________________________

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)**7*csc(d*x+c)**12*(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.33868, size = 124, normalized size = 1.28 \begin{align*} \frac{2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/9240*(2310*a*sin(d*x + c)^7 + 1848*a*sin(d*x + c)^6 - 4620*a*sin(d*x + c)^5 - 3960*a*sin(d*x + c)^4 + 3465*a
*sin(d*x + c)^3 + 3080*a*sin(d*x + c)^2 - 924*a*sin(d*x + c) - 840*a)/(d*sin(d*x + c)^11)