3.674 \(\int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=113 \[ -\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d} \]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*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)

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Rubi [A]  time = 0.133233, antiderivative size = 113, 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 = {2834, 2607, 266, 43, 2606, 270} \[ -\frac{a \cot ^{12}(c+d x)}{12 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^8)/(8*d) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]

Rubi steps

\begin{align*} \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc ^5(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^6(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 )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int x^3 (1+x)^2 \, dx,x,\cot ^2(c+d x)\right )}{2 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 \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}-\frac{a \operatorname{Subst}\left (\int \left (x^3+2 x^4+x^5\right ) \, dx,x,\cot ^2(c+d x)\right )}{2 d}\\ &=-\frac{a \cot ^8(c+d x)}{8 d}-\frac{a \cot ^{10}(c+d x)}{5 d}-\frac{a \cot ^{12}(c+d x)}{12 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 = 0.231248, size = 86, normalized size = 0.76 \[ -\frac{a \csc ^{12}(c+d x) (-45 \sin (c+d x)+1111 \sin (3 (c+d x))+363 \sin (5 (c+d x))+231 \sin (7 (c+d x))+3003 \cos (2 (c+d x))+1155 \cos (4 (c+d x))+385 \cos (6 (c+d x))+1617)}{73920 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

-(a*Csc[c + d*x]^12*(1617 + 3003*Cos[2*(c + d*x)] + 1155*Cos[4*(c + d*x)] + 385*Cos[6*(c + d*x)] - 45*Sin[c +
d*x] + 1111*Sin[3*(c + d*x)] + 363*Sin[5*(c + d*x)] + 231*Sin[7*(c + d*x)]))/(73920*d)

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Maple [B]  time = 0.06, size = 212, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( 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 ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{12\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{30\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{120\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \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)^13*(a+a*sin(d*x+c)),x)

[Out]

1/d*(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)*sin(d*x+c))+a*(-1/12/sin(d*x+c)^12*cos(d*x+c)^8-1/30/sin(d*x+c)^10*cos
(d*x+c)^8-1/120/sin(d*x+c)^8*cos(d*x+c)^8))

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Maxima [A]  time = 1.03558, size = 124, normalized size = 1.1 \begin{align*} \frac{1848 \, a \sin \left (d x + c\right )^{7} + 1540 \, a \sin \left (d x + c\right )^{6} - 3960 \, a \sin \left (d x + c\right )^{5} - 3465 \, a \sin \left (d x + c\right )^{4} + 3080 \, a \sin \left (d x + c\right )^{3} + 2772 \, a \sin \left (d x + c\right )^{2} - 840 \, a \sin \left (d x + c\right ) - 770 \, a}{9240 \, d \sin \left (d x + c\right )^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/9240*(1848*a*sin(d*x + c)^7 + 1540*a*sin(d*x + c)^6 - 3960*a*sin(d*x + c)^5 - 3465*a*sin(d*x + c)^4 + 3080*a
*sin(d*x + c)^3 + 2772*a*sin(d*x + c)^2 - 840*a*sin(d*x + c) - 770*a)/(d*sin(d*x + c)^12)

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Fricas [A]  time = 1.25645, size = 421, normalized size = 3.73 \begin{align*} -\frac{1540 \, a \cos \left (d x + c\right )^{6} - 1155 \, a \cos \left (d x + c\right )^{4} + 462 \, a \cos \left (d x + c\right )^{2} + 8 \,{\left (231 \, a \cos \left (d x + c\right )^{6} - 198 \, a \cos \left (d x + c\right )^{4} + 88 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 77 \, a}{9240 \,{\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, 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)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/9240*(1540*a*cos(d*x + c)^6 - 1155*a*cos(d*x + c)^4 + 462*a*cos(d*x + c)^2 + 8*(231*a*cos(d*x + c)^6 - 198*
a*cos(d*x + c)^4 + 88*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 77*a)/(d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^10 +
15*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*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)**7*csc(d*x+c)**13*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.36302, size = 124, normalized size = 1.1 \begin{align*} \frac{1848 \, a \sin \left (d x + c\right )^{7} + 1540 \, a \sin \left (d x + c\right )^{6} - 3960 \, a \sin \left (d x + c\right )^{5} - 3465 \, a \sin \left (d x + c\right )^{4} + 3080 \, a \sin \left (d x + c\right )^{3} + 2772 \, a \sin \left (d x + c\right )^{2} - 840 \, a \sin \left (d x + c\right ) - 770 \, a}{9240 \, d \sin \left (d x + c\right )^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/9240*(1848*a*sin(d*x + c)^7 + 1540*a*sin(d*x + c)^6 - 3960*a*sin(d*x + c)^5 - 3465*a*sin(d*x + c)^4 + 3080*a
*sin(d*x + c)^3 + 2772*a*sin(d*x + c)^2 - 840*a*sin(d*x + c) - 770*a)/(d*sin(d*x + c)^12)