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

Optimal. Leaf size=97 \[ -\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^5(c+d x)}{5 d} \]

[Out]

-1/5*a*csc(d*x+c)^5/d-1/6*a*csc(d*x+c)^6/d+2/7*a*csc(d*x+c)^7/d+1/4*a*csc(d*x+c)^8/d-1/9*a*csc(d*x+c)^9/d-1/10
*a*csc(d*x+c)^10/d

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Rubi [A]  time = 0.08, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) + (2*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) -
(a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2836

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

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 88, normalized size = 0.91 \[ -\frac {a \csc ^9(c+d x)}{9 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \left (6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right )}{60 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(a*Csc[c + d*x]^5)/d + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*(10*Csc[c + d*x]^6 - 15
*Csc[c + d*x]^8 + 6*Csc[c + d*x]^10))/(60*d)

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fricas [A]  time = 0.82, size = 122, normalized size = 1.26 \[ \frac {210 \, a \cos \left (d x + c\right )^{4} - 105 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (63 \, a \cos \left (d x + c\right )^{4} - 36 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 21 \, a}{1260 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/1260*(210*a*cos(d*x + c)^4 - 105*a*cos(d*x + c)^2 + 4*(63*a*cos(d*x + c)^4 - 36*a*cos(d*x + c)^2 + 8*a)*sin(
d*x + c) + 21*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)

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giac [A]  time = 0.26, size = 70, normalized size = 0.72 \[ -\frac {252 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, a \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, a \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1260*(252*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*a*sin(d*x + c)^3 - 315*a*sin(d*x + c)^2 + 140*a*sin
(d*x + c) + 126*a)/(d*sin(d*x + c)^10)

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maple [B]  time = 0.37, size = 184, normalized size = 1.90 \[ \frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{6}\left (d x +c \right )}{21 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{6}\left (d x +c \right )}{20 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{6}\left (d x +c \right )}{60 \sin \left (d x +c \right )^{6}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/9/sin(d*x+c)^9*cos(d*x+c)^6-1/21/sin(d*x+c)^7*cos(d*x+c)^6-1/105/sin(d*x+c)^5*cos(d*x+c)^6+1/315/si
n(d*x+c)^3*cos(d*x+c)^6-1/105/sin(d*x+c)*cos(d*x+c)^6-1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+a*
(-1/10/sin(d*x+c)^10*cos(d*x+c)^6-1/20/sin(d*x+c)^8*cos(d*x+c)^6-1/60/sin(d*x+c)^6*cos(d*x+c)^6))

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maxima [A]  time = 0.45, size = 70, normalized size = 0.72 \[ -\frac {252 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 360 \, a \sin \left (d x + c\right )^{3} - 315 \, a \sin \left (d x + c\right )^{2} + 140 \, a \sin \left (d x + c\right ) + 126 \, a}{1260 \, d \sin \left (d x + c\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^11*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/1260*(252*a*sin(d*x + c)^5 + 210*a*sin(d*x + c)^4 - 360*a*sin(d*x + c)^3 - 315*a*sin(d*x + c)^2 + 140*a*sin
(d*x + c) + 126*a)/(d*sin(d*x + c)^10)

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mupad [B]  time = 8.86, size = 70, normalized size = 0.72 \[ -\frac {252\,a\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-360\,a\,{\sin \left (c+d\,x\right )}^3-315\,a\,{\sin \left (c+d\,x\right )}^2+140\,a\,\sin \left (c+d\,x\right )+126\,a}{1260\,d\,{\sin \left (c+d\,x\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^5*(a + a*sin(c + d*x)))/sin(c + d*x)^11,x)

[Out]

-(126*a + 140*a*sin(c + d*x) - 315*a*sin(c + d*x)^2 - 360*a*sin(c + d*x)^3 + 210*a*sin(c + d*x)^4 + 252*a*sin(
c + d*x)^5)/(1260*d*sin(c + d*x)^10)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**11*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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