3.216 \(\int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=73 \[ -\frac {a^3 \csc ^7(c+d x)}{7 d}-\frac {a^3 \csc ^6(c+d x)}{2 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^4(c+d x)}{4 d} \]

[Out]

-1/4*a^3*csc(d*x+c)^4/d-3/5*a^3*csc(d*x+c)^5/d-1/2*a^3*csc(d*x+c)^6/d-1/7*a^3*csc(d*x+c)^7/d

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Rubi [A]  time = 0.07, antiderivative size = 73, 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 = {2833, 12, 43} \[ -\frac {a^3 \csc ^7(c+d x)}{7 d}-\frac {a^3 \csc ^6(c+d x)}{2 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

-(a^3*Csc[c + d*x]^4)/(4*d) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(2*d) - (a^3*Csc[c + d*x]^7)
/(7*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 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 2833

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 73, normalized size = 1.00 \[ -\frac {a^3 \csc ^7(c+d x)}{7 d}-\frac {a^3 \csc ^6(c+d x)}{2 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/4*(a^3*Csc[c + d*x]^4)/d - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(2*d) - (a^3*Csc[c + d*x]^7)
/(7*d)

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fricas [A]  time = 0.63, size = 93, normalized size = 1.27 \[ -\frac {84 \, a^{3} \cos \left (d x + c\right )^{2} - 104 \, a^{3} + 35 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3}\right )} \sin \left (d x + c\right )}{140 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/140*(84*a^3*cos(d*x + c)^2 - 104*a^3 + 35*(a^3*cos(d*x + c)^2 - 3*a^3)*sin(d*x + c))/((d*cos(d*x + c)^6 - 3
*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))

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giac [A]  time = 0.19, size = 56, normalized size = 0.77 \[ -\frac {35 \, a^{3} \sin \left (d x + c\right )^{3} + 84 \, a^{3} \sin \left (d x + c\right )^{2} + 70 \, a^{3} \sin \left (d x + c\right ) + 20 \, a^{3}}{140 \, d \sin \left (d x + c\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/140*(35*a^3*sin(d*x + c)^3 + 84*a^3*sin(d*x + c)^2 + 70*a^3*sin(d*x + c) + 20*a^3)/(d*sin(d*x + c)^7)

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maple [A]  time = 0.19, size = 49, normalized size = 0.67 \[ \frac {a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{6}}-\frac {3}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x)

[Out]

a^3/d*(-1/2/sin(d*x+c)^6-3/5/sin(d*x+c)^5-1/7/sin(d*x+c)^7-1/4/sin(d*x+c)^4)

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maxima [A]  time = 0.38, size = 56, normalized size = 0.77 \[ -\frac {35 \, a^{3} \sin \left (d x + c\right )^{3} + 84 \, a^{3} \sin \left (d x + c\right )^{2} + 70 \, a^{3} \sin \left (d x + c\right ) + 20 \, a^{3}}{140 \, d \sin \left (d x + c\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^8*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/140*(35*a^3*sin(d*x + c)^3 + 84*a^3*sin(d*x + c)^2 + 70*a^3*sin(d*x + c) + 20*a^3)/(d*sin(d*x + c)^7)

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mupad [B]  time = 8.66, size = 56, normalized size = 0.77 \[ -\frac {35\,a^3\,{\sin \left (c+d\,x\right )}^3+84\,a^3\,{\sin \left (c+d\,x\right )}^2+70\,a^3\,\sin \left (c+d\,x\right )+20\,a^3}{140\,d\,{\sin \left (c+d\,x\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^3)/sin(c + d*x)^8,x)

[Out]

-(70*a^3*sin(c + d*x) + 20*a^3 + 84*a^3*sin(c + d*x)^2 + 35*a^3*sin(c + d*x)^3)/(140*d*sin(c + d*x)^7)

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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)*csc(d*x+c)**8*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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