\(\int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx\) [216]

Optimal result

Integrand size = 27, antiderivative size = 73 \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{2 d}-\frac {a^3 \csc ^7(c+d x)}{7 d} \]

[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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 45} \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^7(c+d x)}{7 d}-\frac {a^3 \csc ^6(c+d x)}{2 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^4(c+d x)}{4 d} \]

[In]

Int[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)

Rule 12

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

Rule 45

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 2912

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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^8 (a+x)^3}{x^8} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^7 \text {Subst}\left (\int \frac {(a+x)^3}{x^8} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^7 \text {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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{2 d}-\frac {a^3 \csc ^7(c+d x)}{7 d} \]

[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)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-\frac {a^{3} \left (\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\)	\(50\)
default	\(-\frac {a^{3} \left (\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\)	\(50\)
parallelrisch	\(-\frac {a^{3} \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-2688 \cos \left (2 d x +2 c \right )+35 \sin \left (7 d x +7 c \right )-245 \sin \left (5 d x +5 c \right )+4935 \sin \left (d x +c \right )+175 \sin \left (3 d x +3 c \right )+3968\right )}{1146880 d}\)	\(85\)
risch	\(-\frac {4 a^{3} \left (168 i {\mathrm e}^{9 i \left (d x +c \right )}+35 \,{\mathrm e}^{10 i \left (d x +c \right )}-496 i {\mathrm e}^{7 i \left (d x +c \right )}-385 \,{\mathrm e}^{8 i \left (d x +c \right )}+168 i {\mathrm e}^{5 i \left (d x +c \right )}+385 \,{\mathrm e}^{6 i \left (d x +c \right )}-35 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{35 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\)	\(103\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {84 \, a^{3} \cos \left (d x + c\right )^{2} - 104 \, a^{3} + 35 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3}\right )} \sin \left (d x + c\right )}{140 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[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))

Sympy [F(-1)]

Timed out. \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {35 \, a^{3} \sin \left (d x + c\right )^{3} + 84 \, a^{3} \sin \left (d x + c\right )^{2} + 70 \, a^{3} \sin \left (d x + c\right ) + 20 \, a^{3}}{140 \, d \sin \left (d x + c\right )^{7}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {35 \, a^{3} \sin \left (d x + c\right )^{3} + 84 \, a^{3} \sin \left (d x + c\right )^{2} + 70 \, a^{3} \sin \left (d x + c\right ) + 20 \, a^{3}}{140 \, d \sin \left (d x + c\right )^{7}} \]

[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)

Mupad [B] (verification not implemented)

Time = 9.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {35\,a^3\,{\sin \left (c+d\,x\right )}^3+84\,a^3\,{\sin \left (c+d\,x\right )}^2+70\,a^3\,\sin \left (c+d\,x\right )+20\,a^3}{140\,d\,{\sin \left (c+d\,x\right )}^7} \]

[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)