\(\int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx\) [671]

Optimal result

Integrand size = 27, antiderivative size = 81 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \]

[Out]

-1/8*a*cot(d*x+c)^8/d+1/3*a*csc(d*x+c)^3/d-3/5*a*csc(d*x+c)^5/d+3/7*a*csc(d*x+c)^7/d-1/9*a*csc(d*x+c)^9/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2686, 276, 2687, 30} \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{3 d} \]

[In]

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

[Out]

-1/8*(a*Cot[c + d*x]^8)/d + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d)
- (a*Csc[c + d*x]^9)/(9*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 276

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 2686

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 2687

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 2913

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])

Rubi steps \begin{align*} \text {integral}& = a \int \cot ^7(c+d x) \csc ^2(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^7 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \text {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 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)}{9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \]

[In]

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

[Out]

-1/8*(a*Cot[c + d*x]^8)/d + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d)
- (a*Csc[c + d*x]^9)/(9*d)

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(88\)
default	\(-\frac {a \left (\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(88\)
parallelrisch	\(\frac {a \left (\sec ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-5824512 \cos \left (2 d x +2 c \right )-29295 \sin \left (9 d x +9 c \right )-381465 \sin \left (7 d x +7 c \right )-409500 \sin \left (5 d x +5 c \right )-860160 \cos \left (6 d x +6 c \right )+824670 \sin \left (d x +c \right )-2055060 \sin \left (3 d x +3 c \right )-1032192 \cos \left (4 d x +4 c \right )-1458176\right )}{42278584320 d}\)	\(116\)
risch	\(-\frac {2 a \left (420 i {\mathrm e}^{15 i \left (d x +c \right )}+315 \,{\mathrm e}^{16 i \left (d x +c \right )}+504 i {\mathrm e}^{13 i \left (d x +c \right )}-315 \,{\mathrm e}^{14 i \left (d x +c \right )}+2844 i {\mathrm e}^{11 i \left (d x +c \right )}+2205 \,{\mathrm e}^{12 i \left (d x +c \right )}+1424 i {\mathrm e}^{9 i \left (d x +c \right )}-2205 \,{\mathrm e}^{10 i \left (d x +c \right )}+2844 i {\mathrm e}^{7 i \left (d x +c \right )}+2205 \,{\mathrm e}^{8 i \left (d x +c \right )}+504 i {\mathrm e}^{5 i \left (d x +c \right )}-2205 \,{\mathrm e}^{6 i \left (d x +c \right )}+420 i {\mathrm e}^{3 i \left (d x +c \right )}+315 \,{\mathrm e}^{4 i \left (d x +c \right )}-315 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}\)	\(193\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^10*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/9*csc(d*x+c)^9+1/8*csc(d*x+c)^8-3/7*csc(d*x+c)^7-1/2*csc(d*x+c)^6+3/5*csc(d*x+c)^5+3/4*csc(d*x+c)^4-1/
3*csc(d*x+c)^3-1/2*csc(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.72 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {840 \, a \cos \left (d x + c\right )^{6} - 1008 \, a \cos \left (d x + c\right )^{4} + 576 \, a \cos \left (d x + c\right )^{2} + 315 \, {\left (4 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 128 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(840*a*cos(d*x + c)^6 - 1008*a*cos(d*x + c)^4 + 576*a*cos(d*x + c)^2 + 315*(4*a*cos(d*x + c)^6 - 6*a*c
os(d*x + c)^4 + 4*a*cos(d*x + c)^2 - a)*sin(d*x + c) - 128*a)/((d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*co
s(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**10*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1260 \, a \sin \left (d x + c\right )^{7} + 840 \, a \sin \left (d x + c\right )^{6} - 1890 \, a \sin \left (d x + c\right )^{5} - 1512 \, a \sin \left (d x + c\right )^{4} + 1260 \, a \sin \left (d x + c\right )^{3} + 1080 \, a \sin \left (d x + c\right )^{2} - 315 \, a \sin \left (d x + c\right ) - 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/2520*(1260*a*sin(d*x + c)^7 + 840*a*sin(d*x + c)^6 - 1890*a*sin(d*x + c)^5 - 1512*a*sin(d*x + c)^4 + 1260*a*
sin(d*x + c)^3 + 1080*a*sin(d*x + c)^2 - 315*a*sin(d*x + c) - 280*a)/(d*sin(d*x + c)^9)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1260 \, a \sin \left (d x + c\right )^{7} + 840 \, a \sin \left (d x + c\right )^{6} - 1890 \, a \sin \left (d x + c\right )^{5} - 1512 \, a \sin \left (d x + c\right )^{4} + 1260 \, a \sin \left (d x + c\right )^{3} + 1080 \, a \sin \left (d x + c\right )^{2} - 315 \, a \sin \left (d x + c\right ) - 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^10*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2520*(1260*a*sin(d*x + c)^7 + 840*a*sin(d*x + c)^6 - 1890*a*sin(d*x + c)^5 - 1512*a*sin(d*x + c)^4 + 1260*a*
sin(d*x + c)^3 + 1080*a*sin(d*x + c)^2 - 315*a*sin(d*x + c) - 280*a)/(d*sin(d*x + c)^9)

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{2}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^10,x)

[Out]

-(a/9 + (a*sin(c + d*x))/8 - (3*a*sin(c + d*x)^2)/7 - (a*sin(c + d*x)^3)/2 + (3*a*sin(c + d*x)^4)/5 + (3*a*sin
(c + d*x)^5)/4 - (a*sin(c + d*x)^6)/3 - (a*sin(c + d*x)^7)/2)/(d*sin(c + d*x)^9)