\(\int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx\) [673]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{10 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]

[Out]

-1/8*a*cot(d*x+c)^8/d-1/10*a*cot(d*x+c)^10/d+1/5*a*csc(d*x+c)^5/d-3/7*a*csc(d*x+c)^7/d+1/3*a*csc(d*x+c)^9/d-1/
11*a*csc(d*x+c)^11/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2686, 276, 2687, 14} \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^{10}(c+d x)}{10 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^5(c+d x)}{5 d} \]

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

-1/8*(a*Cot[c + d*x]^8)/d - (a*Cot[c + d*x]^10)/(10*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 ^4(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^5(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^4 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int x^7 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \text {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 \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{10 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] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.33 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{2 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {3 a \csc ^8(c+d x)}{8 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]),x]

[Out]

(a*Csc[c + d*x]^4)/(4*d) + (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(2*d) - (3*a*Csc[c + d*x]^7)/(7*d) +
(3*a*Csc[c + d*x]^8)/(8*d) + (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d
)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}-\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\)	\(88\)
default	\(-\frac {a \left (\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}-\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\)	\(88\)
parallelrisch	\(\frac {a \left (\sec ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-27197440-55869440 \cos \left (2 d x +2 c \right )-200739 \sin \left (9 d x +9 c \right )-3727185 \sin \left (7 d x +7 c \right )-7741965 \sin \left (5 d x +5 c \right )-7569408 \cos \left (6 d x +6 c \right )-1807806 \sin \left (d x +c \right )-17632230 \sin \left (3 d x +3 c \right )-19464192 \cos \left (4 d x +4 c \right )+18249 \sin \left (11 d x +11 c \right )\right )}{2480343613440 d}\)	\(127\)
risch	\(\frac {4 a \left (1848 i {\mathrm e}^{17 i \left (d x +c \right )}+1155 \,{\mathrm e}^{18 i \left (d x +c \right )}+4752 i {\mathrm e}^{15 i \left (d x +c \right )}+1155 \,{\mathrm e}^{16 i \left (d x +c \right )}+13640 i {\mathrm e}^{13 i \left (d x +c \right )}+5775 \,{\mathrm e}^{14 i \left (d x +c \right )}+13280 i {\mathrm e}^{11 i \left (d x +c \right )}-1617 \,{\mathrm e}^{12 i \left (d x +c \right )}+13640 i {\mathrm e}^{9 i \left (d x +c \right )}+1617 \,{\mathrm e}^{10 i \left (d x +c \right )}+4752 i {\mathrm e}^{7 i \left (d x +c \right )}-5775 \,{\mathrm e}^{8 i \left (d x +c \right )}+1848 i {\mathrm e}^{5 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}-1155 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{1155 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\)	\(193\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/11*csc(d*x+c)^11+1/10*csc(d*x+c)^10-1/3*csc(d*x+c)^9-3/8*csc(d*x+c)^8+3/7*csc(d*x+c)^7+1/2*csc(d*x+c)^
6-1/5*csc(d*x+c)^5-1/4*csc(d*x+c)^4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1848 \, a \cos \left (d x + c\right )^{6} - 1584 \, a \cos \left (d x + c\right )^{4} + 704 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 128 \, a}{9240 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/9240*(1848*a*cos(d*x + c)^6 - 1584*a*cos(d*x + c)^4 + 704*a*cos(d*x + c)^2 + 231*(10*a*cos(d*x + c)^6 - 10*a
*cos(d*x + c)^4 + 5*a*cos(d*x + c)^2 - a)*sin(d*x + c) - 128*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)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**12*(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 = 0.95 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/9240*(2310*a*sin(d*x + c)^7 + 1848*a*sin(d*x + c)^6 - 4620*a*sin(d*x + c)^5 - 3960*a*sin(d*x + c)^4 + 3465*a
*sin(d*x + c)^3 + 3080*a*sin(d*x + c)^2 - 924*a*sin(d*x + c) - 840*a)/(d*sin(d*x + c)^11)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^12*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/9240*(2310*a*sin(d*x + c)^7 + 1848*a*sin(d*x + c)^6 - 4620*a*sin(d*x + c)^5 - 3960*a*sin(d*x + c)^4 + 3465*a
*sin(d*x + c)^3 + 3080*a*sin(d*x + c)^2 - 924*a*sin(d*x + c) - 840*a)/(d*sin(d*x + c)^11)

Mupad [B] (verification not implemented)

Time = 10.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{4}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{2}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{3}+\frac {a\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^12,x)

[Out]

-(a/11 + (a*sin(c + d*x))/10 - (a*sin(c + d*x)^2)/3 - (3*a*sin(c + d*x)^3)/8 + (3*a*sin(c + d*x)^4)/7 + (a*sin
(c + d*x)^5)/2 - (a*sin(c + d*x)^6)/5 - (a*sin(c + d*x)^7)/4)/(d*sin(c + d*x)^11)