\(\int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx\) [674]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]

[Out]

-1/8*a*cot(d*x+c)^8/d-1/5*a*cot(d*x+c)^10/d-1/12*a*cot(d*x+c)^12/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 = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2913, 2687, 272, 45, 2686, 276} \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^{12}(c+d x)}{12 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^5(c+d x)}{5 d} \]

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

-1/8*(a*Cot[c + d*x]^8)/d - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*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 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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 ^5(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^6(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 )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int x^3 (1+x)^2 \, dx,x,\cot ^2(c+d x)\right )}{2 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 \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}-\frac {a \text {Subst}\left (\int \left (x^3+2 x^4+x^5\right ) \, dx,x,\cot ^2(c+d x)\right )}{2 d} \\ & = -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 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 = 1.14 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^6(c+d x)}{6 d}-\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^8(c+d x)}{8 d}+\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{12}(c+d x)}{12 d} \]

[In]

Integrate[Cot[c + d*x]^7*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) - (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) + (3*a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^12)/(
12*d)

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\)	\(88\)
default	\(-\frac {a \left (\frac {\left (\csc ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\csc ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\)	\(88\)
parallelrisch	\(\frac {a \left (\sec ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-221939718-376472712 \cos \left (2 d x +2 c \right )-1428042 \cos \left (8 d x +8 c \right )-30277632 \sin \left (7 d x +7 c \right )-47579136 \sin \left (5 d x +5 c \right )-45702580 \cos \left (6 d x +6 c \right )+5898240 \sin \left (d x +c \right )-145620992 \sin \left (3 d x +3 c \right )-162098475 \cos \left (4 d x +4 c \right )-21637 \cos \left (12 d x +12 c \right )+259644 \cos \left (10 d x +10 c \right )\right )}{39685497815040 d}\)	\(138\)
risch	\(\frac {32 i a \left (385 i {\mathrm e}^{18 i \left (d x +c \right )}+231 \,{\mathrm e}^{19 i \left (d x +c \right )}+1155 i {\mathrm e}^{16 i \left (d x +c \right )}+363 \,{\mathrm e}^{17 i \left (d x +c \right )}+3003 i {\mathrm e}^{14 i \left (d x +c \right )}+1111 \,{\mathrm e}^{15 i \left (d x +c \right )}+3234 i {\mathrm e}^{12 i \left (d x +c \right )}-45 \,{\mathrm e}^{13 i \left (d x +c \right )}+3003 i {\mathrm e}^{10 i \left (d x +c \right )}+45 \,{\mathrm e}^{11 i \left (d x +c \right )}+1155 i {\mathrm e}^{8 i \left (d x +c \right )}-1111 \,{\mathrm e}^{9 i \left (d x +c \right )}+385 i {\mathrm e}^{6 i \left (d x +c \right )}-363 \,{\mathrm e}^{7 i \left (d x +c \right )}-231 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{1155 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{12}}\)	\(194\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/12*csc(d*x+c)^12+1/11*csc(d*x+c)^11-3/10*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
)^7-1/6*csc(d*x+c)^6-1/5*csc(d*x+c)^5)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.35 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {1540 \, a \cos \left (d x + c\right )^{6} - 1155 \, a \cos \left (d x + c\right )^{4} + 462 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (231 \, a \cos \left (d x + c\right )^{6} - 198 \, a \cos \left (d x + c\right )^{4} + 88 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 77 \, a}{9240 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/9240*(1540*a*cos(d*x + c)^6 - 1155*a*cos(d*x + c)^4 + 462*a*cos(d*x + c)^2 + 8*(231*a*cos(d*x + c)^6 - 198*
a*cos(d*x + c)^4 + 88*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 77*a)/(d*cos(d*x + c)^12 - 6*d*cos(d*x + c)^10 +
15*d*cos(d*x + c)^8 - 20*d*cos(d*x + c)^6 + 15*d*cos(d*x + c)^4 - 6*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**13*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1848 \, a \sin \left (d x + c\right )^{7} + 1540 \, a \sin \left (d x + c\right )^{6} - 3960 \, a \sin \left (d x + c\right )^{5} - 3465 \, a \sin \left (d x + c\right )^{4} + 3080 \, a \sin \left (d x + c\right )^{3} + 2772 \, a \sin \left (d x + c\right )^{2} - 840 \, a \sin \left (d x + c\right ) - 770 \, a}{9240 \, d \sin \left (d x + c\right )^{12}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/9240*(1848*a*sin(d*x + c)^7 + 1540*a*sin(d*x + c)^6 - 3960*a*sin(d*x + c)^5 - 3465*a*sin(d*x + c)^4 + 3080*a
*sin(d*x + c)^3 + 2772*a*sin(d*x + c)^2 - 840*a*sin(d*x + c) - 770*a)/(d*sin(d*x + c)^12)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1848 \, a \sin \left (d x + c\right )^{7} + 1540 \, a \sin \left (d x + c\right )^{6} - 3960 \, a \sin \left (d x + c\right )^{5} - 3465 \, a \sin \left (d x + c\right )^{4} + 3080 \, a \sin \left (d x + c\right )^{3} + 2772 \, a \sin \left (d x + c\right )^{2} - 840 \, a \sin \left (d x + c\right ) - 770 \, a}{9240 \, d \sin \left (d x + c\right )^{12}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^13*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/9240*(1848*a*sin(d*x + c)^7 + 1540*a*sin(d*x + c)^6 - 3960*a*sin(d*x + c)^5 - 3465*a*sin(d*x + c)^4 + 3080*a
*sin(d*x + c)^3 + 2772*a*sin(d*x + c)^2 - 840*a*sin(d*x + c) - 770*a)/(d*sin(d*x + c)^12)

Mupad [B] (verification not implemented)

Time = 10.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{7}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{10}+\frac {a\,\sin \left (c+d\,x\right )}{11}+\frac {a}{12}}{d\,{\sin \left (c+d\,x\right )}^{12}} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^13,x)

[Out]

-(a/12 + (a*sin(c + d*x))/11 - (3*a*sin(c + d*x)^2)/10 - (a*sin(c + d*x)^3)/3 + (3*a*sin(c + d*x)^4)/8 + (3*a*
sin(c + d*x)^5)/7 - (a*sin(c + d*x)^6)/6 - (a*sin(c + d*x)^7)/5)/(d*sin(c + d*x)^12)