\(\int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx\) [670]

Optimal result

Integrand size = 27, antiderivative size = 74 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 74, 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, 2687, 30, 2686, 200} \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {a \csc (c+d x)}{d} \]

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 30

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && 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 (c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^2(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 \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 (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d} \]

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\csc ^{3}\left (d x +c \right )-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}-\csc \left (d x +c \right )\right )}{d}\)	\(84\)
default	\(-\frac {a \left (\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\csc ^{3}\left (d x +c \right )-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}-\csc \left (d x +c \right )\right )}{d}\)	\(84\)
parallelrisch	\(\frac {a \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-319480 \cos \left (2 d x +2 c \right )-3255 \cos \left (8 d x +8 c \right )-71680 \sin \left (7 d x +7 c \right )+215040 \sin \left (5 d x +5 c \right )-45640 \cos \left (6 d x +6 c \right )+1050624 \sin \left (d x +c \right )-759808 \sin \left (3 d x +3 c \right )-91140 \cos \left (4 d x +4 c \right )-113925\right )}{1174405120 d}\)	\(116\)
risch	\(\frac {2 i a \left (35 i {\mathrm e}^{14 i \left (d x +c \right )}+35 \,{\mathrm e}^{15 i \left (d x +c \right )}-105 \,{\mathrm e}^{13 i \left (d x +c \right )}+245 i {\mathrm e}^{10 i \left (d x +c \right )}+371 \,{\mathrm e}^{11 i \left (d x +c \right )}-513 \,{\mathrm e}^{9 i \left (d x +c \right )}+245 i {\mathrm e}^{6 i \left (d x +c \right )}+513 \,{\mathrm e}^{7 i \left (d x +c \right )}-371 \,{\mathrm e}^{5 i \left (d x +c \right )}+35 i {\mathrm e}^{2 i \left (d x +c \right )}+105 \,{\mathrm e}^{3 i \left (d x +c \right )}-35 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{35 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}\)	\(158\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/8*csc(d*x+c)^8+1/7*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+csc(d*x+c)^3-1/2*cs
c(d*x+c)^2-csc(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.77 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {140 \, a \cos \left (d x + c\right )^{6} - 210 \, a \cos \left (d x + c\right )^{4} + 140 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{6} - 70 \, a \cos \left (d x + c\right )^{4} + 56 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 35 \, a}{280 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/280*(140*a*cos(d*x + c)^6 - 210*a*cos(d*x + c)^4 + 140*a*cos(d*x + c)^2 + 8*(35*a*cos(d*x + c)^6 - 70*a*cos
(d*x + c)^4 + 56*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) - 35*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)

Sympy [F(-1)]

Timed out. \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**9*(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 = 1.24 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/280*(280*a*sin(d*x + c)^7 + 140*a*sin(d*x + c)^6 - 280*a*sin(d*x + c)^5 - 210*a*sin(d*x + c)^4 + 168*a*sin(d
*x + c)^3 + 140*a*sin(d*x + c)^2 - 40*a*sin(d*x + c) - 35*a)/(d*sin(d*x + c)^8)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {280 \, a \sin \left (d x + c\right )^{7} + 140 \, a \sin \left (d x + c\right )^{6} - 280 \, a \sin \left (d x + c\right )^{5} - 210 \, a \sin \left (d x + c\right )^{4} + 168 \, a \sin \left (d x + c\right )^{3} + 140 \, a \sin \left (d x + c\right )^{2} - 40 \, a \sin \left (d x + c\right ) - 35 \, a}{280 \, d \sin \left (d x + c\right )^{8}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^9*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/280*(280*a*sin(d*x + c)^7 + 140*a*sin(d*x + c)^6 - 280*a*sin(d*x + c)^5 - 210*a*sin(d*x + c)^4 + 168*a*sin(d
*x + c)^3 + 140*a*sin(d*x + c)^2 - 40*a*sin(d*x + c) - 35*a)/(d*sin(d*x + c)^8)

Mupad [B] (verification not implemented)

Time = 10.37 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23 \[ \int \cot ^7(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {-a\,{\sin \left (c+d\,x\right )}^7-\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}+a\,{\sin \left (c+d\,x\right )}^5+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {a\,\sin \left (c+d\,x\right )}{7}+\frac {a}{8}}{d\,{\sin \left (c+d\,x\right )}^8} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^9,x)

[Out]

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