\(\int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx\) [691]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 73, 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 = {2914, 2686, 276, 2687, 30} \[ \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}+\frac {2 \csc ^5(c+d x)}{5 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]

[In]

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

[Out]

Cot[c + d*x]^6/(6*a*d) - Csc[c + d*x]^3/(3*a*d) + (2*Csc[c + d*x]^5)/(5*a*d) - Csc[c + d*x]^7/(7*a*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 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^5(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac {\int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a} \\ & = \frac {\text {Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a d} \\ & = \frac {\cot ^6(c+d x)}{6 a d}-\frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a d} \\ & = \frac {\cot ^6(c+d x)}{6 a d}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {2 \csc ^5(c+d x)}{5 a d}-\frac {\csc ^7(c+d x)}{7 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^2(c+d x) \left (105-70 \csc (c+d x)-105 \csc ^2(c+d x)+84 \csc ^3(c+d x)+35 \csc ^4(c+d x)-30 \csc ^5(c+d x)\right )}{210 a d} \]

[In]

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

[Out]

(Csc[c + d*x]^2*(105 - 70*Csc[c + d*x] - 105*Csc[c + d*x]^2 + 84*Csc[c + d*x]^3 + 35*Csc[c + d*x]^4 - 30*Csc[c
+ d*x]^5))/(210*a*d)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d a}\)	\(70\)
default	\(-\frac {\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {2 \left (\csc ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{2}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{2}}{d a}\)	\(70\)
parallelrisch	\(-\frac {\left (7168 \cos \left (2 d x +2 c \right )-385 \sin \left (7 d x +7 c \right )-4025 \sin \left (5 d x +5 c \right )-8925 \sin \left (d x +c \right )-1365 \sin \left (3 d x +3 c \right )+8960 \cos \left (4 d x +4 c \right )+14592\right ) \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{27525120 a d}\)	\(96\)
risch	\(-\frac {2 \left (-140 i {\mathrm e}^{11 i \left (d x +c \right )}+105 \,{\mathrm e}^{12 i \left (d x +c \right )}-112 i {\mathrm e}^{9 i \left (d x +c \right )}-105 \,{\mathrm e}^{10 i \left (d x +c \right )}-456 i {\mathrm e}^{7 i \left (d x +c \right )}+350 \,{\mathrm e}^{8 i \left (d x +c \right )}-112 i {\mathrm e}^{5 i \left (d x +c \right )}-350 \,{\mathrm e}^{6 i \left (d x +c \right )}-140 i {\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{4 i \left (d x +c \right )}-105 \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{105 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\)	\(149\)
norman	\(\frac {-\frac {1}{896 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{672 d a}+\frac {7 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d a}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 d a}-\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {7 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}-\frac {7 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 d a}+\frac {7 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d a}+\frac {\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )}{672 d a}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{896 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(242\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^8/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/d/a*(1/7*csc(d*x+c)^7-1/6*csc(d*x+c)^6-2/5*csc(d*x+c)^5+1/2*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 = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.42 \[ \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {70 \, \cos \left (d x + c\right )^{4} - 56 \, \cos \left (d x + c\right )^{2} - 35 \, {\left (3 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 16}{210 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/210*(70*cos(d*x + c)^4 - 56*cos(d*x + c)^2 - 35*(3*cos(d*x + c)^4 - 3*cos(d*x + c)^2 + 1)*sin(d*x + c) + 16)
/((a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**8/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {105 \, \sin \left (d x + c\right )^{5} - 70 \, \sin \left (d x + c\right )^{4} - 105 \, \sin \left (d x + c\right )^{3} + 84 \, \sin \left (d x + c\right )^{2} + 35 \, \sin \left (d x + c\right ) - 30}{210 \, a d \sin \left (d x + c\right )^{7}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/210*(105*sin(d*x + c)^5 - 70*sin(d*x + c)^4 - 105*sin(d*x + c)^3 + 84*sin(d*x + c)^2 + 35*sin(d*x + c) - 30)
/(a*d*sin(d*x + c)^7)

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {105 \, \sin \left (d x + c\right )^{5} - 70 \, \sin \left (d x + c\right )^{4} - 105 \, \sin \left (d x + c\right )^{3} + 84 \, \sin \left (d x + c\right )^{2} + 35 \, \sin \left (d x + c\right ) - 30}{210 \, a d \sin \left (d x + c\right )^{7}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/210*(105*sin(d*x + c)^5 - 70*sin(d*x + c)^4 - 105*sin(d*x + c)^3 + 84*sin(d*x + c)^2 + 35*sin(d*x + c) - 30)
/(a*d*sin(d*x + c)^7)

Mupad [B] (verification not implemented)

Time = 10.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^7(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {105\,{\sin \left (c+d\,x\right )}^5-70\,{\sin \left (c+d\,x\right )}^4-105\,{\sin \left (c+d\,x\right )}^3+84\,{\sin \left (c+d\,x\right )}^2+35\,\sin \left (c+d\,x\right )-30}{210\,a\,d\,{\sin \left (c+d\,x\right )}^7} \]

[In]

int(cos(c + d*x)^7/(sin(c + d*x)^8*(a + a*sin(c + d*x))),x)

[Out]

(35*sin(c + d*x) + 84*sin(c + d*x)^2 - 105*sin(c + d*x)^3 - 70*sin(c + d*x)^4 + 105*sin(c + d*x)^5 - 30)/(210*
a*d*sin(c + d*x)^7)