\(\int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx\) [667]

Optimal result

Integrand size = 25, antiderivative size = 115 \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \csc (c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc ^3(c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {3 a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d} \]

[Out]

-3*a*csc(d*x+c)/d+3/2*a*csc(d*x+c)^2/d+a*csc(d*x+c)^3/d-1/4*a*csc(d*x+c)^4/d-1/5*a*csc(d*x+c)^5/d+3*a*ln(sin(d
*x+c))/d-a*sin(d*x+c)/d-1/2*a*sin(d*x+c)^2/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2915, 12, 90} \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin (c+d x)}{d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {3 a \csc (c+d x)}{d}+\frac {3 a \log (\sin (c+d x))}{d} \]

[In]

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

[Out]

(-3*a*Csc[c + d*x])/d + (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[
c + d*x]^5)/(5*d) + (3*a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \csc (c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc ^3(c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {3 a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d} \]

[In]

Integrate[Cos[c + d*x]*Cot[c + d*x]^6*(a + a*Sin[c + d*x]),x]

[Out]

(-3*a*Csc[c + d*x])/d + (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[
c + d*x]^5)/(5*d) + (3*a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{8}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )\right )}{d}\)	\(179\)
default	\(\frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{8}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )\right )}{d}\)	\(179\)
parallelrisch	\(\frac {3 \left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (-\sin \left (5 d x +5 c \right )+5 \sin \left (3 d x +3 c \right )-10 \sin \left (d x +c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sin \left (5 d x +5 c \right )-5 \sin \left (3 d x +3 c \right )+10 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {137 \sin \left (5 d x +5 c \right )}{192}+\frac {\sin \left (7 d x +7 c \right )}{24}+\frac {47 \cos \left (2 d x +2 c \right )}{6}-3 \cos \left (4 d x +4 c \right )+\frac {\cos \left (6 d x +6 c \right )}{6}-\frac {97 \sin \left (d x +c \right )}{96}+\frac {63 \sin \left (3 d x +3 c \right )}{64}-\frac {91}{15}\right ) \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{512 d}\)	\(189\)
risch	\(-3 i a x +\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {6 i a c}{d}-\frac {2 i a \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}-40 \,{\mathrm e}^{7 i \left (d x +c \right )}-15 i {\mathrm e}^{8 i \left (d x +c \right )}+66 \,{\mathrm e}^{5 i \left (d x +c \right )}+35 i {\mathrm e}^{6 i \left (d x +c \right )}-40 \,{\mathrm e}^{3 i \left (d x +c \right )}-35 i {\mathrm e}^{4 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}+15 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{5 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(219\)
norman	\(\frac {-\frac {a}{160 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {13 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {9 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {161 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {175 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {175 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {161 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {9 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {13 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {83 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(274\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^6*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-1/4/sin(d*x+c)^4*cos(d*x+c)^8+1/2/sin(d*x+c)^2*cos(d*x+c)^8+1/2*cos(d*x+c)^6+3/4*cos(d*x+c)^4+3/2*cos
(d*x+c)^2+3*ln(sin(d*x+c)))+a*(-1/5/sin(d*x+c)^5*cos(d*x+c)^8+1/5/sin(d*x+c)^3*cos(d*x+c)^8-1/sin(d*x+c)*cos(d
*x+c)^8-(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.37 \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {20 \, a \cos \left (d x + c\right )^{6} - 120 \, a \cos \left (d x + c\right )^{4} + 160 \, a \cos \left (d x + c\right )^{2} + 60 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 5 \, {\left (2 \, a \cos \left (d x + c\right )^{6} - 5 \, a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + 4 \, a\right )} \sin \left (d x + c\right ) - 64 \, a}{20 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/20*(20*a*cos(d*x + c)^6 - 120*a*cos(d*x + c)^4 + 160*a*cos(d*x + c)^2 + 60*(a*cos(d*x + c)^4 - 2*a*cos(d*x +
c)^2 + a)*log(1/2*sin(d*x + c))*sin(d*x + c) + 5*(2*a*cos(d*x + c)^6 - 5*a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^
2 + 4*a)*sin(d*x + c) - 64*a)/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**6*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 20 \, a \sin \left (d x + c\right ) + \frac {60 \, a \sin \left (d x + c\right )^{4} - 30 \, a \sin \left (d x + c\right )^{3} - 20 \, a \sin \left (d x + c\right )^{2} + 5 \, a \sin \left (d x + c\right ) + 4 \, a}{\sin \left (d x + c\right )^{5}}}{20 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/20*(10*a*sin(d*x + c)^2 - 60*a*log(sin(d*x + c)) + 20*a*sin(d*x + c) + (60*a*sin(d*x + c)^4 - 30*a*sin(d*x
+ c)^3 - 20*a*sin(d*x + c)^2 + 5*a*sin(d*x + c) + 4*a)/sin(d*x + c)^5)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 20 \, a \sin \left (d x + c\right ) + \frac {137 \, a \sin \left (d x + c\right )^{5} + 60 \, a \sin \left (d x + c\right )^{4} - 30 \, a \sin \left (d x + c\right )^{3} - 20 \, a \sin \left (d x + c\right )^{2} + 5 \, a \sin \left (d x + c\right ) + 4 \, a}{\sin \left (d x + c\right )^{5}}}{20 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/20*(10*a*sin(d*x + c)^2 - 60*a*log(abs(sin(d*x + c))) + 20*a*sin(d*x + c) + (137*a*sin(d*x + c)^5 + 60*a*si
n(d*x + c)^4 - 30*a*sin(d*x + c)^3 - 20*a*sin(d*x + c)^2 + 5*a*sin(d*x + c) + 4*a)/sin(d*x + c)^5)/d

Mupad [B] (verification not implemented)

Time = 10.48 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.44 \[ \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {19\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {102\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+137\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {39\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {161\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^6,x)

[Out]

(5*a*tan(c/2 + (d*x)/2)^2)/(16*d) - (3*a*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (19*a*tan(c/2 + (d*x)/2))/(16*d) +
(3*a*tan(c/2 + (d*x)/2)^3)/(32*d) - (a*tan(c/2 + (d*x)/2)^4)/(64*d) - (a*tan(c/2 + (d*x)/2)^5)/(160*d) + (3*a
*log(tan(c/2 + (d*x)/2)))/d - (a/5 + (a*tan(c/2 + (d*x)/2))/2 - (13*a*tan(c/2 + (d*x)/2)^2)/5 - 9*a*tan(c/2 +
(d*x)/2)^3 + (161*a*tan(c/2 + (d*x)/2)^4)/5 - (39*a*tan(c/2 + (d*x)/2)^5)/2 + 137*a*tan(c/2 + (d*x)/2)^6 + 54*
a*tan(c/2 + (d*x)/2)^7 + 102*a*tan(c/2 + (d*x)/2)^8)/(d*(32*tan(c/2 + (d*x)/2)^5 + 64*tan(c/2 + (d*x)/2)^7 + 3
2*tan(c/2 + (d*x)/2)^9))