\(\int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx\) [662]

Optimal result

Integrand size = 25, antiderivative size = 118 \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^7(c+d x)}{7 d} \]

[Out]

a*ln(sin(d*x+c))/d+a*sin(d*x+c)/d-3/2*a*sin(d*x+c)^2/d-a*sin(d*x+c)^3/d+3/4*a*sin(d*x+c)^4/d+3/5*a*sin(d*x+c)^
5/d-1/6*a*sin(d*x+c)^6/d-1/7*a*sin(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 118, 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} \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^6(c+d x)}{6 d}+\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]

[In]

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

[Out]

(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (3*a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c
+ d*x]^4)/(4*d) + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d) - (a*Sin[c + d*x]^7)/(7*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 (a-x)^3 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (a^6+\frac {a^7}{x}-3 a^5 x-3 a^4 x^2+3 a^3 x^3+3 a^2 x^4-a x^5-x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = \frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^7(c+d x)}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^7(c+d x)}{7 d} \]

[In]

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

[Out]

(a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (3*a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c
+ d*x]^4)/(4*d) + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d) - (a*Sin[c + d*x]^7)/(7*d)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+a \left (\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(85\)
default	\(\frac {\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+a \left (\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(85\)
parallelrisch	\(\frac {a \left (6720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6720 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3500+3045 \cos \left (2 d x +2 c \right )+15 \sin \left (7 d x +7 c \right )+147 \sin \left (5 d x +5 c \right )+35 \cos \left (6 d x +6 c \right )+3675 \sin \left (d x +c \right )+735 \sin \left (3 d x +3 c \right )+420 \cos \left (4 d x +4 c \right )\right )}{6720 d}\)	\(109\)
risch	\(-i a x +\frac {29 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {29 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {2 i a c}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {35 a \sin \left (d x +c \right )}{64 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \cos \left (6 d x +6 c \right )}{192 d}+\frac {7 a \sin \left (5 d x +5 c \right )}{320 d}+\frac {a \cos \left (4 d x +4 c \right )}{16 d}+\frac {7 a \sin \left (3 d x +3 c \right )}{64 d}\)	\(149\)
norman	\(\frac {\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {86 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {424 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {86 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {4 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {104 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {104 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {6 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {18 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {18 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(273\)

[In]

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

[Out]

1/d*(1/7*a*(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)+a*(1/6*cos(d*x+c)^6+1/4*cos(d*x+c)
^4+1/2*cos(d*x+c)^2+ln(sin(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {70 \, a \cos \left (d x + c\right )^{6} + 105 \, a \cos \left (d x + c\right )^{4} + 210 \, a \cos \left (d x + c\right )^{2} + 420 \, a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{420 \, d} \]

[In]

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

[Out]

1/420*(70*a*cos(d*x + c)^6 + 105*a*cos(d*x + c)^4 + 210*a*cos(d*x + c)^2 + 420*a*log(1/2*sin(d*x + c)) + 12*(5
*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 + 8*a*cos(d*x + c)^2 + 16*a)*sin(d*x + c))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 252 \, a \sin \left (d x + c\right )^{5} - 315 \, a \sin \left (d x + c\right )^{4} + 420 \, a \sin \left (d x + c\right )^{3} + 630 \, a \sin \left (d x + c\right )^{2} - 420 \, a \log \left (\sin \left (d x + c\right )\right ) - 420 \, a \sin \left (d x + c\right )}{420 \, d} \]

[In]

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

[Out]

-1/420*(60*a*sin(d*x + c)^7 + 70*a*sin(d*x + c)^6 - 252*a*sin(d*x + c)^5 - 315*a*sin(d*x + c)^4 + 420*a*sin(d*
x + c)^3 + 630*a*sin(d*x + c)^2 - 420*a*log(sin(d*x + c)) - 420*a*sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 252 \, a \sin \left (d x + c\right )^{5} - 315 \, a \sin \left (d x + c\right )^{4} + 420 \, a \sin \left (d x + c\right )^{3} + 630 \, a \sin \left (d x + c\right )^{2} - 420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 420 \, a \sin \left (d x + c\right )}{420 \, d} \]

[In]

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

[Out]

-1/420*(60*a*sin(d*x + c)^7 + 70*a*sin(d*x + c)^6 - 252*a*sin(d*x + c)^5 - 315*a*sin(d*x + c)^4 + 420*a*sin(d*
x + c)^3 + 630*a*sin(d*x + c)^2 - 420*a*log(abs(sin(d*x + c))) - 420*a*sin(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 10.75 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.36 \[ \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^4}{4\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^6}{6\,d}+\frac {16\,a\,\sin \left (c+d\,x\right )}{35\,d}+\frac {8\,a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d} \]

[In]

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

[Out]

(a*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (a*log(1/cos(c/2 + (d*x)/2)^2))/d + (a*cos(c + d*x)^2)/(2*d
) + (a*cos(c + d*x)^4)/(4*d) + (a*cos(c + d*x)^6)/(6*d) + (16*a*sin(c + d*x))/(35*d) + (8*a*cos(c + d*x)^2*sin
(c + d*x))/(35*d) + (6*a*cos(c + d*x)^4*sin(c + d*x))/(35*d) + (a*cos(c + d*x)^6*sin(c + d*x))/(7*d)