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

Optimal. Leaf size=118 \[ -\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^6(c+d x)}{6 d}+\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]

[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

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Rubi [A]  time = 0.08, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2836, 12, 88} \[ -\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^6(c+d x)}{6 d}+\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[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 88

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 2836

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*x)/b
)^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*} \int \cos ^6(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^3 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {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*}

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Mathematica [A]  time = 0.13, size = 106, normalized size = 0.90 \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {a \left (-2 \sin ^6(c+d x)+9 \sin ^4(c+d x)-18 \sin ^2(c+d x)+12 \log (\sin (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d) + (a*(12*Log
[Sin[c + d*x]] - 18*Sin[c + d*x]^2 + 9*Sin[c + d*x]^4 - 2*Sin[c + d*x]^6))/(12*d)

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fricas [A]  time = 0.82, size = 96, normalized size = 0.81 \[ \frac {70 \, a \cos \left (d x + c\right )^{6} + 105 \, a \cos \left (d x + c\right )^{4} + 210 \, a \cos \left (d x + c\right )^{2} + 420 \, a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.27, size = 92, normalized size = 0.78 \[ -\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 252 \, a \sin \left (d x + c\right )^{5} - 315 \, a \sin \left (d x + c\right )^{4} + 420 \, a \sin \left (d x + c\right )^{3} + 630 \, a \sin \left (d x + c\right )^{2} - 420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 420 \, a \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.34, size = 128, normalized size = 1.08 \[ \frac {16 a \sin \left (d x +c \right )}{35 d}+\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{7 d}+\frac {6 a \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35 d}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{35 d}+\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.34, size = 91, normalized size = 0.77 \[ -\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 252 \, a \sin \left (d x + c\right )^{5} - 315 \, a \sin \left (d x + c\right )^{4} + 420 \, a \sin \left (d x + c\right )^{3} + 630 \, a \sin \left (d x + c\right )^{2} - 420 \, a \log \left (\sin \left (d x + c\right )\right ) - 420 \, a \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 9.16, size = 160, normalized size = 1.36 \[ \frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^4}{4\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^6}{6\,d}+\frac {16\,a\,\sin \left (c+d\,x\right )}{35\,d}+\frac {8\,a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{35\,d}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{35\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )}{7\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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