\(\int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx\) [666]

Optimal result

Integrand size = 27, antiderivative size = 118 \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \csc (c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 90} \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x)}{2 d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {3 a \csc (c+d x)}{d}+\frac {3 a \log (\sin (c+d x))}{d} \]

[In]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a \csc (c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{3 d} \]

[In]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3}\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(161\)
default	\(\frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3}\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(161\)
parallelrisch	\(\frac {a \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (576 \left (-\cos \left (4 d x +4 c \right )-3+4 \cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2304 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )+576 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (4 d x +4 c \right )+232 \sin \left (5 d x +5 c \right )+8 \sin \left (7 d x +7 c \right )+1728 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+396 \cos \left (2 d x +2 c \right )-441 \cos \left (4 d x +4 c \right )+24 \cos \left (6 d x +6 c \right )+5544 \sin \left (d x +c \right )-2424 \sin \left (3 d x +3 c \right )-363\right )}{24576 d}\)	\(197\)
risch	\(-3 i a x -\frac {i a \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {11 i a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {11 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {6 i a c}{d}+\frac {2 i a \left (9 i {\mathrm e}^{6 i \left (d x +c \right )}+9 \,{\mathrm e}^{7 i \left (d x +c \right )}-12 i {\mathrm e}^{4 i \left (d x +c \right )}-23 \,{\mathrm e}^{5 i \left (d x +c \right )}+9 i {\mathrm e}^{2 i \left (d x +c \right )}+23 \,{\mathrm e}^{3 i \left (d x +c \right )}-9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(228\)
norman	\(\frac {-\frac {a}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d}+\frac {17 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {5 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {91 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {35 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {91 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {5 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {17 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {55 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {55 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(274\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^5*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.20 \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {6 \, a \cos \left (d x + c\right )^{6} - 15 \, a \cos \left (d x + c\right )^{4} - 6 \, a \cos \left (d x + c\right )^{2} + 36 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 24 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right ) + 12 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(6*a*cos(d*x + c)^6 - 15*a*cos(d*x + c)^4 - 6*a*cos(d*x + c)^2 + 36*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^
2 + a)*log(1/2*sin(d*x + c)) + 4*(a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 - 24*a*cos(d*x + c)^2 + 16*a)*sin(d*x
+ c) + 12*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**5*(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 = 0.78 \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, a \sin \left (d x + c\right )^{3} + 6 \, a \sin \left (d x + c\right )^{2} - 36 \, a \log \left (\sin \left (d x + c\right )\right ) - 36 \, a \sin \left (d x + c\right ) - \frac {36 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} - 4 \, a \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/12*(4*a*sin(d*x + c)^3 + 6*a*sin(d*x + c)^2 - 36*a*log(sin(d*x + c)) - 36*a*sin(d*x + c) - (36*a*sin(d*x +
c)^3 + 18*a*sin(d*x + c)^2 - 4*a*sin(d*x + c) - 3*a)/sin(d*x + c)^4)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, a \sin \left (d x + c\right )^{3} + 6 \, a \sin \left (d x + c\right )^{2} - 36 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 36 \, a \sin \left (d x + c\right ) + \frac {75 \, a \sin \left (d x + c\right )^{4} - 36 \, a \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/12*(4*a*sin(d*x + c)^3 + 6*a*sin(d*x + c)^2 - 36*a*log(abs(sin(d*x + c))) - 36*a*sin(d*x + c) + (75*a*sin(d
*x + c)^4 - 36*a*sin(d*x + c)^3 - 18*a*sin(d*x + c)^2 + 4*a*sin(d*x + c) + 3*a)/sin(d*x + c)^4)/d

Mupad [B] (verification not implemented)

Time = 10.41 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.46 \[ \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {118\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-27\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {644\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {69\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{4}+160\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {57\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {a}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}+\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^5,x)

[Out]

(11*a*tan(c/2 + (d*x)/2))/(8*d) + ((17*a*tan(c/2 + (d*x)/2)^2)/4 - (2*a*tan(c/2 + (d*x)/2))/3 - a/4 + 20*a*tan
(c/2 + (d*x)/2)^3 + (57*a*tan(c/2 + (d*x)/2)^4)/4 + 160*a*tan(c/2 + (d*x)/2)^5 - (69*a*tan(c/2 + (d*x)/2)^6)/4
+ (644*a*tan(c/2 + (d*x)/2)^7)/3 - 27*a*tan(c/2 + (d*x)/2)^8 + 118*a*tan(c/2 + (d*x)/2)^9)/(d*(16*tan(c/2 + (
d*x)/2)^4 + 48*tan(c/2 + (d*x)/2)^6 + 48*tan(c/2 + (d*x)/2)^8 + 16*tan(c/2 + (d*x)/2)^10)) - (3*a*log(tan(c/2
+ (d*x)/2)^2 + 1))/d + (5*a*tan(c/2 + (d*x)/2)^2)/(16*d) - (a*tan(c/2 + (d*x)/2)^3)/(24*d) - (a*tan(c/2 + (d*x
)/2)^4)/(64*d) + (3*a*log(tan(c/2 + (d*x)/2)))/d