\(\int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\) [527]

Optimal result

Integrand size = 21, antiderivative size = 131 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc ^3(c+d x)}{d}-\frac {a^3 \csc ^4(c+d x)}{4 d}-\frac {5 a^3 \log (\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2786, 90} \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \csc ^4(c+d x)}{4 d}-\frac {a^3 \csc ^3(c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}+\frac {5 a^3 \csc (c+d x)}{d}-\frac {5 a^3 \log (\sin (c+d x))}{d} \]

[In]

Int[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

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

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 2786

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

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

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (60 \csc (c+d x)-6 \csc ^2(c+d x)-12 \csc ^3(c+d x)-3 \csc ^4(c+d x)-60 \log (\sin (c+d x))+12 \sin (c+d x)+18 \sin ^2(c+d x)+4 \sin ^3(c+d x)\right )}{12 d} \]

[In]

Integrate[Cot[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(60*Csc[c + d*x] - 6*Csc[c + d*x]^2 - 12*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 - 60*Log[Sin[c + d*x]] + 12*Si
n[c + d*x] + 18*Sin[c + d*x]^2 + 4*Sin[c + d*x]^3))/(12*d)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.50

method	result	size
parallelrisch	\(\frac {a^{3} \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (960 \left (3+\cos \left (4 d x +4 c \right )-4 \cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )-960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (4 d x +4 c \right )+152 \sin \left (5 d x +5 c \right )-8 \sin \left (7 d x +7 c \right )-2880 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1428 \cos \left (2 d x +2 c \right )+615 \cos \left (4 d x +4 c \right )-72 \cos \left (6 d x +6 c \right )+5464 \sin \left (d x +c \right )-2568 \sin \left (3 d x +3 c \right )+501\right )}{24576 d}\)	\(197\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(210\)
default	\(\frac {a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(210\)
risch	\(5 i a^{3} x +\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {5 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {5 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {10 i a^{3} c}{d}+\frac {2 i a^{3} \left (-i {\mathrm e}^{6 i \left (d x +c \right )}+5 \,{\mathrm e}^{7 i \left (d x +c \right )}+4 i {\mathrm e}^{4 i \left (d x +c \right )}-11 \,{\mathrm e}^{5 i \left (d x +c \right )}-i {\mathrm e}^{2 i \left (d x +c \right )}+11 \,{\mathrm e}^{3 i \left (d x +c \right )}-5 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(248\)
norman	\(\frac {-\frac {a^{3}}{64 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {15 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {7 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {81 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {115 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {81 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {7 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {15 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {113 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {113 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a^{3} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(304\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/24576*a^3*csc(1/2*d*x+1/2*c)^4*sec(1/2*d*x+1/2*c)^4*(960*(3+cos(4*d*x+4*c)-4*cos(2*d*x+2*c))*ln(sec(1/2*d*x+
1/2*c)^2)+3840*ln(tan(1/2*d*x+1/2*c))*cos(2*d*x+2*c)-960*ln(tan(1/2*d*x+1/2*c))*cos(4*d*x+4*c)+152*sin(5*d*x+5
*c)-8*sin(7*d*x+7*c)-2880*ln(tan(1/2*d*x+1/2*c))-1428*cos(2*d*x+2*c)+615*cos(4*d*x+4*c)-72*cos(6*d*x+6*c)+5464
*sin(d*x+c)-2568*sin(3*d*x+3*c)+501)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.21 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {18 \, a^{3} \cos \left (d x + c\right )^{6} - 45 \, a^{3} \cos \left (d x + c\right )^{4} + 30 \, a^{3} \cos \left (d x + c\right )^{2} + 60 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 6 \, a^{3} \cos \left (d x + c\right )^{4} + 24 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/12*(18*a^3*cos(d*x + c)^6 - 45*a^3*cos(d*x + c)^4 + 30*a^3*cos(d*x + c)^2 + 60*(a^3*cos(d*x + c)^4 - 2*a^3*
cos(d*x + c)^2 + a^3)*log(1/2*sin(d*x + c)) + 4*(a^3*cos(d*x + c)^6 - 6*a^3*cos(d*x + c)^4 + 24*a^3*cos(d*x +
c)^2 - 16*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**5*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 18 \, a^{3} \sin \left (d x + c\right )^{2} - 60 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 12 \, a^{3} \sin \left (d x + c\right ) + \frac {3 \, {\left (20 \, a^{3} \sin \left (d x + c\right )^{3} - 2 \, a^{3} \sin \left (d x + c\right )^{2} - 4 \, a^{3} \sin \left (d x + c\right ) - a^{3}\right )}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/12*(4*a^3*sin(d*x + c)^3 + 18*a^3*sin(d*x + c)^2 - 60*a^3*log(sin(d*x + c)) + 12*a^3*sin(d*x + c) + 3*(20*a^
3*sin(d*x + c)^3 - 2*a^3*sin(d*x + c)^2 - 4*a^3*sin(d*x + c) - a^3)/sin(d*x + c)^4)/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 18 \, a^{3} \sin \left (d x + c\right )^{2} - 60 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, a^{3} \sin \left (d x + c\right ) + \frac {125 \, a^{3} \sin \left (d x + c\right )^{4} + 60 \, a^{3} \sin \left (d x + c\right )^{3} - 6 \, a^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) - 3 \, a^{3}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/12*(4*a^3*sin(d*x + c)^3 + 18*a^3*sin(d*x + c)^2 - 60*a^3*log(abs(sin(d*x + c))) + 12*a^3*sin(d*x + c) + (12
5*a^3*sin(d*x + c)^4 + 60*a^3*sin(d*x + c)^3 - 6*a^3*sin(d*x + c)^2 - 12*a^3*sin(d*x + c) - 3*a^3)/sin(d*x + c
)^4)/d

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.46 \[ \int \cot ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {66\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+93\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {620\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {347\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{4}+128\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {39\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {a^3}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {5\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {17\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {5\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]

[In]

int((cos(c + d*x)^5*(a + a*sin(c + d*x))^3)/sin(c + d*x)^5,x)

[Out]

(28*a^3*tan(c/2 + (d*x)/2)^3 - (15*a^3*tan(c/2 + (d*x)/2)^2)/4 - (39*a^3*tan(c/2 + (d*x)/2)^4)/4 + 128*a^3*tan
(c/2 + (d*x)/2)^5 + (347*a^3*tan(c/2 + (d*x)/2)^6)/4 + (620*a^3*tan(c/2 + (d*x)/2)^7)/3 + 93*a^3*tan(c/2 + (d*
x)/2)^8 + 66*a^3*tan(c/2 + (d*x)/2)^9 - a^3/4 - 2*a^3*tan(c/2 + (d*x)/2))/(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)) - (a^3*tan(c/2 + (d*x)/2)^3)/(8*d) -
(a^3*tan(c/2 + (d*x)/2)^4)/(64*d) - (3*a^3*tan(c/2 + (d*x)/2)^2)/(16*d) - (5*a^3*log(tan(c/2 + (d*x)/2)))/d +
(17*a^3*tan(c/2 + (d*x)/2))/(8*d) + (5*a^3*log(tan(c/2 + (d*x)/2)^2 + 1))/d