\(\int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx\) [665]

Optimal result

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

[Out]

3*a*csc(d*x+c)/d-1/2*a*csc(d*x+c)^2/d-1/3*a*csc(d*x+c)^3/d-3*a*ln(sin(d*x+c))/d+3*a*sin(d*x+c)/d+3/2*a*sin(d*x
+c)^2/d-1/3*a*sin(d*x+c)^3/d-1/4*a*sin(d*x+c)^4/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} \[ \int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^3(c+d x)}{3 d}+\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d}+\frac {3 a \csc (c+d x)}{d}-\frac {3 a \log (\sin (c+d x))}{d} \]

[In]

Int[Cos[c + d*x]^3*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

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

Mathematica [A] (verified)

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

[In]

Integrate[Cos[c + d*x]^3*Cot[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3}\right )}{d}\)	\(143\)
default	\(\frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3}\right )}{d}\)	\(143\)
parallelrisch	\(-\frac {a \left (1728 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )-1728 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )-576 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (3 d x +3 c \right )+576 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (3 d x +3 c \right )-663 \sin \left (d x +c \right )-8 \cos \left (6 d x +6 c \right )-240 \cos \left (4 d x +4 c \right )+2184 \cos \left (2 d x +2 c \right )-3 \sin \left (7 d x +7 c \right )-51 \sin \left (5 d x +5 c \right )+441 \sin \left (3 d x +3 c \right )-1680\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6144 d}\)	\(187\)
risch	\(3 i a x -\frac {a \,{\mathrm e}^{4 i \left (d x +c \right )}}{64 d}-\frac {i a \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {5 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 d}-\frac {11 i a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {11 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {5 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}+\frac {i a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {a \,{\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {6 i a c}{d}+\frac {2 i a \left (9 \,{\mathrm e}^{5 i \left (d x +c \right )}-14 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(235\)
norman	\(\frac {-\frac {a}{24 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {29 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {101 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {231 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {231 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {101 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {29 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {10 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {57 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {57 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(274\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-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*
(-1/3/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)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.18 \[ \int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {32 \, a \cos \left (d x + c\right )^{6} + 192 \, a \cos \left (d x + c\right )^{4} - 768 \, a \cos \left (d x + c\right )^{2} + 288 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \, {\left (8 \, a \cos \left (d x + c\right )^{6} + 24 \, a \cos \left (d x + c\right )^{4} - 51 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 512 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/96*(32*a*cos(d*x + c)^6 + 192*a*cos(d*x + c)^4 - 768*a*cos(d*x + c)^2 + 288*(a*cos(d*x + c)^2 - a)*log(1/2*
sin(d*x + c))*sin(d*x + c) + 3*(8*a*cos(d*x + c)^6 + 24*a*cos(d*x + c)^4 - 51*a*cos(d*x + c)^2 + 3*a)*sin(d*x
+ c) + 512*a)/((d*cos(d*x + c)^2 - d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

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

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 36 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 36 \, a \sin \left (d x + c\right ) - \frac {2 \, {\left (33 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} - 3 \, a \sin \left (d x + c\right ) - 2 \, a\right )}}{\sin \left (d x + c\right )^{3}}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/12*(3*a*sin(d*x + c)^4 + 4*a*sin(d*x + c)^3 - 18*a*sin(d*x + c)^2 + 36*a*log(abs(sin(d*x + c))) - 36*a*sin(
d*x + c) - 2*(33*a*sin(d*x + c)^3 + 18*a*sin(d*x + c)^2 - 3*a*sin(d*x + c) - 2*a)/sin(d*x + c)^3)/d

Mupad [B] (verification not implemented)

Time = 10.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.54 \[ \int \cos ^3(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {59\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+47\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {499\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+60\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {562\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+42\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+90\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^4,x)

[Out]

(11*a*tan(c/2 + (d*x)/2))/(8*d) + (3*a*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (a*tan(c/2 + (d*x)/2)^2)/(8*d) - (a*
tan(c/2 + (d*x)/2)^3)/(24*d) - (3*a*log(tan(c/2 + (d*x)/2)))/d + ((29*a*tan(c/2 + (d*x)/2)^2)/3 - a*tan(c/2 +
(d*x)/2) - a/3 - 4*a*tan(c/2 + (d*x)/2)^3 + 90*a*tan(c/2 + (d*x)/2)^4 + 42*a*tan(c/2 + (d*x)/2)^5 + (562*a*tan
(c/2 + (d*x)/2)^6)/3 + 60*a*tan(c/2 + (d*x)/2)^7 + (499*a*tan(c/2 + (d*x)/2)^8)/3 + 47*a*tan(c/2 + (d*x)/2)^9
+ 59*a*tan(c/2 + (d*x)/2)^10)/(d*(8*tan(c/2 + (d*x)/2)^3 + 32*tan(c/2 + (d*x)/2)^5 + 48*tan(c/2 + (d*x)/2)^7 +
32*tan(c/2 + (d*x)/2)^9 + 8*tan(c/2 + (d*x)/2)^11))