\(\int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [687]

Optimal result

Integrand size = 29, antiderivative size = 97 \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {2 \log (\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90} \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin (c+d x)}{a d}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}+\frac {2 \csc (c+d x)}{a d}+\frac {2 \log (\sin (c+d x))}{a d} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {12 \csc (c+d x)+3 \csc ^2(c+d x)-2 \csc ^3(c+d x)+12 \log (\sin (c+d x))+6 \sin (c+d x)-3 \sin ^2(c+d x)}{6 a d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(-\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )+\frac {1}{3 \sin \left (d x +c \right )^{3}}-2 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )}}{d a}\)	\(67\)
default	\(-\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )+\frac {1}{3 \sin \left (d x +c \right )^{3}}-2 \ln \left (\sin \left (d x +c \right )\right )-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )}}{d a}\)	\(67\)
parallelrisch	\(\frac {-768 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+768 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-16 \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\left (-3 \cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )+6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+24 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-3 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (384 \cos \left (d x +c \right )-1152\right ) \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}\)	\(187\)
risch	\(-\frac {2 i x}{a}+\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d a}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d a}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}-\frac {4 i c}{a d}+\frac {2 i \left (6 \,{\mathrm e}^{5 i \left (d x +c \right )}-8 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i {\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}-3 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\)	\(189\)
norman	\(\frac {-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {7 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {7 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {37 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {37 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {103 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}+\frac {103 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\)	\(298\)

[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-sin(d*x+c)+1/3/sin(d*x+c)^3-2*ln(sin(d*x+c))-1/2/sin(d*x+c)^2-2/sin(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {12 \, \cos \left (d x + c\right )^{4} - 24 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 48 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (2 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 32}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

-1/12*(12*cos(d*x + c)^4 - 24*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c))*sin(d*x + c) - 48*cos(d*x + c)^2 - 3*
(2*cos(d*x + c)^4 - 3*cos(d*x + c)^2 - 1)*sin(d*x + c) + 32)/((a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))

Sympy [F(-1)]

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

[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 = 73, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {3 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} - \frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {12 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {3 \, {\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac {22 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \]

[In]

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

[Out]

1/6*(12*log(abs(sin(d*x + c)))/a - 3*(a*sin(d*x + c)^2 - 2*a*sin(d*x + c))/a^2 - (22*sin(d*x + c)^3 - 12*sin(d
*x + c)^2 - 3*sin(d*x + c) + 2)/(a*sin(d*x + c)^3))/d

Mupad [B] (verification not implemented)

Time = 10.91 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.28 \[ \int \frac {\cos ^3(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}+\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}+\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]

[In]

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

[Out]

tan(c/2 + (d*x)/2)^2/(8*a*d) - tan(c/2 + (d*x)/2)^3/(24*a*d) + (2*log(tan(c/2 + (d*x)/2)))/(a*d) + (7*tan(c/2
+ (d*x)/2))/(8*a*d) + (tan(c/2 + (d*x)/2) + (19*tan(c/2 + (d*x)/2)^2)/3 + 2*tan(c/2 + (d*x)/2)^3 + (89*tan(c/2
+ (d*x)/2)^4)/3 - 15*tan(c/2 + (d*x)/2)^5 + 23*tan(c/2 + (d*x)/2)^6 - 1/3)/(d*(8*a*tan(c/2 + (d*x)/2)^3 + 16*
a*tan(c/2 + (d*x)/2)^5 + 8*a*tan(c/2 + (d*x)/2)^7)) - (2*log(tan(c/2 + (d*x)/2)^2 + 1))/(a*d)