\(\int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx\) [688]

Optimal result

Integrand size = 29, antiderivative size = 94 \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{a d}+\frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc ^4(c+d x)}{4 a d}+\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 94, 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} \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin (c+d x)}{a d}-\frac {\csc ^4(c+d x)}{4 a d}+\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{a d}-\frac {2 \csc (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]

[In]

Int[(Cos[c + d*x]^2*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

(-2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d) + Log[Sin[c +
d*x]]/(a*d) - Sin[c + d*x]/(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^5 (a-x)^3 (a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (-1+\frac {a^5}{x^5}-\frac {a^4}{x^4}-\frac {2 a^3}{x^3}+\frac {2 a^2}{x^2}+\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {2 \csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{a d}+\frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc ^4(c+d x)}{4 a d}+\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {24 \csc (c+d x)-12 \csc ^2(c+d x)-4 \csc ^3(c+d x)+3 \csc ^4(c+d x)-12 \log (\sin (c+d x))+12 \sin (c+d x)}{12 a d} \]

[In]

Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]

[Out]

-1/12*(24*Csc[c + d*x] - 12*Csc[c + d*x]^2 - 4*Csc[c + d*x]^3 + 3*Csc[c + d*x]^4 - 12*Log[Sin[c + d*x]] + 12*S
in[c + d*x])/(a*d)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {-\sin \left (d x +c \right )+\frac {1}{3 \sin \left (d x +c \right )^{3}}+\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )}}{d a}\)	\(62\)
default	\(\frac {-\sin \left (d x +c \right )+\frac {1}{3 \sin \left (d x +c \right )^{3}}+\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{2}}-\frac {2}{\sin \left (d x +c \right )}}{d a}\)	\(62\)
risch	\(-\frac {i x}{a}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d a}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d a}-\frac {2 i c}{a d}-\frac {4 i \left (-3 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}+3 i {\mathrm e}^{4 i \left (d x +c \right )}-7 \,{\mathrm e}^{5 i \left (d x +c \right )}-3 i {\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\)	\(177\)
parallelrisch	\(\frac {-384 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+96 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-324\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 \cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )-3 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+36 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+36 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-50 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+456 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-228 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}\)	\(192\)
norman	\(\frac {-\frac {1}{64 a d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}+\frac {19 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {61 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {61 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {91 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {91 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d a}-\frac {107 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {107 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\)	\(297\)

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {12 \, \cos \left (d x + c\right )^{2} - 12 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 9}{12 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]

[In]

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

[Out]

-1/12*(12*cos(d*x + c)^2 - 12*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*sin(d*x + c)) + 4*(3*cos(d*x + c
)^4 - 12*cos(d*x + c)^2 + 8)*sin(d*x + c) - 9)/(a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)

Sympy [F(-1)]

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

[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 = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {12 \, \sin \left (d x + c\right )}{a} - \frac {24 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{a \sin \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {12 \, \sin \left (d x + c\right )}{a} - \frac {25 \, \sin \left (d x + c\right )^{4} + 24 \, \sin \left (d x + c\right )^{3} - 12 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) + 3}{a \sin \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

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

[Out]

1/12*(12*log(abs(sin(d*x + c)))/a - 12*sin(d*x + c)/a - (25*sin(d*x + c)^4 + 24*sin(d*x + c)^3 - 12*sin(d*x +
c)^2 - 4*sin(d*x + c) + 3)/(a*sin(d*x + c)^4))/d

Mupad [B] (verification not implemented)

Time = 11.18 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.28 \[ \int \frac {\cos ^2(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {-46\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {1}{4}}{d\,\left (16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]

[In]

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

[Out]

(3*tan(c/2 + (d*x)/2)^2)/(16*a*d) + tan(c/2 + (d*x)/2)^3/(24*a*d) - tan(c/2 + (d*x)/2)^4/(64*a*d) + log(tan(c/
2 + (d*x)/2))/(a*d) + ((2*tan(c/2 + (d*x)/2))/3 + (11*tan(c/2 + (d*x)/2)^2)/4 - (40*tan(c/2 + (d*x)/2)^3)/3 +
3*tan(c/2 + (d*x)/2)^4 - 46*tan(c/2 + (d*x)/2)^5 - 1/4)/(d*(16*a*tan(c/2 + (d*x)/2)^4 + 16*a*tan(c/2 + (d*x)/2
)^6)) - (7*tan(c/2 + (d*x)/2))/(8*a*d) - log(tan(c/2 + (d*x)/2)^2 + 1)/(a*d)