\(\int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [563]

Optimal result

Integrand size = 21, antiderivative size = 120 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {12 \csc (c+d x)}{a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {4 \csc ^3(c+d x)}{3 a^4 d}-\frac {\csc ^4(c+d x)}{4 a^4 d}+\frac {16 \log (\sin (c+d x))}{a^4 d}-\frac {16 \log (1+\sin (c+d x))}{a^4 d}+\frac {4}{d \left (a^4+a^4 \sin (c+d x)\right )} \]

[Out]

12*csc(d*x+c)/a^4/d-4*csc(d*x+c)^2/a^4/d+4/3*csc(d*x+c)^3/a^4/d-1/4*csc(d*x+c)^4/a^4/d+16*ln(sin(d*x+c))/a^4/d
-16*ln(1+sin(d*x+c))/a^4/d+4/d/(a^4+a^4*sin(d*x+c))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 120, 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} \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {\csc ^4(c+d x)}{4 a^4 d}+\frac {4 \csc ^3(c+d x)}{3 a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {12 \csc (c+d x)}{a^4 d}+\frac {16 \log (\sin (c+d x))}{a^4 d}-\frac {16 \log (\sin (c+d x)+1)}{a^4 d} \]

[In]

Int[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]

[Out]

(12*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (4*Csc[c + d*x]^3)/(3*a^4*d) - Csc[c + d*x]^4/(4*a^4*
d) + (16*Log[Sin[c + d*x]])/(a^4*d) - (16*Log[1 + Sin[c + d*x]])/(a^4*d) + 4/(d*(a^4 + a^4*Sin[c + d*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 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}{x^5 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{x^5}-\frac {4}{a x^4}+\frac {8}{a^2 x^3}-\frac {12}{a^3 x^2}+\frac {16}{a^4 x}-\frac {4}{a^3 (a+x)^2}-\frac {16}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {12 \csc (c+d x)}{a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {4 \csc ^3(c+d x)}{3 a^4 d}-\frac {\csc ^4(c+d x)}{4 a^4 d}+\frac {16 \log (\sin (c+d x))}{a^4 d}-\frac {16 \log (1+\sin (c+d x))}{a^4 d}+\frac {4}{d \left (a^4+a^4 \sin (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {144 \csc (c+d x)-48 \csc ^2(c+d x)+16 \csc ^3(c+d x)-3 \csc ^4(c+d x)+192 \log (\sin (c+d x))-192 \log (1+\sin (c+d x))+\frac {48}{1+\sin (c+d x)}}{12 a^4 d} \]

[In]

Integrate[Cot[c + d*x]^5/(a + a*Sin[c + d*x])^4,x]

[Out]

(144*Csc[c + d*x] - 48*Csc[c + d*x]^2 + 16*Csc[c + d*x]^3 - 3*Csc[c + d*x]^4 + 192*Log[Sin[c + d*x]] - 192*Log
[1 + Sin[c + d*x]] + 48/(1 + Sin[c + d*x]))/(12*a^4*d)

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(\frac {-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {4}{3 \sin \left (d x +c \right )^{3}}-\frac {4}{\sin \left (d x +c \right )^{2}}+\frac {12}{\sin \left (d x +c \right )}+16 \ln \left (\sin \left (d x +c \right )\right )+\frac {4}{1+\sin \left (d x +c \right )}-16 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{4}}\)	\(81\)
default	\(\frac {-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {4}{3 \sin \left (d x +c \right )^{3}}-\frac {4}{\sin \left (d x +c \right )^{2}}+\frac {12}{\sin \left (d x +c \right )}+16 \ln \left (\sin \left (d x +c \right )\right )+\frac {4}{1+\sin \left (d x +c \right )}-16 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{4}}\)	\(81\)
risch	\(\frac {4 i \left (24 i {\mathrm e}^{8 i \left (d x +c \right )}+24 \,{\mathrm e}^{9 i \left (d x +c \right )}-85 i {\mathrm e}^{6 i \left (d x +c \right )}-80 \,{\mathrm e}^{7 i \left (d x +c \right )}+85 i {\mathrm e}^{4 i \left (d x +c \right )}+106 \,{\mathrm e}^{5 i \left (d x +c \right )}-24 i {\mathrm e}^{2 i \left (d x +c \right )}-80 \,{\mathrm e}^{3 i \left (d x +c \right )}+24 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} d \,a^{4}}-\frac {32 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{4}}+\frac {16 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{4}}\)	\(183\)
parallelrisch	\(\frac {-6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3072 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+26 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-143 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+26 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+872 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-143 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+872 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-3624 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}\)	\(188\)
norman	\(\frac {-\frac {96 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {96 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1}{64 a d}+\frac {11 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}-\frac {43 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {129 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {129 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {43 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {6397 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {6397 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {35569 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {35569 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}}-\frac {32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}\)	\(322\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d/a^4*(-1/4/sin(d*x+c)^4+4/3/sin(d*x+c)^3-4/sin(d*x+c)^2+12/sin(d*x+c)+16*ln(sin(d*x+c))+4/(1+sin(d*x+c))-16
*ln(1+sin(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (116) = 232\).

Time = 0.28 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.96 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {192 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 192 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 192 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (96 \, \cos \left (d x + c\right )^{2} - 109\right )} \sin \left (d x + c\right ) + 157}{12 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/12*(192*cos(d*x + c)^4 - 352*cos(d*x + c)^2 + 192*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*c
os(d*x + c)^2 + 1)*sin(d*x + c) + 1)*log(1/2*sin(d*x + c)) - 192*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + (cos(d*x
+ c)^4 - 2*cos(d*x + c)^2 + 1)*sin(d*x + c) + 1)*log(sin(d*x + c) + 1) - (96*cos(d*x + c)^2 - 109)*sin(d*x +
c) + 157)/(a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^2 + a^4*d + (a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c
)^2 + a^4*d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**5/(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {192 \, \sin \left (d x + c\right )^{4} + 96 \, \sin \left (d x + c\right )^{3} - 32 \, \sin \left (d x + c\right )^{2} + 13 \, \sin \left (d x + c\right ) - 3}{a^{4} \sin \left (d x + c\right )^{5} + a^{4} \sin \left (d x + c\right )^{4}} - \frac {192 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {192 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/12*((192*sin(d*x + c)^4 + 96*sin(d*x + c)^3 - 32*sin(d*x + c)^2 + 13*sin(d*x + c) - 3)/(a^4*sin(d*x + c)^5 +
a^4*sin(d*x + c)^4) - 192*log(sin(d*x + c) + 1)/a^4 + 192*log(sin(d*x + c))/a^4)/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.82 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {6144 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3072 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {1536 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{2}} + \frac {6400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1248 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 204 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 32 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {3 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 32 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 204 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1248 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{16}}}{192 \, d} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/192*(6144*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 3072*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 - 1536*(6*tan(1/
2*d*x + 1/2*c)^2 + 11*tan(1/2*d*x + 1/2*c) + 6)/(a^4*(tan(1/2*d*x + 1/2*c) + 1)^2) + (6400*tan(1/2*d*x + 1/2*c
)^4 - 1248*tan(1/2*d*x + 1/2*c)^3 + 204*tan(1/2*d*x + 1/2*c)^2 - 32*tan(1/2*d*x + 1/2*c) + 3)/(a^4*tan(1/2*d*x
+ 1/2*c)^4) + (3*a^12*tan(1/2*d*x + 1/2*c)^4 - 32*a^12*tan(1/2*d*x + 1/2*c)^3 + 204*a^12*tan(1/2*d*x + 1/2*c)
^2 - 1248*a^12*tan(1/2*d*x + 1/2*c))/a^16)/d

Mupad [B] (verification not implemented)

Time = 10.87 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.94 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6\,a^4\,d}-\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^4\,d}+\frac {16\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {32\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}+\frac {-24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+191\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {218\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {143\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{12}+\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6}-\frac {1}{4}}{d\,\left (16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+32\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^4\,d} \]

[In]

int(cos(c + d*x)^5/(sin(c + d*x)^5*(a + a*sin(c + d*x))^4),x)

[Out]

tan(c/2 + (d*x)/2)^3/(6*a^4*d) - (17*tan(c/2 + (d*x)/2)^2)/(16*a^4*d) - tan(c/2 + (d*x)/2)^4/(64*a^4*d) + (16*
log(tan(c/2 + (d*x)/2)))/(a^4*d) - (32*log(tan(c/2 + (d*x)/2) + 1))/(a^4*d) + ((13*tan(c/2 + (d*x)/2))/6 - (14
3*tan(c/2 + (d*x)/2)^2)/12 + (218*tan(c/2 + (d*x)/2)^3)/3 + 191*tan(c/2 + (d*x)/2)^4 - 24*tan(c/2 + (d*x)/2)^5
- 1/4)/(d*(16*a^4*tan(c/2 + (d*x)/2)^4 + 32*a^4*tan(c/2 + (d*x)/2)^5 + 16*a^4*tan(c/2 + (d*x)/2)^6)) + (13*ta
n(c/2 + (d*x)/2))/(2*a^4*d)