\(\int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [651]

Optimal result

Integrand size = 27, antiderivative size = 93 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {15 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \]

[Out]

-15/8*arctanh(cos(d*x+c))/a^3/d+4*cot(d*x+c)/a^3/d+cot(d*x+c)^3/a^3/d-15/8*cot(d*x+c)*csc(d*x+c)/a^3/d-1/4*cot
(d*x+c)*csc(d*x+c)^3/a^3/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2948, 2836, 3852, 8, 3853, 3855} \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {15 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]

[In]

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

[Out]

(-15*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + (4*Cot[c + d*x])/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (15*Cot[c + d*x]*C
sc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2836

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 2948

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[a^(2*m), Int[(d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f,
n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^5(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \csc ^2(c+d x)+3 a^3 \csc ^3(c+d x)-3 a^3 \csc ^4(c+d x)+a^3 \csc ^5(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \csc ^2(c+d x) \, dx}{a^3}+\frac {\int \csc ^5(c+d x) \, dx}{a^3}+\frac {3 \int \csc ^3(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^4(c+d x) \, dx}{a^3} \\ & = -\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^3}+\frac {3 \int \csc (c+d x) \, dx}{2 a^3}+\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}+\frac {3 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {3 \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {3 \int \csc (c+d x) \, dx}{8 a^3} \\ & = -\frac {15 \text {arctanh}(\cos (c+d x))}{8 a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.34 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^4(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (46 \cos (c+d x)+120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^4(c+d x)+6 \cos (3 (c+d x)) (-5+8 \sin (c+d x))-56 \sin (2 (c+d x))\right )}{64 a^3 d (1+\sin (c+d x))^3} \]

[In]

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

[Out]

-1/64*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(46*Cos[c + d*x] + 120*(Log[Cos[(c + d*x)/2]] -
Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + 6*Cos[3*(c + d*x)]*(-5 + 8*Sin[c + d*x]) - 56*Sin[2*(c + d*x)]))/(a^3*
d*(1 + Sin[c + d*x])^3)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-104 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+104 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{3} d}\)	\(120\)
derivativedivides	\(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+30 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {26}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{16 d \,a^{3}}\)	\(124\)
default	\(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+30 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {26}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{16 d \,a^{3}}\)	\(124\)
risch	\(\frac {15 \,{\mathrm e}^{7 i \left (d x +c \right )}-23 \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i {\mathrm e}^{6 i \left (d x +c \right )}-23 \,{\mathrm e}^{3 i \left (d x +c \right )}-72 i {\mathrm e}^{4 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}+88 i {\mathrm e}^{2 i \left (d x +c \right )}-24 i}{4 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{3}}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{3}}\)	\(146\)
norman	\(\frac {-\frac {1}{64 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {17 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {17 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {3 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {3 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}+\frac {60 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2175 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {273 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {87 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {291 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {1017 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {1839 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}\)	\(336\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/64*(tan(1/2*d*x+1/2*c)^4-cot(1/2*d*x+1/2*c)^4-8*tan(1/2*d*x+1/2*c)^3+8*cot(1/2*d*x+1/2*c)^3+32*tan(1/2*d*x+1
/2*c)^2-32*cot(1/2*d*x+1/2*c)^2-104*tan(1/2*d*x+1/2*c)+120*ln(tan(1/2*d*x+1/2*c))+104*cot(1/2*d*x+1/2*c))/a^3/
d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.60 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {30 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 34 \, \cos \left (d x + c\right )}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/16*(30*cos(d*x + c)^3 - 15*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) + 15*(cos(d*x
+ c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) - 16*(3*cos(d*x + c)^3 - 4*cos(d*x + c))*sin(d*x
+ c) - 34*cos(d*x + c))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**5/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (87) = 174\).

Time = 0.23 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.10 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {104 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {32 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{3}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {32 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {104 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{3} \sin \left (d x + c\right )^{4}}}{64 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/64*((104*sin(d*x + c)/(cos(d*x + c) + 1) - 32*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 8*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 - sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/a^3 - 120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - (8*
sin(d*x + c)/(cos(d*x + c) + 1) - 32*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 104*sin(d*x + c)^3/(cos(d*x + c) +
1)^3 - 1)*(cos(d*x + c) + 1)^4/(a^3*sin(d*x + c)^4))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.68 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 104 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 104 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{64 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/64*(120*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (250*tan(1/2*d*x + 1/2*c)^4 - 104*tan(1/2*d*x + 1/2*c)^3 + 32*t
an(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 1)/(a^3*tan(1/2*d*x + 1/2*c)^4) + (a^9*tan(1/2*d*x + 1/2*c)^4
- 8*a^9*tan(1/2*d*x + 1/2*c)^3 + 32*a^9*tan(1/2*d*x + 1/2*c)^2 - 104*a^9*tan(1/2*d*x + 1/2*c))/a^12)/d

Mupad [B] (verification not implemented)

Time = 10.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.62 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^3\,d}+\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^3\,d}-\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{4}\right )}{16\,a^3\,d} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^5*(a + a*sin(c + d*x))^3),x)

[Out]

tan(c/2 + (d*x)/2)^2/(2*a^3*d) - tan(c/2 + (d*x)/2)^3/(8*a^3*d) + tan(c/2 + (d*x)/2)^4/(64*a^3*d) + (15*log(ta
n(c/2 + (d*x)/2)))/(8*a^3*d) - (13*tan(c/2 + (d*x)/2))/(8*a^3*d) + (cot(c/2 + (d*x)/2)^4*(2*tan(c/2 + (d*x)/2)
- 8*tan(c/2 + (d*x)/2)^2 + 26*tan(c/2 + (d*x)/2)^3 - 1/4))/(16*a^3*d)