\(\int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [637]

Optimal result

Integrand size = 27, antiderivative size = 73 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{a^2}-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\cos (c+d x)}{a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d} \]

[Out]

-x/a^2-arctanh(cos(d*x+c))/a^2/d+cos(d*x+c)/a^2/d-1/3*cos(d*x+c)^3/a^2/d-cos(d*x+c)*sin(d*x+c)/a^2/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2954, 2952, 2715, 8, 2672, 327, 212, 2645, 30} \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos (c+d x)}{a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac {x}{a^2} \]

[In]

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

[Out]

-(x/a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a^2*d) - Cos[c + d*x]^3/(3*a^2*d) - (Cos[c + d*x]*Sin
[c + d*x])/(a^2*d)

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 2715

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

Rule 2952

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

Rule 2954

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

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

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {-9 \cos (c+d x)+\cos (3 (c+d x))+6 \left (2 \left (c+d x+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sin (2 (c+d x))\right )}{12 a^2 d} \]

[In]

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

[Out]

-1/12*(-9*Cos[c + d*x] + Cos[3*(c + d*x)] + 6*(2*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + S
in[2*(c + d*x)]))/(a^2*d)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {-12 d x -6 \sin \left (2 d x +2 c \right )-\cos \left (3 d x +3 c \right )+9 \cos \left (d x +c \right )+12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8}{12 d \,a^{2}}\)	\(57\)
derivativedivides	\(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 \left (-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {1}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\)	\(87\)
default	\(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 \left (-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {1}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\)	\(87\)
risch	\(-\frac {x}{a^{2}}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}-\frac {\cos \left (3 d x +3 c \right )}{12 d \,a^{2}}-\frac {\sin \left (2 d x +2 c \right )}{2 d \,a^{2}}\)	\(115\)
norman	\(\frac {-\frac {35 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {8 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {35 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {8 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4}{3 a d}-\frac {3 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x}{a}+\frac {6 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {76 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {18 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {34 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {24 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {68 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {28 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {46 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {10 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\)	\(511\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/12*(-12*d*x-6*sin(2*d*x+2*c)-cos(3*d*x+3*c)+9*cos(d*x+c)+12*ln(tan(1/2*d*x+1/2*c))+8)/d/a^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \, \cos \left (d x + c\right )^{3} + 6 \, d x + 6 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right ) + 3 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{6 \, a^{2} d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\cos ^{6}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)/(a+a*sin(d*x+c))**2,x)

[Out]

Integral(cos(c + d*x)**6*csc(c + d*x)/(sin(c + d*x)**2 + 2*sin(c + d*x) + 1), x)/a**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (71) = 142\).

Time = 0.38 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.58 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {6 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{3 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*(2*(3*sin(d*x + c)/(cos(d*x + c) + 1) - 6*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 3*sin(d*x + c)^5/(cos(d*x
+ c) + 1)^5 - 2)/(a^2 + 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + 6*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - 3*log(sin(d*x +
c)/(cos(d*x + c) + 1))/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} - \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/3*(3*(d*x + c)/a^2 - 3*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - 2*(3*tan(1/2*d*x + 1/2*c)^5 + 6*tan(1/2*d*x + 1
/2*c)^2 - 3*tan(1/2*d*x + 1/2*c) + 2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^2))/d

Mupad [B] (verification not implemented)

Time = 10.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.29 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2\,\mathrm {atan}\left (\frac {4}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4}-\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4}\right )}{a^2\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4}{3}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)*(a + a*sin(c + d*x))^2),x)

[Out]

(2*atan(4/(4*tan(c/2 + (d*x)/2) + 4) - (4*tan(c/2 + (d*x)/2))/(4*tan(c/2 + (d*x)/2) + 4)))/(a^2*d) + log(tan(c
/2 + (d*x)/2))/(a^2*d) + (4*tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2) + 2*tan(c/2 + (d*x)/2)^5 + 4/3)/(d*(3*
a^2*tan(c/2 + (d*x)/2)^2 + 3*a^2*tan(c/2 + (d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2))