\(\int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx\) [592]

Optimal result

Integrand size = 27, antiderivative size = 161 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 x}{8}-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{3 d} \]

[Out]

5/8*a^2*x-a^2*arctanh(cos(d*x+c))/d+a^2*cos(d*x+c)/d+1/3*a^2*cos(d*x+c)^3/d+1/5*a^2*cos(d*x+c)^5/d-1/7*a^2*cos
(d*x+c)^7/d+5/8*a^2*cos(d*x+c)*sin(d*x+c)/d+5/12*a^2*cos(d*x+c)^3*sin(d*x+c)/d+1/3*a^2*cos(d*x+c)^5*sin(d*x+c)
/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2952, 2715, 8, 2672, 308, 212, 2645, 30} \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac {5 a^2 \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac {5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {5 a^2 x}{8} \]

[In]

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

[Out]

(5*a^2*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c +
d*x]^5)/(5*d) - (a^2*Cos[c + d*x]^7)/(7*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a^2*Cos[c + d*x]^3*
Sin[c + d*x])/(12*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(3*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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (2 a^2 \cos ^6(c+d x)+a^2 \cos ^5(c+d x) \cot (c+d x)+a^2 \cos ^6(c+d x) \sin (c+d x)\right ) \, dx \\ & = a^2 \int \cos ^5(c+d x) \cot (c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin (c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \, dx \\ & = \frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac {1}{3} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \frac {x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac {1}{4} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (-1-x^2-x^4+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{3 d}+\frac {1}{8} \left (5 a^2\right ) \int 1 \, dx-\frac {a^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {5 a^2 x}{8}-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {5 a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.54 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.70 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (4200 c+4200 d x+8715 \cos (c+d x)+665 \cos (3 (c+d x))-21 \cos (5 (c+d x))-15 \cos (7 (c+d x))-6720 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6720 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3150 \sin (2 (c+d x))+630 \sin (4 (c+d x))+70 \sin (6 (c+d x))\right )}{6720 d} \]

[In]

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

[Out]

(a^2*(4200*c + 4200*d*x + 8715*Cos[c + d*x] + 665*Cos[3*(c + d*x)] - 21*Cos[5*(c + d*x)] - 15*Cos[7*(c + d*x)]
- 6720*Log[Cos[(c + d*x)/2]] + 6720*Log[Sin[(c + d*x)/2]] + 3150*Sin[2*(c + d*x)] + 630*Sin[4*(c + d*x)] + 70
*Sin[6*(c + d*x)]))/(6720*d)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.63

method	result	size
parallelrisch	\(\frac {a^{2} \left (4200 d x +8715 \cos \left (d x +c \right )+6720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \cos \left (7 d x +7 c \right )+3150 \sin \left (2 d x +2 c \right )+70 \sin \left (6 d x +6 c \right )+630 \sin \left (4 d x +4 c \right )-21 \cos \left (5 d x +5 c \right )+665 \cos \left (3 d x +3 c \right )+9344\right )}{6720 d}\)	\(101\)
derivativedivides	\(\frac {-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+2 a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(114\)
default	\(\frac {-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+2 a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(114\)
risch	\(\frac {5 a^{2} x}{8}+\frac {83 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {83 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{2} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{96 d}-\frac {a^{2} \cos \left (5 d x +5 c \right )}{320 d}+\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{32 d}+\frac {19 a^{2} \cos \left (3 d x +3 c \right )}{192 d}+\frac {15 a^{2} \sin \left (2 d x +2 c \right )}{32 d}\)	\(183\)
norman	\(\frac {\frac {4 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a^{2} x}{8}+\frac {292 a^{2}}{105 d}+\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {7 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {85 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {85 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {7 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {11 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {175 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {175 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a^{2} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {24 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {116 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {172 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {176 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {232 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(395\)

[In]

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

[Out]

1/6720*a^2*(4200*d*x+8715*cos(d*x+c)+6720*ln(tan(1/2*d*x+1/2*c))-15*cos(7*d*x+7*c)+3150*sin(2*d*x+2*c)+70*sin(
6*d*x+6*c)+630*sin(4*d*x+4*c)-21*cos(5*d*x+5*c)+665*cos(3*d*x+3*c)+9344)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {120 \, a^{2} \cos \left (d x + c\right )^{7} - 168 \, a^{2} \cos \left (d x + c\right )^{5} - 280 \, a^{2} \cos \left (d x + c\right )^{3} - 525 \, a^{2} d x - 840 \, a^{2} \cos \left (d x + c\right ) + 420 \, a^{2} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 420 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 35 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{5} + 10 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]

[In]

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

[Out]

-1/840*(120*a^2*cos(d*x + c)^7 - 168*a^2*cos(d*x + c)^5 - 280*a^2*cos(d*x + c)^3 - 525*a^2*d*x - 840*a^2*cos(d
*x + c) + 420*a^2*log(1/2*cos(d*x + c) + 1/2) - 420*a^2*log(-1/2*cos(d*x + c) + 1/2) - 35*(8*a^2*cos(d*x + c)^
5 + 10*a^2*cos(d*x + c)^3 + 15*a^2*cos(d*x + c))*sin(d*x + c))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.76 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {480 \, a^{2} \cos \left (d x + c\right )^{7} - 112 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{3360 \, d} \]

[In]

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

[Out]

-1/3360*(480*a^2*cos(d*x + c)^7 - 112*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log(cos(d*x
+ c) + 1) + 15*log(cos(d*x + c) - 1))*a^2 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 4
8*sin(2*d*x + 2*c))*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.52 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {525 \, {\left (d x + c\right )} a^{2} + 840 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 1680 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 980 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 10080 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 2975 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 16240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 24640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2975 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14448 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 980 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6496 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1168 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{840 \, d} \]

[In]

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

[Out]

1/840*(525*(d*x + c)*a^2 + 840*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 2*(1155*a^2*tan(1/2*d*x + 1/2*c)^13 - 1680
*a^2*tan(1/2*d*x + 1/2*c)^12 + 980*a^2*tan(1/2*d*x + 1/2*c)^11 - 10080*a^2*tan(1/2*d*x + 1/2*c)^10 + 2975*a^2*
tan(1/2*d*x + 1/2*c)^9 - 16240*a^2*tan(1/2*d*x + 1/2*c)^8 - 24640*a^2*tan(1/2*d*x + 1/2*c)^6 - 2975*a^2*tan(1/
2*d*x + 1/2*c)^5 - 14448*a^2*tan(1/2*d*x + 1/2*c)^4 - 980*a^2*tan(1/2*d*x + 1/2*c)^3 - 6496*a^2*tan(1/2*d*x +
1/2*c)^2 - 1155*a^2*tan(1/2*d*x + 1/2*c) - 1168*a^2)/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d

Mupad [B] (verification not implemented)

Time = 12.18 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.39 \[ \int \cos ^5(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {5\,a^2\,\mathrm {atan}\left (\frac {25\,a^4}{16\,\left (\frac {5\,a^4}{2}-\frac {25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {5\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {5\,a^4}{2}-\frac {25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {-\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+24\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {85\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {116\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {176\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {85\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {172\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {232\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {11\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {292\,a^2}{105}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(a^2*log(tan(c/2 + (d*x)/2)))/d + (5*a^2*atan((25*a^4)/(16*((5*a^4)/2 - (25*a^4*tan(c/2 + (d*x)/2))/16)) + (5*
a^4*tan(c/2 + (d*x)/2))/(2*((5*a^4)/2 - (25*a^4*tan(c/2 + (d*x)/2))/16))))/(4*d) + ((232*a^2*tan(c/2 + (d*x)/2
)^2)/15 + (7*a^2*tan(c/2 + (d*x)/2)^3)/3 + (172*a^2*tan(c/2 + (d*x)/2)^4)/5 + (85*a^2*tan(c/2 + (d*x)/2)^5)/12
+ (176*a^2*tan(c/2 + (d*x)/2)^6)/3 + (116*a^2*tan(c/2 + (d*x)/2)^8)/3 - (85*a^2*tan(c/2 + (d*x)/2)^9)/12 + 24
*a^2*tan(c/2 + (d*x)/2)^10 - (7*a^2*tan(c/2 + (d*x)/2)^11)/3 + 4*a^2*tan(c/2 + (d*x)/2)^12 - (11*a^2*tan(c/2 +
(d*x)/2)^13)/4 + (292*a^2)/105 + (11*a^2*tan(c/2 + (d*x)/2))/4)/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*
x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^
12 + tan(c/2 + (d*x)/2)^14 + 1))