\(\int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\) [387]

Optimal result

Integrand size = 29, antiderivative size = 118 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=a^2 x-\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{4 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d} \]

[Out]

a^2*x-3/4*a^2*arctanh(cos(d*x+c))/d+a^2*cot(d*x+c)/d-1/3*a^2*cot(d*x+c)^3/d-1/5*a^2*cot(d*x+c)^5/d+3/4*a^2*cot
(d*x+c)*csc(d*x+c)/d-1/2*a^2*cot(d*x+c)^3*csc(d*x+c)/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 3554, 8, 2691, 3855, 2687, 30} \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+a^2 x \]

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]

[Out]

a^2*x - (3*a^2*ArcTanh[Cos[c + d*x]])/(4*d) + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c +
d*x]^5)/(5*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x])/(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 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 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 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.69 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (480 c+480 d x+272 \cot \left (\frac {1}{2} (c+d x)\right )+150 \csc ^2\left (\frac {1}{2} (c+d x)\right )-360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-150 \sec ^2\left (\frac {1}{2} (c+d x)\right )+15 \sec ^4\left (\frac {1}{2} (c+d x)\right )-8 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+96 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+\frac {1}{2} \csc ^4\left (\frac {1}{2} (c+d x)\right ) (-30+\sin (c+d x))-\frac {3}{2} \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-272 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{480 d} \]

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(480*c + 480*d*x + 272*Cot[(c + d*x)/2] + 150*Csc[(c + d*x)/2]^2 - 360*Log[Cos[(c + d*x)/2]] + 360*Log[Si
n[(c + d*x)/2]] - 150*Sec[(c + d*x)/2]^2 + 15*Sec[(c + d*x)/2]^4 - 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 96*Cs
c[c + d*x]^5*Sin[(c + d*x)/2]^6 + (Csc[(c + d*x)/2]^4*(-30 + Sin[c + d*x]))/2 - (3*Csc[(c + d*x)/2]^6*Sin[c +
d*x])/2 - 272*Tan[(c + d*x)/2]))/(480*d)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\)	\(130\)
default	\(\frac {a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\)	\(130\)
parallelrisch	\(\frac {a^{2} \left (3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+480 d x +360 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-270 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+270 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}\)	\(152\)
risch	\(a^{2} x -\frac {a^{2} \left (-60 i {\mathrm e}^{8 i \left (d x +c \right )}+75 \,{\mathrm e}^{9 i \left (d x +c \right )}+360 i {\mathrm e}^{6 i \left (d x +c \right )}-30 \,{\mathrm e}^{7 i \left (d x +c \right )}-320 i {\mathrm e}^{4 i \left (d x +c \right )}+280 i {\mathrm e}^{2 i \left (d x +c \right )}+30 \,{\mathrm e}^{3 i \left (d x +c \right )}-68 i-75 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{30 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}\)	\(163\)
norman	\(\frac {a^{2} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{2} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a^{2}}{160 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d}-\frac {11 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {3 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {257 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {53 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {53 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {257 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}-\frac {3 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {11 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+2 a^{2} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {15 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {15 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}\)	\(350\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+2*a^2*(-1/4/sin(d*x+c)^4*cos(d*x+c)^5+1/8/sin(d*x+c)^2*cos(d*x+c
)^5+1/8*cos(d*x+c)^3+3/8*cos(d*x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c)))-1/5*a^2/sin(d*x+c)^5*cos(d*x+c)^5)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (108) = 216\).

Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.03 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {136 \, a^{2} \cos \left (d x + c\right )^{5} - 280 \, a^{2} \cos \left (d x + c\right )^{3} + 120 \, a^{2} \cos \left (d x + c\right ) - 45 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 45 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (4 \, a^{2} d x \cos \left (d x + c\right )^{4} - 8 \, a^{2} d x \cos \left (d x + c\right )^{2} - 5 \, a^{2} \cos \left (d x + c\right )^{3} + 4 \, a^{2} d x + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/120*(136*a^2*cos(d*x + c)^5 - 280*a^2*cos(d*x + c)^3 + 120*a^2*cos(d*x + c) - 45*(a^2*cos(d*x + c)^4 - 2*a^2
*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 45*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^
2 + a^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 30*(4*a^2*d*x*cos(d*x + c)^4 - 8*a^2*d*x*cos(d*x + c)^2 -
5*a^2*cos(d*x + c)^3 + 4*a^2*d*x + 3*a^2*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2
+ d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**6*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {40 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{2} - 15 \, a^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {24 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{120 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/120*(40*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^2 - 15*a^2*(2*(5*cos(d*x + c)^3 - 3*cos(d*x
+ c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 24*a^2/ta
n(d*x + c)^5)/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.75 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, {\left (d x + c\right )} a^{2} + 360 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {822 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/480*(3*a^2*tan(1/2*d*x + 1/2*c)^5 + 15*a^2*tan(1/2*d*x + 1/2*c)^4 + 5*a^2*tan(1/2*d*x + 1/2*c)^3 - 120*a^2*t
an(1/2*d*x + 1/2*c)^2 + 480*(d*x + c)*a^2 + 360*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 270*a^2*tan(1/2*d*x + 1/2
*c) - (822*a^2*tan(1/2*d*x + 1/2*c)^5 - 270*a^2*tan(1/2*d*x + 1/2*c)^4 - 120*a^2*tan(1/2*d*x + 1/2*c)^3 + 5*a^
2*tan(1/2*d*x + 1/2*c)^2 + 15*a^2*tan(1/2*d*x + 1/2*c) + 3*a^2)/tan(1/2*d*x + 1/2*c)^5)/d

Mupad [B] (verification not implemented)

Time = 10.14 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.33 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {2\,a^2\,\mathrm {atan}\left (\frac {4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {9\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^6,x)

[Out]

(a^2*cot(c/2 + (d*x)/2)^2)/(4*d) - (a^2*cot(c/2 + (d*x)/2)^3)/(96*d) - (a^2*cot(c/2 + (d*x)/2)^4)/(32*d) - (a^
2*cot(c/2 + (d*x)/2)^5)/(160*d) - (a^2*tan(c/2 + (d*x)/2)^2)/(4*d) + (a^2*tan(c/2 + (d*x)/2)^3)/(96*d) + (a^2*
tan(c/2 + (d*x)/2)^4)/(32*d) + (a^2*tan(c/2 + (d*x)/2)^5)/(160*d) + (2*a^2*atan((4*cos(c/2 + (d*x)/2) + 3*sin(
c/2 + (d*x)/2))/(3*cos(c/2 + (d*x)/2) - 4*sin(c/2 + (d*x)/2))))/d + (3*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d
*x)/2)))/(4*d) + (9*a^2*cot(c/2 + (d*x)/2))/(16*d) - (9*a^2*tan(c/2 + (d*x)/2))/(16*d)