\(\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx\) [284]

Optimal result

Integrand size = 29, antiderivative size = 124 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \]

[Out]

3/16*a^2*arctanh(cos(d*x+c))/d-2/3*a^2*cot(d*x+c)^3/d-2/5*a^2*cot(d*x+c)^5/d+3/16*a^2*cot(d*x+c)*csc(d*x+c)/d-
5/24*a^2*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^2*cot(d*x+c)*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3853, 3855, 2687, 14} \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{16 d} \]

[In]

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

[Out]

(3*a^2*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (3*a^2*Co
t[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^
5)/(6*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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}& = \int \left (a^2 \cot ^2(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^2(c+d x) \csc ^4(c+d x)+a^2 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{6} a^2 \int \csc ^5(c+d x) \, dx-\frac {1}{4} a^2 \int \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{8} a^2 \int \csc (c+d x) \, dx-\frac {1}{8} a^2 \int \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{16} a^2 \int \csc (c+d x) \, dx \\ & = \frac {3 a^2 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \csc ^6(c+d x) \left (-1500 \cos (c+d x)+130 \cos (3 (c+d x))+90 \cos (5 (c+d x))+450 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-675 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+270 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-45 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-450 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+675 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-270 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-960 \sin (2 (c+d x))-384 \sin (4 (c+d x))+64 \sin (6 (c+d x))\right )}{7680 d} \]

[In]

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

[Out]

(a^2*Csc[c + d*x]^6*(-1500*Cos[c + d*x] + 130*Cos[3*(c + d*x)] + 90*Cos[5*(c + d*x)] + 450*Log[Cos[(c + d*x)/2
]] - 675*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 270*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 45*Cos[6*(c + d
*x)]*Log[Cos[(c + d*x)/2]] - 450*Log[Sin[(c + d*x)/2]] + 675*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 270*Cos[
4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 45*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 960*Sin[2*(c + d*x)] - 384*Si
n[4*(c + d*x)] + 64*Sin[6*(c + d*x)]))/(7680*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.35

method	result	size
risch	\(-\frac {a^{2} \left (45 \,{\mathrm e}^{11 i \left (d x +c \right )}+65 \,{\mathrm e}^{9 i \left (d x +c \right )}-750 \,{\mathrm e}^{7 i \left (d x +c \right )}+960 i {\mathrm e}^{8 i \left (d x +c \right )}-750 \,{\mathrm e}^{5 i \left (d x +c \right )}-640 i {\mathrm e}^{6 i \left (d x +c \right )}+65 \,{\mathrm e}^{3 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}-384 i {\mathrm e}^{2 i \left (d x +c \right )}+64 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\)	\(168\)
parallelrisch	\(-\frac {\left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {24 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {24 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+9 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{384 d}\)	\(172\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )+a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{16}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(200\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )+a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{16}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(200\)
norman	\(\frac {-\frac {a^{2}}{384 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d}-\frac {11 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {11 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {17 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {5 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {5 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {17 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {11 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {11 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a^{2} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\)	\(339\)

[In]

int(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/120*a^2*(45*exp(11*I*(d*x+c))+65*exp(9*I*(d*x+c))-750*exp(7*I*(d*x+c))+960*I*exp(8*I*(d*x+c))-750*exp(5*I*(
d*x+c))-640*I*exp(6*I*(d*x+c))+65*exp(3*I*(d*x+c))+45*exp(I*(d*x+c))-384*I*exp(2*I*(d*x+c))+64*I)/d/(exp(2*I*(
d*x+c))-1)^6+3/16*a^2/d*ln(exp(I*(d*x+c))+1)-3/16*a^2/d*ln(exp(I*(d*x+c))-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (112) = 224\).

Time = 0.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.83 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {90 \, a^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{2} \cos \left (d x + c\right )^{3} - 90 \, a^{2} \cos \left (d x + c\right ) - 45 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 64 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{5} - 5 \, a^{2} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/480*(90*a^2*cos(d*x + c)^5 - 80*a^2*cos(d*x + c)^3 - 90*a^2*cos(d*x + c) - 45*(a^2*cos(d*x + c)^6 - 3*a^2*c
os(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) + 45*(a^2*cos(d*x + c)^6 - 3*a^2*cos(d
*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2) + 64*(2*a^2*cos(d*x + c)^5 - 5*a^2*cos(d*
x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**2*csc(d*x+c)**7*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.51 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {5 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {64 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{2}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/480*(5*a^2*(2*(3*cos(d*x + c)^5 - 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3
*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) + 30*a^2*(2*(cos(d*x + c)^3 + cos(d*
x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) + 64*(5*tan(d
*x + c)^2 + 3)*a^2/tan(d*x + c)^5)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (112) = 224\).

Time = 0.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.84 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {882 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(5*a^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*tan(1/2*d*x + 1/2*c)^5 + 45*a^2*tan(1/2*d*x + 1/2*c)^4 + 40*a^2*
tan(1/2*d*x + 1/2*c)^3 - 15*a^2*tan(1/2*d*x + 1/2*c)^2 - 360*a^2*log(abs(tan(1/2*d*x + 1/2*c))) - 240*a^2*tan(
1/2*d*x + 1/2*c) + (882*a^2*tan(1/2*d*x + 1/2*c)^6 + 240*a^2*tan(1/2*d*x + 1/2*c)^5 + 15*a^2*tan(1/2*d*x + 1/2
*c)^4 - 40*a^2*tan(1/2*d*x + 1/2*c)^3 - 45*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*a^2*tan(1/2*d*x + 1/2*c) - 5*a^2)/t
an(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 11.02 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2\,\left (5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+360\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int((cos(c + d*x)^2*(a + a*sin(c + d*x))^2)/sin(c + d*x)^7,x)

[Out]

-(a^2*(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 - 24*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 24*co
s(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 45*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 40*cos(c/2 + (d*x)/2)
^3*sin(c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 240*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d
*x)/2)^7 - 240*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 15*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 40*c
os(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 45*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 360*log(sin(c/2 + (
d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6))/(1920*d*cos(c/2 + (d*x)/2)^6*sin(c/2 +
(d*x)/2)^6)