\(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx\) [389]

Optimal result

Integrand size = 29, antiderivative size = 176 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {11 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d} \]

[Out]

-11/128*a^2*arctanh(cos(d*x+c))/d-2/5*a^2*cot(d*x+c)^5/d-2/7*a^2*cot(d*x+c)^7/d-11/128*a^2*cot(d*x+c)*csc(d*x+
c)/d+7/64*a^2*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^2*cot(d*x+c)^3*csc(d*x+c)^3/d+1/16*a^2*cot(d*x+c)*csc(d*x+c)^5/d
-1/8*a^2*cot(d*x+c)^3*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3853, 3855, 2687, 14} \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {11 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

(-11*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (11*a^
2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(6*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*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 ^4(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^4(c+d x)+a^2 \cot ^4(c+d x) \csc ^5(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {1}{8} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{2} a^2 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{16} a^2 \int \csc ^5(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^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{64} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx+\frac {1}{16} a^2 \int \csc (c+d x) \, dx \\ & = -\frac {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 ^7(c+d x)}{7 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{128} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = -\frac {11 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.37 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.65 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^8(c+d x) \left (158270 \cos (c+d x)+77210 \cos (3 (c+d x))-18130 \cos (5 (c+d x))-2310 \cos (7 (c+d x))+40425 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-64680 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32340 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9240 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1155 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-40425 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64680 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-32340 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9240 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1155 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+86016 \sin (2 (c+d x))+64512 \sin (4 (c+d x))+12288 \sin (6 (c+d x))-1536 \sin (8 (c+d x))\right )}{1720320 d} \]

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/1720320*(a^2*Csc[c + d*x]^8*(158270*Cos[c + d*x] + 77210*Cos[3*(c + d*x)] - 18130*Cos[5*(c + d*x)] - 2310*C
os[7*(c + d*x)] + 40425*Log[Cos[(c + d*x)/2]] - 64680*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 32340*Cos[4*(c
+ d*x)]*Log[Cos[(c + d*x)/2]] - 9240*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 1155*Cos[8*(c + d*x)]*Log[Cos[(c
+ d*x)/2]] - 40425*Log[Sin[(c + d*x)/2]] + 64680*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 32340*Cos[4*(c + d*
x)]*Log[Sin[(c + d*x)/2]] + 9240*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1155*Cos[8*(c + d*x)]*Log[Sin[(c + d
*x)/2]] + 86016*Sin[2*(c + d*x)] + 64512*Sin[4*(c + d*x)] + 12288*Sin[6*(c + d*x)] - 1536*Sin[8*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.83

method	result	size
parallelrisch	\(-\frac {2261 \left (-\frac {540672 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2261}+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {1103 \cos \left (3 d x +3 c \right )}{2261}-\frac {37 \cos \left (5 d x +5 c \right )}{323}-\frac {33 \cos \left (7 d x +7 c \right )}{2261}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {9216 \cos \left (d x +c \right )}{2261}+\frac {3072 \cos \left (3 d x +3 c \right )}{1615}+\frac {3072 \cos \left (5 d x +5 c \right )}{11305}-\frac {3072 \cos \left (7 d x +7 c \right )}{79135}\right ) \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{6291456 d}\)	\(146\)
risch	\(\frac {a^{2} \left (1155 \,{\mathrm e}^{15 i \left (d x +c \right )}+9065 \,{\mathrm e}^{13 i \left (d x +c \right )}-38605 \,{\mathrm e}^{11 i \left (d x +c \right )}+53760 i {\mathrm e}^{12 i \left (d x +c \right )}-79135 \,{\mathrm e}^{9 i \left (d x +c \right )}-79135 \,{\mathrm e}^{7 i \left (d x +c \right )}+53760 i {\mathrm e}^{8 i \left (d x +c \right )}-38605 \,{\mathrm e}^{5 i \left (d x +c \right )}-86016 i {\mathrm e}^{6 i \left (d x +c \right )}+9065 \,{\mathrm e}^{3 i \left (d x +c \right )}-10752 i {\mathrm e}^{4 i \left (d x +c \right )}+1155 \,{\mathrm e}^{i \left (d x +c \right )}-12288 i {\mathrm e}^{2 i \left (d x +c \right )}+1536 i\right )}{6720 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}+\frac {11 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}-\frac {11 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\)	\(214\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(256\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(256\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-2261/6291456*(-540672/2261*ln(tan(1/2*d*x+1/2*c))+(sec(1/2*d*x+1/2*c)*(cos(d*x+c)+1103/2261*cos(3*d*x+3*c)-37
/323*cos(5*d*x+5*c)-33/2261*cos(7*d*x+7*c))*csc(1/2*d*x+1/2*c)+9216/2261*cos(d*x+c)+3072/1615*cos(3*d*x+3*c)+3
072/11305*cos(5*d*x+5*c)-3072/79135*cos(7*d*x+7*c))*sec(1/2*d*x+1/2*c)^7*csc(1/2*d*x+1/2*c)^7)*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.54 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {2310 \, a^{2} \cos \left (d x + c\right )^{7} + 490 \, a^{2} \cos \left (d x + c\right )^{5} - 8470 \, a^{2} \cos \left (d x + c\right )^{3} + 2310 \, a^{2} \cos \left (d x + c\right ) - 1155 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1155 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1536 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{7} - 7 \, a^{2} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{26880 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/26880*(2310*a^2*cos(d*x + c)^7 + 490*a^2*cos(d*x + c)^5 - 8470*a^2*cos(d*x + c)^3 + 2310*a^2*cos(d*x + c) -
1155*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*c
os(d*x + c) + 1/2) + 1155*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x +
c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2) + 1536*(2*a^2*cos(d*x + c)^7 - 7*a^2*cos(d*x + c)^5)*sin(d*x + c))/(d
*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**9*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {105 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1536 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/26880*(105*a^2*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*cos(d*x + c))/(cos(d*x + c)^
8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c)
- 1)) + 280*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)) - 1536*(7*tan(d*x + c)^2 + 5)*a^2
/tan(d*x + c)^7)/d

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.66 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {105 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 480 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 18480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 10080 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {50226 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 10080 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{215040 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/215040*(105*a^2*tan(1/2*d*x + 1/2*c)^8 + 480*a^2*tan(1/2*d*x + 1/2*c)^7 + 560*a^2*tan(1/2*d*x + 1/2*c)^6 - 6
72*a^2*tan(1/2*d*x + 1/2*c)^5 - 2520*a^2*tan(1/2*d*x + 1/2*c)^4 - 3360*a^2*tan(1/2*d*x + 1/2*c)^3 - 1680*a^2*t
an(1/2*d*x + 1/2*c)^2 + 18480*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 10080*a^2*tan(1/2*d*x + 1/2*c) - (50226*a^2
*tan(1/2*d*x + 1/2*c)^8 + 10080*a^2*tan(1/2*d*x + 1/2*c)^7 - 1680*a^2*tan(1/2*d*x + 1/2*c)^6 - 3360*a^2*tan(1/
2*d*x + 1/2*c)^5 - 2520*a^2*tan(1/2*d*x + 1/2*c)^4 - 672*a^2*tan(1/2*d*x + 1/2*c)^3 + 560*a^2*tan(1/2*d*x + 1/
2*c)^2 + 480*a^2*tan(1/2*d*x + 1/2*c) + 105*a^2)/tan(1/2*d*x + 1/2*c)^8)/d

Mupad [B] (verification not implemented)

Time = 10.34 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{448\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{448\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {11\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}-\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d}+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d} \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^9,x)

[Out]

(a^2*cot(c/2 + (d*x)/2)^2)/(128*d) + (a^2*cot(c/2 + (d*x)/2)^3)/(64*d) + (3*a^2*cot(c/2 + (d*x)/2)^4)/(256*d)
+ (a^2*cot(c/2 + (d*x)/2)^5)/(320*d) - (a^2*cot(c/2 + (d*x)/2)^6)/(384*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(448*d)
- (a^2*cot(c/2 + (d*x)/2)^8)/(2048*d) - (a^2*tan(c/2 + (d*x)/2)^2)/(128*d) - (a^2*tan(c/2 + (d*x)/2)^3)/(64*d
) - (3*a^2*tan(c/2 + (d*x)/2)^4)/(256*d) - (a^2*tan(c/2 + (d*x)/2)^5)/(320*d) + (a^2*tan(c/2 + (d*x)/2)^6)/(38
4*d) + (a^2*tan(c/2 + (d*x)/2)^7)/(448*d) + (a^2*tan(c/2 + (d*x)/2)^8)/(2048*d) + (11*a^2*log(tan(c/2 + (d*x)/
2)))/(128*d) - (3*a^2*cot(c/2 + (d*x)/2))/(64*d) + (3*a^2*tan(c/2 + (d*x)/2))/(64*d)