\(\int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\) [390]

Optimal result

Integrand size = 29, antiderivative size = 168 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d} \]

[Out]

-3/64*a^2*arctanh(cos(d*x+c))/d-2/5*a^2*cot(d*x+c)^5/d-3/7*a^2*cot(d*x+c)^7/d-1/9*a^2*cot(d*x+c)^9/d-3/64*a^2*
cot(d*x+c)*csc(d*x+c)/d-1/32*a^2*cot(d*x+c)*csc(d*x+c)^3/d+1/8*a^2*cot(d*x+c)*csc(d*x+c)^5/d-1/4*a^2*cot(d*x+c
)^3*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 14, 2691, 3853, 3855, 276} \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d} \]

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]

[Out]

(-3*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (3*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot
[c + d*x]^9)/(9*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (a^
2*Cot[c + d*x]*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(4*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 276

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

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 ^4(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^5(c+d x)+a^2 \cot ^4(c+d x) \csc ^6(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx \\ & = -\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}-\frac {1}{4} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \text {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {1}{8} a^2 \int \csc ^5(c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {1}{32} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx \\ & = -\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {1}{64} \left (3 a^2\right ) \int \csc (c+d x) \, dx \\ & = -\frac {3 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.73 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.86 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^9(c+d x) \left (451584 \cos (c+d x)+155904 \cos (3 (c+d x))-20736 \cos (5 (c+d x))-14976 \cos (7 (c+d x))+1664 \cos (9 (c+d x))+119070 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-119070 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+212940 \sin (2 (c+d x))-79380 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+79380 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+195300 \sin (4 (c+d x))+34020 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-34020 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+16380 \sin (6 (c+d x))-8505 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+8505 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-1890 \sin (8 (c+d x))+945 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))-945 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{5160960 d} \]

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]

[Out]

-1/5160960*(a^2*Csc[c + d*x]^9*(451584*Cos[c + d*x] + 155904*Cos[3*(c + d*x)] - 20736*Cos[5*(c + d*x)] - 14976
*Cos[7*(c + d*x)] + 1664*Cos[9*(c + d*x)] + 119070*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 119070*Log[Sin[(c + d*
x)/2]]*Sin[c + d*x] + 212940*Sin[2*(c + d*x)] - 79380*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] + 79380*Log[Sin[(
c + d*x)/2]]*Sin[3*(c + d*x)] + 195300*Sin[4*(c + d*x)] + 34020*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 34020
*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 16380*Sin[6*(c + d*x)] - 8505*Log[Cos[(c + d*x)/2]]*Sin[7*(c + d*x)]
+ 8505*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 1890*Sin[8*(c + d*x)] + 945*Log[Cos[(c + d*x)/2]]*Sin[9*(c +
d*x)] - 945*Log[Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(-\frac {\left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {9 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {9 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {45 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {45 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {72 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {72 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-36 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+36 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+162 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-162 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-216 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{4608 d}\)	\(198\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )}{d}\)	\(220\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35 \sin \left (d x +c \right )^{5}}\right )+2 a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{5}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right )^{5}}\right )}{d}\)	\(220\)
risch	\(\frac {a^{2} \left (945 \,{\mathrm e}^{17 i \left (d x +c \right )}-120960 i {\mathrm e}^{10 i \left (d x +c \right )}-8190 \,{\mathrm e}^{15 i \left (d x +c \right )}+40320 i {\mathrm e}^{14 i \left (d x +c \right )}-97650 \,{\mathrm e}^{13 i \left (d x +c \right )}-19584 i {\mathrm e}^{4 i \left (d x +c \right )}-106470 \,{\mathrm e}^{11 i \left (d x +c \right )}-330624 i {\mathrm e}^{8 i \left (d x +c \right )}+14976 i {\mathrm e}^{2 i \left (d x +c \right )}+106470 \,{\mathrm e}^{7 i \left (d x +c \right )}-8064 i {\mathrm e}^{6 i \left (d x +c \right )}+97650 \,{\mathrm e}^{5 i \left (d x +c \right )}-147840 i {\mathrm e}^{12 i \left (d x +c \right )}+8190 \,{\mathrm e}^{3 i \left (d x +c \right )}-1664 i-945 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{10080 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}\)	\(238\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/4608*(cot(1/2*d*x+1/2*c)^9-tan(1/2*d*x+1/2*c)^9+9/2*cot(1/2*d*x+1/2*c)^8-9/2*tan(1/2*d*x+1/2*c)^8+45/7*cot(
1/2*d*x+1/2*c)^7-45/7*tan(1/2*d*x+1/2*c)^7-72/5*cot(1/2*d*x+1/2*c)^5+72/5*tan(1/2*d*x+1/2*c)^5-36*cot(1/2*d*x+
1/2*c)^4+36*tan(1/2*d*x+1/2*c)^4-48*cot(1/2*d*x+1/2*c)^3+48*tan(1/2*d*x+1/2*c)^3+162*cot(1/2*d*x+1/2*c)-162*ta
n(1/2*d*x+1/2*c)-216*ln(tan(1/2*d*x+1/2*c)))*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {3328 \, a^{2} \cos \left (d x + c\right )^{9} - 14976 \, a^{2} \cos \left (d x + c\right )^{7} + 16128 \, a^{2} \cos \left (d x + c\right )^{5} + 945 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 945 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 630 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{7} - 11 \, a^{2} \cos \left (d x + c\right )^{5} - 11 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/40320*(3328*a^2*cos(d*x + c)^9 - 14976*a^2*cos(d*x + c)^7 + 16128*a^2*cos(d*x + c)^5 + 945*(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)*si
n(d*x + c) - 945*(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)*sin(d*x + c) - 630*(3*a^2*cos(d*x + c)^7 - 11*a^2*cos(d*x + c)^5 - 11*a^2*cos(
d*x + c)^3 + 3*a^2*cos(d*x + c))*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)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**10*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {315 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1152 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}} - \frac {128 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{40320 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/40320*(315*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)) - 1152*(7*tan(d*x + c)^2 + 5)*a^2/tan(d*x + c)^7 - 128*(63*tan(d*x + c)^4 + 90*tan(d*x + c)^2 + 35)*a^2
/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.55 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15120 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {42774 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 70 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{322560 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^10*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/322560*(70*a^2*tan(1/2*d*x + 1/2*c)^9 + 315*a^2*tan(1/2*d*x + 1/2*c)^8 + 450*a^2*tan(1/2*d*x + 1/2*c)^7 - 10
08*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 + 15120*a^2*
log(abs(tan(1/2*d*x + 1/2*c))) + 11340*a^2*tan(1/2*d*x + 1/2*c) - (42774*a^2*tan(1/2*d*x + 1/2*c)^9 + 11340*a^
2*tan(1/2*d*x + 1/2*c)^8 - 3360*a^2*tan(1/2*d*x + 1/2*c)^6 - 2520*a^2*tan(1/2*d*x + 1/2*c)^5 - 1008*a^2*tan(1/
2*d*x + 1/2*c)^4 + 450*a^2*tan(1/2*d*x + 1/2*c)^2 + 315*a^2*tan(1/2*d*x + 1/2*c) + 70*a^2)/tan(1/2*d*x + 1/2*c
)^9)/d

Mupad [B] (verification not implemented)

Time = 13.30 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.30 \[ \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\left (70\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+315\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )}{322560\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

[In]

int((cos(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^10,x)

[Out]

(a^2*(70*sin(c/2 + (d*x)/2)^18 - 70*cos(c/2 + (d*x)/2)^18 + 315*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^17 - 315
*cos(c/2 + (d*x)/2)^17*sin(c/2 + (d*x)/2) + 450*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^16 - 1008*cos(c/2 + (d
*x)/2)^4*sin(c/2 + (d*x)/2)^14 - 2520*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^13 - 3360*cos(c/2 + (d*x)/2)^6*s
in(c/2 + (d*x)/2)^12 + 11340*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^10 - 11340*cos(c/2 + (d*x)/2)^10*sin(c/2
+ (d*x)/2)^8 + 3360*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^6 + 2520*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)
^5 + 1008*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^4 - 450*cos(c/2 + (d*x)/2)^16*sin(c/2 + (d*x)/2)^2 + 15120*
log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^9))/(322560*d*cos(c/2 + (d*
x)/2)^9*sin(c/2 + (d*x)/2)^9)