\(\int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx\) [293]

Optimal result

Integrand size = 29, antiderivative size = 124 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d} \]

[Out]

7/16*a^3*arctanh(cos(d*x+c))/d-4/3*a^3*cot(d*x+c)^3/d-3/5*a^3*cot(d*x+c)^5/d+7/16*a^3*cot(d*x+c)*csc(d*x+c)/d-
17/24*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^3*cot(d*x+c)*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 30, 2691, 3853, 3855, 14} \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]

[In]

Int[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

(7*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (7*a^3*Co
t[c + d*x]*Csc[c + d*x])/(16*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*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 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 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^3 \cot ^2(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^4(c+d x)+a^3 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{6} a^3 \int \csc ^5(c+d x) \, dx-\frac {1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{8} a^3 \int \csc ^3(c+d x) \, dx-\frac {1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {3 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {1}{16} a^3 \int \csc (c+d x) \, dx \\ & = \frac {7 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(252\) vs. \(2(124)=248\).

Time = 8.41 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.03 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (\csc ^6\left (\frac {1}{2} (c+d x)\right ) (18+5 \csc (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (34+90 \csc (c+d x))-2 \csc ^2\left (\frac {1}{2} (c+d x)\right ) (176+105 \csc (c+d x))-840 \csc (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+(97+159 \cos (c+d x)+44 \cos (2 (c+d x))) \sec ^6\left (\frac {1}{2} (c+d x)\right )+840 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )-1440 \csc ^5(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-320 \csc ^7(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x) (1+\sin (c+d x))^3}{1920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

Integrate[Cot[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/1920*(a^3*(Csc[(c + d*x)/2]^6*(18 + 5*Csc[c + d*x]) + Csc[(c + d*x)/2]^4*(34 + 90*Csc[c + d*x]) - 2*Csc[(c
+ d*x)/2]^2*(176 + 105*Csc[c + d*x]) - 840*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (97
+ 159*Cos[c + d*x] + 44*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6 + 840*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 1440*Cs
c[c + d*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6)*Sin[c + d*x]*(1 + Sin[c + d*x])^3)/(d
*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.39

method	result	size
parallelrisch	\(-\frac {\left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {36 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {36 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+21 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+28 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+168 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{3}}{384 d}\)	\(172\)
risch	\(-\frac {a^{3} \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}+365 \,{\mathrm e}^{9 i \left (d x +c \right )}-240 i {\mathrm e}^{10 i \left (d x +c \right )}-1110 \,{\mathrm e}^{7 i \left (d x +c \right )}+2160 i {\mathrm e}^{8 i \left (d x +c \right )}-1110 \,{\mathrm e}^{5 i \left (d x +c \right )}-1760 i {\mathrm e}^{6 i \left (d x +c \right )}+365 \,{\mathrm e}^{3 i \left (d x +c \right )}+480 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}-816 i {\mathrm e}^{2 i \left (d x +c \right )}+176 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\)	\(192\)
derivativedivides	\(\frac {-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{3}}+3 a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )+a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{16}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(222\)
default	\(\frac {-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{3}}+3 a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )+a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{16}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\)	\(222\)

[In]

int(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/384*(cot(1/2*d*x+1/2*c)^6-tan(1/2*d*x+1/2*c)^6+36/5*cot(1/2*d*x+1/2*c)^5-36/5*tan(1/2*d*x+1/2*c)^5+21*cot(1
/2*d*x+1/2*c)^4-21*tan(1/2*d*x+1/2*c)^4+28*cot(1/2*d*x+1/2*c)^3-28*tan(1/2*d*x+1/2*c)^3-3*cot(1/2*d*x+1/2*c)^2
+3*tan(1/2*d*x+1/2*c)^2-120*cot(1/2*d*x+1/2*c)+120*tan(1/2*d*x+1/2*c)+168*ln(tan(1/2*d*x+1/2*c)))*a^3/d

Fricas [B] (verification not implemented)

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

Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.83 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {210 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \, a^{3} \cos \left (d x + c\right )^{3} - 210 \, a^{3} \cos \left (d x + c\right ) - 105 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 32 \, {\left (11 \, a^{3} \cos \left (d x + c\right )^{5} - 20 \, a^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/480*(210*a^3*cos(d*x + c)^5 - 80*a^3*cos(d*x + c)^3 - 210*a^3*cos(d*x + c) - 105*(a^3*cos(d*x + c)^6 - 3*a^
3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 105*(a^3*cos(d*x + c)^6 - 3*a^3*c
os(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d*x + c) + 1/2) + 32*(11*a^3*cos(d*x + c)^5 - 20*a^3*
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. \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**2*csc(d*x+c)**7*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.61 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {5 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 90 \, a^{3} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {160 \, a^{3}}{\tan \left (d x + c\right )^{3}} + \frac {96 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/480*(5*a^3*(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)) + 90*a^3*(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)) + 160*a^3/tan
(d*x + c)^3 + 96*(5*tan(d*x + c)^2 + 3)*a^3/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 = 0.39 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.84 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 840 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2058 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^7*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/1920*(5*a^3*tan(1/2*d*x + 1/2*c)^6 + 36*a^3*tan(1/2*d*x + 1/2*c)^5 + 105*a^3*tan(1/2*d*x + 1/2*c)^4 + 140*a^
3*tan(1/2*d*x + 1/2*c)^3 - 15*a^3*tan(1/2*d*x + 1/2*c)^2 - 840*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 600*a^3*ta
n(1/2*d*x + 1/2*c) + (2058*a^3*tan(1/2*d*x + 1/2*c)^6 + 600*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x +
1/2*c)^4 - 140*a^3*tan(1/2*d*x + 1/2*c)^3 - 105*a^3*tan(1/2*d*x + 1/2*c)^2 - 36*a^3*tan(1/2*d*x + 1/2*c) - 5*a
^3)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 10.45 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \cot ^2(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3\,\left (5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-36\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+36\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int((cos(c + d*x)^2*(a + a*sin(c + d*x))^3)/sin(c + d*x)^7,x)

[Out]

-(a^3*(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 - 36*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 36*co
s(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 105*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 140*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 + 600*cos(c/2 + (d*x)/2)^5*sin(c/2 +
(d*x)/2)^7 - 600*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 + 14
0*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 105*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 840*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)