Integrand size = 21, antiderivative size = 165 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {13 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {152 a^3 \tan (c+d x)}{15 d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {76 a^6 \sec (c+d x) \tan (c+d x)}{15 d \left (a^3-a^3 \cos (c+d x)\right )} \]
13/2*a^3*arctanh(sin(d*x+c))/d+152/15*a^3*tan(d*x+c)/d+13/2*a^3*sec(d*x+c) *tan(d*x+c)/d-1/5*a^6*sec(d*x+c)*tan(d*x+c)/d/(a-a*cos(d*x+c))^3-11/15*a^5 *sec(d*x+c)*tan(d*x+c)/d/(a-a*cos(d*x+c))^2-76/15*a^6*sec(d*x+c)*tan(d*x+c )/d/(a^3-a^3*cos(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(353\) vs. \(2(165)=330\).
Time = 2.01 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.14 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (24960 \cos ^2(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\csc \left (\frac {c}{2}\right ) \csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec (c) \left (-1235 \sin \left (\frac {d x}{2}\right )+3805 \sin \left (\frac {3 d x}{2}\right )+4329 \sin \left (c-\frac {d x}{2}\right )-1989 \sin \left (c+\frac {d x}{2}\right )-3575 \sin \left (2 c+\frac {d x}{2}\right )+475 \sin \left (c+\frac {3 d x}{2}\right )+2005 \sin \left (2 c+\frac {3 d x}{2}\right )+2275 \sin \left (3 c+\frac {3 d x}{2}\right )-2673 \sin \left (c+\frac {5 d x}{2}\right )+105 \sin \left (2 c+\frac {5 d x}{2}\right )-1593 \sin \left (3 c+\frac {5 d x}{2}\right )-975 \sin \left (4 c+\frac {5 d x}{2}\right )+1325 \sin \left (2 c+\frac {7 d x}{2}\right )-255 \sin \left (3 c+\frac {7 d x}{2}\right )+875 \sin \left (4 c+\frac {7 d x}{2}\right )+195 \sin \left (5 c+\frac {7 d x}{2}\right )-304 \sin \left (3 c+\frac {9 d x}{2}\right )+90 \sin \left (4 c+\frac {9 d x}{2}\right )-214 \sin \left (5 c+\frac {9 d x}{2}\right )\right )\right )}{30720 d} \]
-1/30720*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^2*(2496 0*Cos[c + d*x]^2*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Csc[c/2]*Csc[(c + d*x)/2]^5*Sec[c]*(-1235*S in[(d*x)/2] + 3805*Sin[(3*d*x)/2] + 4329*Sin[c - (d*x)/2] - 1989*Sin[c + ( d*x)/2] - 3575*Sin[2*c + (d*x)/2] + 475*Sin[c + (3*d*x)/2] + 2005*Sin[2*c + (3*d*x)/2] + 2275*Sin[3*c + (3*d*x)/2] - 2673*Sin[c + (5*d*x)/2] + 105*S in[2*c + (5*d*x)/2] - 1593*Sin[3*c + (5*d*x)/2] - 975*Sin[4*c + (5*d*x)/2] + 1325*Sin[2*c + (7*d*x)/2] - 255*Sin[3*c + (7*d*x)/2] + 875*Sin[4*c + (7 *d*x)/2] + 195*Sin[5*c + (7*d*x)/2] - 304*Sin[3*c + (9*d*x)/2] + 90*Sin[4* c + (9*d*x)/2] - 214*Sin[5*c + (9*d*x)/2])))/d
Time = 1.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.048, Rules used = {3042, 4360, 25, 25, 3042, 25, 3348, 25, 3042, 3245, 3042, 3457, 3042, 3457, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \csc ^6(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^6(c+d x) \sec ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \csc ^6(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^6(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle -a^6 \int -\frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^6 \int \frac {\sec ^3(c+d x)}{(a-a \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle a^6 \left (\frac {\int \frac {(4 \cos (c+d x) a+7 a) \sec ^3(c+d x)}{(a-a \cos (c+d x))^2}dx}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\int \frac {4 \sin \left (c+d x+\frac {\pi }{2}\right ) a+7 a}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle a^6 \left (\frac {\frac {\int \frac {\left (33 \cos (c+d x) a^2+43 a^2\right ) \sec ^3(c+d x)}{a-a \cos (c+d x)}dx}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\frac {\int \frac {33 \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+43 a^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {\int \left (152 \cos (c+d x) a^3+195 a^3\right ) \sec ^3(c+d x)dx}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {\int \frac {152 \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+195 a^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {195 a^3 \int \sec ^3(c+d x)dx+152 a^3 \int \sec ^2(c+d x)dx}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {152 a^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+195 a^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {195 a^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {152 a^3 \int 1d(-\tan (c+d x))}{d}}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {195 a^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {152 a^3 \tan (c+d x)}{d}}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {195 a^3 \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {152 a^3 \tan (c+d x)}{d}}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {195 a^3 \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {152 a^3 \tan (c+d x)}{d}}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle a^6 \left (\frac {\frac {\frac {195 a^3 \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {152 a^3 \tan (c+d x)}{d}}{a^2}-\frac {76 a^2 \tan (c+d x) \sec (c+d x)}{d (a-a \cos (c+d x))}}{3 a^2}-\frac {11 a \tan (c+d x) \sec (c+d x)}{3 d (a-a \cos (c+d x))^2}}{5 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}\right )\) |
a^6*(-1/5*(Sec[c + d*x]*Tan[c + d*x])/(d*(a - a*Cos[c + d*x])^3) + ((-11*a *Sec[c + d*x]*Tan[c + d*x])/(3*d*(a - a*Cos[c + d*x])^2) + ((-76*a^2*Sec[c + d*x]*Tan[c + d*x])/(d*(a - a*Cos[c + d*x])) + ((152*a^3*Tan[c + d*x])/d + 195*a^3*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d )))/a^2)/(3*a^2))/(5*a^2))
3.1.54.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.40 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83
| method | result | size |
| parallelrisch | \(\frac {777 a^{3} \left (\frac {520 \left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{777}+\frac {520 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{777}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\cos \left (d x +c \right )-\frac {174 \cos \left (2 d x +2 c \right )}{259}+\frac {239 \cos \left (3 d x +3 c \right )}{777}-\frac {152 \cos \left (4 d x +4 c \right )}{2331}-\frac {1354}{2331}\right )\right )}{80 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(137\) |
| norman | \(\frac {-\frac {a^{3}}{20 d}-\frac {17 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{30 d}-\frac {97 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 d}+\frac {131 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6 d}-\frac {51 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(154\) |
| risch | \(-\frac {i a^{3} \left (195 \,{\mathrm e}^{8 i \left (d x +c \right )}-975 \,{\mathrm e}^{7 i \left (d x +c \right )}+2275 \,{\mathrm e}^{6 i \left (d x +c \right )}-3575 \,{\mathrm e}^{5 i \left (d x +c \right )}+4329 \,{\mathrm e}^{4 i \left (d x +c \right )}-3805 \,{\mathrm e}^{3 i \left (d x +c \right )}+2673 \,{\mathrm e}^{2 i \left (d x +c \right )}-1325 \,{\mathrm e}^{i \left (d x +c \right )}+304\right )}{15 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(169\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )}{d}\) | \(241\) |
| default | \(\frac {a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )}{d}\) | \(241\) |
777/80*a^3*(520/777*(-cos(2*d*x+2*c)-1)*ln(tan(1/2*d*x+1/2*c)-1)+520/777*( 1+cos(2*d*x+2*c))*ln(tan(1/2*d*x+1/2*c)+1)+cot(1/2*d*x+1/2*c)*csc(1/2*d*x+ 1/2*c)^4*(cos(d*x+c)-174/259*cos(2*d*x+2*c)+239/777*cos(3*d*x+3*c)-152/233 1*cos(4*d*x+4*c)-1354/2331))/d/(1+cos(2*d*x+2*c))
Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {608 \, a^{3} \cos \left (d x + c\right )^{5} - 826 \, a^{3} \cos \left (d x + c\right )^{4} - 476 \, a^{3} \cos \left (d x + c\right )^{3} + 868 \, a^{3} \cos \left (d x + c\right )^{2} - 120 \, a^{3} \cos \left (d x + c\right ) - 30 \, a^{3} - 195 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 195 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
-1/60*(608*a^3*cos(d*x + c)^5 - 826*a^3*cos(d*x + c)^4 - 476*a^3*cos(d*x + c)^3 + 868*a^3*cos(d*x + c)^2 - 120*a^3*cos(d*x + c) - 30*a^3 - 195*(a^3* cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(sin(d*x + c) + 1)*sin(d*x + c) + 195*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^3 + a^ 3*cos(d*x + c)^2)*log(-sin(d*x + c) + 1)*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + d*cos(d*x + c)^2)*sin(d*x + c))
Timed out. \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.38 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^{3} {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 70 \, \sin \left (d x + c\right )^{4} - 14 \, \sin \left (d x + c\right )^{2} - 6\right )}}{\sin \left (d x + c\right )^{7} - \sin \left (d x + c\right )^{5}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a^{3} {\left (\frac {15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
-1/60*(a^3*(2*(105*sin(d*x + c)^6 - 70*sin(d*x + c)^4 - 14*sin(d*x + c)^2 - 6)/(sin(d*x + c)^7 - sin(d*x + c)^5) - 105*log(sin(d*x + c) + 1) + 105*l og(sin(d*x + c) - 1)) + 6*a^3*(2*(15*sin(d*x + c)^4 + 5*sin(d*x + c)^2 + 3 )/sin(d*x + c)^5 - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) + 36*a^3*((15*tan(d*x + c)^4 + 5*tan(d*x + c)^2 + 1)/tan(d*x + c)^5 - 5*tan( d*x + c)) + 4*(15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 + 3)*a^3/tan(d*x + c) ^5)/d
Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {390 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 390 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {60 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{60 \, d} \]
1/60*(390*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 390*a^3*log(abs(tan(1/2 *d*x + 1/2*c) - 1)) - 60*(5*a^3*tan(1/2*d*x + 1/2*c)^3 - 7*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 - (465*a^3*tan(1/2*d*x + 1/2*c)^ 4 + 40*a^3*tan(1/2*d*x + 1/2*c)^2 + 3*a^3)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 17.71 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {13\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {51\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {262\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {388\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {a^3}{5}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]