\(\int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx\) [56]

Optimal result

Integrand size = 21, antiderivative size = 232 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {17 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {16 a^3 \cot (c+d x)}{d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \csc (c+d x)}{2 d}-\frac {17 a^3 \csc ^3(c+d x)}{6 d}-\frac {17 a^3 \csc ^5(c+d x)}{10 d}-\frac {17 a^3 \csc ^7(c+d x)}{14 d}-\frac {17 a^3 \csc ^9(c+d x)}{18 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d} \]

[Out]

17/2*a^3*arctanh(sin(d*x+c))/d-16*a^3*cot(d*x+c)/d-34/3*a^3*cot(d*x+c)^3/d-36/5*a^3*cot(d*x+c)^5/d-19/7*a^3*co
t(d*x+c)^7/d-4/9*a^3*cot(d*x+c)^9/d-17/2*a^3*csc(d*x+c)/d-17/6*a^3*csc(d*x+c)^3/d-17/10*a^3*csc(d*x+c)^5/d-17/
14*a^3*csc(d*x+c)^7/d-17/18*a^3*csc(d*x+c)^9/d+1/2*a^3*csc(d*x+c)^9*sec(d*x+c)^2/d+3*a^3*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2952, 3852, 2701, 308, 213, 2700, 276, 294} \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {17 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {16 a^3 \cot (c+d x)}{d}-\frac {17 a^3 \csc ^9(c+d x)}{18 d}-\frac {17 a^3 \csc ^7(c+d x)}{14 d}-\frac {17 a^3 \csc ^5(c+d x)}{10 d}-\frac {17 a^3 \csc ^3(c+d x)}{6 d}-\frac {17 a^3 \csc (c+d x)}{2 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d} \]

[In]

Int[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]

[Out]

(17*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (16*a^3*Cot[c + d*x])/d - (34*a^3*Cot[c + d*x]^3)/(3*d) - (36*a^3*Cot[c
+ d*x]^5)/(5*d) - (19*a^3*Cot[c + d*x]^7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Csc[c + d*x])/(2*d)
- (17*a^3*Csc[c + d*x]^3)/(6*d) - (17*a^3*Csc[c + d*x]^5)/(10*d) - (17*a^3*Csc[c + d*x]^7)/(14*d) - (17*a^3*Cs
c[c + d*x]^9)/(18*d) + (a^3*Csc[c + d*x]^9*Sec[c + d*x]^2)/(2*d) + (3*a^3*Tan[c + d*x])/d

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, 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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^{10}(c+d x) \sec ^3(c+d x) \, dx \\ & = \int \left (a^3 \csc ^{10}(c+d x)+3 a^3 \csc ^{10}(c+d x) \sec (c+d x)+3 a^3 \csc ^{10}(c+d x) \sec ^2(c+d x)+a^3 \csc ^{10}(c+d x) \sec ^3(c+d x)\right ) \, dx \\ & = a^3 \int \csc ^{10}(c+d x) \, dx+a^3 \int \csc ^{10}(c+d x) \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^{10}(c+d x) \sec (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^{10}(c+d x) \sec ^2(c+d x) \, dx \\ & = -\frac {a^3 \text {Subst}\left (\int \frac {x^{12}}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {a^3 \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^{10}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {6 a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^{10}}+\frac {5}{x^8}+\frac {10}{x^6}+\frac {10}{x^4}+\frac {5}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1+x^2+x^4+x^6+x^8+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (11 a^3\right ) \text {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = -\frac {16 a^3 \cot (c+d x)}{d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^3(c+d x)}{d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {3 a^3 \csc ^7(c+d x)}{7 d}-\frac {a^3 \csc ^9(c+d x)}{3 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (11 a^3\right ) \text {Subst}\left (\int \left (1+x^2+x^4+x^6+x^8+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = \frac {3 a^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {16 a^3 \cot (c+d x)}{d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \csc (c+d x)}{2 d}-\frac {17 a^3 \csc ^3(c+d x)}{6 d}-\frac {17 a^3 \csc ^5(c+d x)}{10 d}-\frac {17 a^3 \csc ^7(c+d x)}{14 d}-\frac {17 a^3 \csc ^9(c+d x)}{18 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {\left (11 a^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = \frac {17 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {16 a^3 \cot (c+d x)}{d}-\frac {34 a^3 \cot ^3(c+d x)}{3 d}-\frac {36 a^3 \cot ^5(c+d x)}{5 d}-\frac {19 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {17 a^3 \csc (c+d x)}{2 d}-\frac {17 a^3 \csc ^3(c+d x)}{6 d}-\frac {17 a^3 \csc ^5(c+d x)}{10 d}-\frac {17 a^3 \csc ^7(c+d x)}{14 d}-\frac {17 a^3 \csc ^9(c+d x)}{18 d}+\frac {a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1000\) vs. \(2(232)=464\).

Time = 12.60 (sec) , antiderivative size = 1000, normalized size of antiderivative = 4.31 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {9833 \cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{80640 d}-\frac {979 \cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{53760 d}-\frac {5 \cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{2016 d}-\frac {\cos ^3(c+d x) \cot \left (\frac {c}{2}\right ) \csc ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{4608 d}-\frac {17 \cos ^3(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{16 d}+\frac {17 \cos ^3(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3}{16 d}+\frac {197147 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{161280 d}+\frac {9833 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{80640 d}+\frac {979 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{53760 d}+\frac {5 \cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{2016 d}+\frac {\cos ^3(c+d x) \csc \left (\frac {c}{2}\right ) \csc ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{4608 d}-\frac {35 \cos ^3(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{1536 d}-\frac {\cos ^3(c+d x) \sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin \left (\frac {d x}{2}\right )}{1536 d}+\frac {\cos (c+d x) \sec (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \sin (d x)}{16 d}+\frac {\cos ^2(c+d x) \sec (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 (\sin (c)+6 \sin (d x))}{16 d}-\frac {\cos ^3(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \tan \left (\frac {c}{2}\right )}{1536 d} \]

[In]

Integrate[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]

[Out]

(-9833*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^2*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(80640*d) - (
979*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(53760*d) - (5*C
os[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^6*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(2016*d) - (Cos[c + d
*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^8*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(4608*d) - (17*Cos[c + d*x]^3
*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(16*d) + (17*Cos[c
+ d*x]^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(16*d) + (1
97147*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(16
1280*d) + (9833*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(
d*x)/2])/(80640*d) + (979*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x
])^3*Sin[(d*x)/2])/(53760*d) + (5*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^7*Sec[c/2 + (d*x)/2]^6*(a + a*Sec
[c + d*x])^3*Sin[(d*x)/2])/(2016*d) + (Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^9*Sec[c/2 + (d*x)/2]^6*(a +
a*Sec[c + d*x])^3*Sin[(d*x)/2])/(4608*d) - (35*Cos[c + d*x]^3*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(a + a*Sec[c + d*x
])^3*Sin[(d*x)/2])/(1536*d) - (Cos[c + d*x]^3*Sec[c/2]*Sec[c/2 + (d*x)/2]^9*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2
])/(1536*d) + (Cos[c + d*x]*Sec[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[d*x])/(16*d) + (Cos[c + d*x
]^2*Sec[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(Sin[c] + 6*Sin[d*x]))/(16*d) - (Cos[c + d*x]^3*Sec[c/2
+ (d*x)/2]^8*(a + a*Sec[c + d*x])^3*Tan[c/2])/(1536*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.75 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {i a^{3} \left (5355 \,{\mathrm e}^{15 i \left (d x +c \right )}-32130 \,{\mathrm e}^{14 i \left (d x +c \right )}+73185 \,{\mathrm e}^{13 i \left (d x +c \right )}-64260 \,{\mathrm e}^{12 i \left (d x +c \right )}-34629 \,{\mathrm e}^{11 i \left (d x +c \right )}+157794 \,{\mathrm e}^{10 i \left (d x +c \right )}-207111 \,{\mathrm e}^{9 i \left (d x +c \right )}+125256 \,{\mathrm e}^{8 i \left (d x +c \right )}+62713 \,{\mathrm e}^{7 i \left (d x +c \right )}-175518 \,{\mathrm e}^{6 i \left (d x +c \right )}+171707 \,{\mathrm e}^{5 i \left (d x +c \right )}-80132 \,{\mathrm e}^{4 i \left (d x +c \right )}-37919 \,{\mathrm e}^{3 i \left (d x +c \right )}+78974 \,{\mathrm e}^{2 i \left (d x +c \right )}-42261 \,{\mathrm e}^{i \left (d x +c \right )}+7936\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\)	\(259\)
parallelrisch	\(-\frac {1673 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\frac {31314}{11711}+\frac {765 \left (\frac {\sin \left (8 d x +8 c \right )}{14}-\frac {3 \sin \left (7 d x +7 c \right )}{7}-\sin \left (5 d x +5 c \right )+\sin \left (6 d x +6 c \right )-\frac {\sin \left (4 d x +4 c \right )}{7}+\frac {13 \sin \left (3 d x +3 c \right )}{7}+\frac {17 \sin \left (d x +c \right )}{7}-3 \sin \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{478}+\frac {765 \left (\frac {3 \sin \left (7 d x +7 c \right )}{7}-\frac {\sin \left (8 d x +8 c \right )}{14}+\frac {\sin \left (4 d x +4 c \right )}{7}-\frac {13 \sin \left (3 d x +3 c \right )}{7}-\sin \left (6 d x +6 c \right )+\sin \left (5 d x +5 c \right )-\frac {17 \sin \left (d x +c \right )}{7}+3 \sin \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{478}-\frac {18453 \cos \left (7 d x +7 c \right )}{23422}+\cos \left (6 d x +6 c \right )-\frac {36098 \cos \left (4 d x +4 c \right )}{11711}-\frac {72199 \cos \left (d x +c \right )}{23422}-\frac {633 \cos \left (2 d x +2 c \right )}{1673}+\frac {2519 \cos \left (5 d x +5 c \right )}{3346}+\frac {68539 \cos \left (3 d x +3 c \right )}{23422}+\frac {1984 \cos \left (8 d x +8 c \right )}{11711}\right ) a^{3} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{92160 d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(325\)
derivativedivides	\(\frac {a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )^{2}}-\frac {11}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{2}}-\frac {11}{35 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {11}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {11}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {11}{2 \sin \left (d x +c \right )}+\frac {11 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16}{63 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32}{63 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128}{63 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {256 \cot \left (d x +c \right )}{63}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\)	\(353\)
default	\(\frac {a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )^{2}}-\frac {11}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )^{2}}-\frac {11}{35 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {11}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {11}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {11}{2 \sin \left (d x +c \right )}+\frac {11 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10}{63 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16}{63 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32}{63 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128}{63 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {256 \cot \left (d x +c \right )}{63}\right )+3 a^{3} \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\)	\(353\)

[In]

int(csc(d*x+c)^10*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/315*I*a^3*(5355*exp(15*I*(d*x+c))-32130*exp(14*I*(d*x+c))+73185*exp(13*I*(d*x+c))-64260*exp(12*I*(d*x+c))-3
4629*exp(11*I*(d*x+c))+157794*exp(10*I*(d*x+c))-207111*exp(9*I*(d*x+c))+125256*exp(8*I*(d*x+c))+62713*exp(7*I*
(d*x+c))-175518*exp(6*I*(d*x+c))+171707*exp(5*I*(d*x+c))-80132*exp(4*I*(d*x+c))-37919*exp(3*I*(d*x+c))+78974*e
xp(2*I*(d*x+c))-42261*exp(I*(d*x+c))+7936)/d/(exp(I*(d*x+c))-1)^9/(exp(I*(d*x+c))+1)^3/(exp(2*I*(d*x+c))+1)^2-
17/2*a^3/d*ln(exp(I*(d*x+c))-I)+17/2*a^3/d*ln(exp(I*(d*x+c))+I)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.62 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {15872 \, a^{3} \cos \left (d x + c\right )^{8} - 36906 \, a^{3} \cos \left (d x + c\right )^{7} - 8322 \, a^{3} \cos \left (d x + c\right )^{6} + 73402 \, a^{3} \cos \left (d x + c\right )^{5} - 33342 \, a^{3} \cos \left (d x + c\right )^{4} - 34746 \, a^{3} \cos \left (d x + c\right )^{3} + 26702 \, a^{3} \cos \left (d x + c\right )^{2} - 1890 \, a^{3} \cos \left (d x + c\right ) - 630 \, a^{3} - 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 5355 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{1260 \, {\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + 2 \, d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]

[In]

integrate(csc(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/1260*(15872*a^3*cos(d*x + c)^8 - 36906*a^3*cos(d*x + c)^7 - 8322*a^3*cos(d*x + c)^6 + 73402*a^3*cos(d*x + c
)^5 - 33342*a^3*cos(d*x + c)^4 - 34746*a^3*cos(d*x + c)^3 + 26702*a^3*cos(d*x + c)^2 - 1890*a^3*cos(d*x + c) -
630*a^3 - 5355*(a^3*cos(d*x + c)^7 - 3*a^3*cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*a
^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2)*log(sin(d*x + c) + 1)*sin(d*x + c) + 5355*(a^3*cos(d*x + c)^7 - 3*a^3*
cos(d*x + c)^6 + 2*a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 3*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)^7 - 3*d*cos(d*x + c)^6 + 2*d*cos(d*x + c)^5 + 2*d*cos(d*x +
c)^4 - 3*d*cos(d*x + c)^3 + d*cos(d*x + c)^2)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**10*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.33 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^{3} {\left (\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{10} - 2310 \, \sin \left (d x + c\right )^{8} - 462 \, \sin \left (d x + c\right )^{6} - 198 \, \sin \left (d x + c\right )^{4} - 110 \, \sin \left (d x + c\right )^{2} - 70\right )}}{\sin \left (d x + c\right )^{11} - \sin \left (d x + c\right )^{9}} - 3465 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3465 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{6} + 63 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{2} + 35\right )}}{\sin \left (d x + c\right )^{9}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 60 \, a^{3} {\left (\frac {315 \, \tan \left (d x + c\right )^{8} + 210 \, \tan \left (d x + c\right )^{6} + 126 \, \tan \left (d x + c\right )^{4} + 45 \, \tan \left (d x + c\right )^{2} + 7}{\tan \left (d x + c\right )^{9}} - 63 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (315 \, \tan \left (d x + c\right )^{8} + 420 \, \tan \left (d x + c\right )^{6} + 378 \, \tan \left (d x + c\right )^{4} + 180 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{1260 \, d} \]

[In]

integrate(csc(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/1260*(a^3*(2*(3465*sin(d*x + c)^10 - 2310*sin(d*x + c)^8 - 462*sin(d*x + c)^6 - 198*sin(d*x + c)^4 - 110*si
n(d*x + c)^2 - 70)/(sin(d*x + c)^11 - sin(d*x + c)^9) - 3465*log(sin(d*x + c) + 1) + 3465*log(sin(d*x + c) - 1
)) + 6*a^3*(2*(315*sin(d*x + c)^8 + 105*sin(d*x + c)^6 + 63*sin(d*x + c)^4 + 45*sin(d*x + c)^2 + 35)/sin(d*x +
c)^9 - 315*log(sin(d*x + c) + 1) + 315*log(sin(d*x + c) - 1)) + 60*a^3*((315*tan(d*x + c)^8 + 210*tan(d*x + c
)^6 + 126*tan(d*x + c)^4 + 45*tan(d*x + c)^2 + 7)/tan(d*x + c)^9 - 63*tan(d*x + c)) + 4*(315*tan(d*x + c)^8 +
420*tan(d*x + c)^6 + 378*tan(d*x + c)^4 + 180*tan(d*x + c)^2 + 35)*a^3/tan(d*x + c)^9)/d

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.87 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 171360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 171360 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3780 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {20160 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {220185 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 26880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4347 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 540 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{20160 \, d} \]

[In]

integrate(csc(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/20160*(105*a^3*tan(1/2*d*x + 1/2*c)^3 - 171360*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 171360*a^3*log(abs(
tan(1/2*d*x + 1/2*c) - 1)) + 3780*a^3*tan(1/2*d*x + 1/2*c) + 20160*(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 + (220185*a^3*tan(1/2*d*x + 1/2*c)^8 + 26880*a^3*tan(1/2*d*x +
1/2*c)^6 + 4347*a^3*tan(1/2*d*x + 1/2*c)^4 + 540*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3)/tan(1/2*d*x + 1/2*c)^9)
/d

Mupad [B] (verification not implemented)

Time = 14.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {17\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {1019\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {5282\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+\frac {8132\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{15}+\frac {6242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{105}+\frac {3302\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{315}+\frac {94\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}+\frac {a^3}{9}}{d\,\left (64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )} \]

[In]

int((a + a/cos(c + d*x))^3/sin(c + d*x)^10,x)

[Out]

(17*a^3*atanh(tan(c/2 + (d*x)/2)))/d - (a^3*tan(c/2 + (d*x)/2)^3)/(192*d) - (3*a^3*tan(c/2 + (d*x)/2))/(16*d)
- ((94*a^3*tan(c/2 + (d*x)/2)^2)/63 + (3302*a^3*tan(c/2 + (d*x)/2)^4)/315 + (6242*a^3*tan(c/2 + (d*x)/2)^6)/10
5 + (8132*a^3*tan(c/2 + (d*x)/2)^8)/15 - (5282*a^3*tan(c/2 + (d*x)/2)^10)/3 + 1019*a^3*tan(c/2 + (d*x)/2)^12 +
a^3/9)/(d*(64*tan(c/2 + (d*x)/2)^9 - 128*tan(c/2 + (d*x)/2)^11 + 64*tan(c/2 + (d*x)/2)^13))