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

Optimal. Leaf size=232 \[ \frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{19 a^3 \cot ^7(c+d x)}{7 d}-\frac{36 a^3 \cot ^5(c+d x)}{5 d}-\frac{34 a^3 \cot ^3(c+d x)}{3 d}-\frac{16 a^3 \cot (c+d x)}{d}-\frac{17 a^3 \csc ^9(c+d x)}{18 d}-\frac{17 a^3 \csc ^7(c+d x)}{14 d}-\frac{17 a^3 \csc ^5(c+d x)}{10 d}-\frac{17 a^3 \csc ^3(c+d x)}{6 d}-\frac{17 a^3 \csc (c+d x)}{2 d}+\frac{17 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d} \]

[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

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Rubi [A]  time = 0.331666, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3872, 2873, 3767, 2621, 302, 207, 2620, 270, 288} \[ \frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{19 a^3 \cot ^7(c+d x)}{7 d}-\frac{36 a^3 \cot ^5(c+d x)}{5 d}-\frac{34 a^3 \cot ^3(c+d x)}{3 d}-\frac{16 a^3 \cot (c+d x)}{d}-\frac{17 a^3 \csc ^9(c+d x)}{18 d}-\frac{17 a^3 \csc ^7(c+d x)}{14 d}-\frac{17 a^3 \csc ^5(c+d x)}{10 d}-\frac{17 a^3 \csc ^3(c+d x)}{6 d}-\frac{17 a^3 \csc (c+d x)}{2 d}+\frac{17 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2873

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 3767

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 2621

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 302

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 207

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

Rule 2620

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 270

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 288

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]

Rubi steps

\begin{align*} \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\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 \operatorname{Subst}\left (\int \frac{x^{12}}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{a^3 \operatorname{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 ) \operatorname{Subst}\left (\int \frac{x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (11 a^3\right ) \operatorname{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 \tanh ^{-1}(\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 ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=\frac{17 a^3 \tanh ^{-1}(\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]  time = 6.68008, size = 1000, normalized size = 4.31 \[ \frac{\cos ^3(c+d x) \csc \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \csc ^9\left (\frac{c}{2}+\frac{d x}{2}\right )}{4608 d}-\frac{\cos ^3(c+d x) \cot \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \csc ^8\left (\frac{c}{2}+\frac{d x}{2}\right )}{4608 d}+\frac{5 \cos ^3(c+d x) \csc \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \csc ^7\left (\frac{c}{2}+\frac{d x}{2}\right )}{2016 d}-\frac{5 \cos ^3(c+d x) \cot \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \csc ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{2016 d}+\frac{979 \cos ^3(c+d x) \csc \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \csc ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{53760 d}-\frac{979 \cos ^3(c+d x) \cot \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \csc ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{53760 d}+\frac{9833 \cos ^3(c+d x) \csc \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \csc ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{80640 d}-\frac{9833 \cos ^3(c+d x) \cot \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \csc ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{80640 d}+\frac{197147 \cos ^3(c+d x) \csc \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}\right )}{161280 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 ) (\sec (c+d x) a+a)^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 ) (\sec (c+d x) a+a)^3}{16 d}-\frac{\cos ^3(c+d x) \sec \left (\frac{c}{2}\right ) \sec ^9\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^3 \sin \left (\frac{d x}{2}\right )}{1536 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 ) (\sec (c+d x) a+a)^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 ) (\sec (c+d x) a+a)^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 ) (\sec (c+d x) a+a)^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 ) (\sec (c+d x) a+a)^3 \tan \left (\frac{c}{2}\right )}{1536 d} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [B]  time = 0.086, size = 446, normalized size = 1.9 \begin{align*} -{\frac{3968\,{a}^{3}\cot \left ( dx+c \right ) }{315\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{8}}{9\,d}}-{\frac{8\,{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{6}}{63\,d}}-{\frac{16\,{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{105\,d}}-{\frac{64\,{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{315\,d}}-{\frac{{a}^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{3\,{a}^{3}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{3\,{a}^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{3}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{17\,{a}^{3}}{2\,d\sin \left ( dx+c \right ) }}+{\frac{17\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}-{\frac{{a}^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}\cos \left ( dx+c \right ) }}-{\frac{10\,{a}^{3}}{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) }}-{\frac{16\,{a}^{3}}{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) }}-{\frac{32\,{a}^{3}}{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) }}+{\frac{128\,{a}^{3}}{21\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-{\frac{{a}^{3}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{11\,{a}^{3}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{11\,{a}^{3}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{11\,{a}^{3}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{11\,{a}^{3}}{6\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3968/315*a^3*cot(d*x+c)/d-1/9/d*a^3*cot(d*x+c)*csc(d*x+c)^8-8/63/d*a^3*cot(d*x+c)*csc(d*x+c)^6-16/105/d*a^3*c
ot(d*x+c)*csc(d*x+c)^4-64/315/d*a^3*cot(d*x+c)*csc(d*x+c)^2-1/3/d*a^3/sin(d*x+c)^9-3/7/d*a^3/sin(d*x+c)^7-3/5/
d*a^3/sin(d*x+c)^5-1/d*a^3/sin(d*x+c)^3-17/2/d*a^3/sin(d*x+c)+17/2/d*a^3*ln(sec(d*x+c)+tan(d*x+c))-1/3/d*a^3/s
in(d*x+c)^9/cos(d*x+c)-10/21/d*a^3/sin(d*x+c)^7/cos(d*x+c)-16/21/d*a^3/sin(d*x+c)^5/cos(d*x+c)-32/21/d*a^3/sin
(d*x+c)^3/cos(d*x+c)+128/21/d*a^3/sin(d*x+c)/cos(d*x+c)-1/9/d*a^3/sin(d*x+c)^9/cos(d*x+c)^2-11/63/d*a^3/sin(d*
x+c)^7/cos(d*x+c)^2-11/35/d*a^3/sin(d*x+c)^5/cos(d*x+c)^2-11/15/d*a^3/sin(d*x+c)^3/cos(d*x+c)^2+11/6/d*a^3/sin
(d*x+c)/cos(d*x+c)^2

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Maxima [A]  time = 1.02781, size = 416, normalized size = 1.79 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.92144, size = 973, normalized size = 4.19 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.41781, size = 273, normalized size = 1.18 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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