Integrand size = 14, antiderivative size = 125 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{8} b^3 c^2 \csc ^{-1}(c x)+\frac {3 b^2 \left (a+b \csc ^{-1}(c x)\right )}{4 x^2}-\frac {3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{4 x}+\frac {1}{4} c^2 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 x^2} \]
-3/8*b^3*c^2*arccsc(c*x)+3/4*b^2*(a+b*arccsc(c*x))/x^2+1/4*c^2*(a+b*arccsc (c*x))^3-1/2*(a+b*arccsc(c*x))^3/x^2+3/8*b^3*c*(1-1/c^2/x^2)^(1/2)/x-3/4*b *c*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/x
Time = 0.16 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {-4 a^3+6 a b^2-6 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+3 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+6 b \left (-2 a^2+b^2-2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)-6 b^2 \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x+a \left (2-c^2 x^2\right )\right ) \csc ^{-1}(c x)^2+2 b^3 \left (-2+c^2 x^2\right ) \csc ^{-1}(c x)^3-3 b \left (-2 a^2+b^2\right ) c^2 x^2 \arcsin \left (\frac {1}{c x}\right )}{8 x^2} \]
(-4*a^3 + 6*a*b^2 - 6*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b^3*c*Sqrt[1 - 1 /(c^2*x^2)]*x + 6*b*(-2*a^2 + b^2 - 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCs c[c*x] - 6*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x + a*(2 - c^2*x^2))*ArcCsc[c*x] ^2 + 2*b^3*(-2 + c^2*x^2)*ArcCsc[c*x]^3 - 3*b*(-2*a^2 + b^2)*c^2*x^2*ArcSi n[1/(c*x)])/(8*x^2)
Time = 0.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5746, 4904, 3042, 3792, 17, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 5746 |
| \(\displaystyle -c^2 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^3}{c x}d\csc ^{-1}(c x)\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{c^2 x^2}d\csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \left (\frac {1}{2} \int \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\csc ^{-1}(c x)+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}\right )\right )\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \left (-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \left (-\frac {1}{2} b^2 \int \sin \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \left (-\frac {1}{2} b^2 \left (\frac {1}{2} \int 1d\csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -c^2 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{2 c^2 x^2}-\frac {3}{2} b \left (-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \csc ^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{6 b}-\frac {1}{2} b^2 \left (\frac {1}{2} \csc ^{-1}(c x)-\frac {\sqrt {1-\frac {1}{c^2 x^2}}}{2 c x}\right )\right )\right )\) |
-(c^2*((a + b*ArcCsc[c*x])^3/(2*c^2*x^2) - (3*b*(-1/2*(b^2*(-1/2*Sqrt[1 - 1/(c^2*x^2)]/(c*x) + ArcCsc[c*x]/2)) + (b*(a + b*ArcCsc[c*x]))/(2*c^2*x^2) - (Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(2*c*x) + (a + b*ArcCsc[c *x])^3/(6*b)))/2))
3.1.30.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x _)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) , x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n , 0] || LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(109)=218\).
Time = 1.07 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.54
| method | result | size |
| derivativedivides | \(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{3}}{2 c^{2} x^{2}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{4 c x}-\frac {3 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 c x}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{8}+\frac {\operatorname {arccsc}\left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right )\) | \(318\) |
| default | \(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{3}}{2 c^{2} x^{2}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{4 c x}-\frac {3 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 c x}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{8}+\frac {\operatorname {arccsc}\left (c x \right )^{3}}{2}\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}\right )\right )\) | \(318\) |
| parts | \(-\frac {a^{3}}{2 x^{2}}+b^{3} c^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{3}}{2 c^{2} x^{2}}-\frac {3 \operatorname {arccsc}\left (c x \right )^{2} \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{4 c x}-\frac {3 \left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{8 c x}+\frac {3 \,\operatorname {arccsc}\left (c x \right )}{8}+\frac {\operatorname {arccsc}\left (c x \right )^{3}}{2}\right )+3 a \,b^{2} c^{2} \left (\frac {\left (c^{2} x^{2}-1\right ) \operatorname {arccsc}\left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\operatorname {arccsc}\left (c x \right ) \left (\operatorname {arccsc}\left (c x \right ) c x +\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{2 c x}+\frac {\operatorname {arccsc}\left (c x \right )^{2}}{4}+\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-\sqrt {c^{2} x^{2}-1}\right )}{4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} c^{3}}\right )\) | \(321\) |
c^2*(-1/2*a^3/c^2/x^2+b^3*(1/2*(c^2*x^2-1)/c^2/x^2*arccsc(c*x)^3-3/4*arccs c(c*x)^2*(arccsc(c*x)*c*x+((c^2*x^2-1)/c^2/x^2)^(1/2))/c/x-3/4*(c^2*x^2-1) /c^2/x^2*arccsc(c*x)+3/8/c/x*((c^2*x^2-1)/c^2/x^2)^(1/2)+3/8*arccsc(c*x)+1 /2*arccsc(c*x)^3)+3*a*b^2*(1/2*(c^2*x^2-1)/c^2/x^2*arccsc(c*x)^2-1/2*arccs c(c*x)*(arccsc(c*x)*c*x+((c^2*x^2-1)/c^2/x^2)^(1/2))/c/x+1/4*arccsc(c*x)^2 +1/4/c^2/x^2)+3*a^2*b*(-1/2/c^2/x^2*arccsc(c*x)-1/4*(c^2*x^2-1)^(1/2)*(-ar ctan(1/(c^2*x^2-1)^(1/2))*c^2*x^2+(c^2*x^2-1)^(1/2))/((c^2*x^2-1)/c^2/x^2) ^(1/2)/c^3/x^3))
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{3} - 4 \, a^{3} + 6 \, a b^{2} + 6 \, {\left (a b^{2} c^{2} x^{2} - 2 \, a b^{2}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, {\left ({\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 4 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, {\left (2 \, b^{3} \operatorname {arccsc}\left (c x\right )^{2} + 4 \, a b^{2} \operatorname {arccsc}\left (c x\right ) + 2 \, a^{2} b - b^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, x^{2}} \]
1/8*(2*(b^3*c^2*x^2 - 2*b^3)*arccsc(c*x)^3 - 4*a^3 + 6*a*b^2 + 6*(a*b^2*c^ 2*x^2 - 2*a*b^2)*arccsc(c*x)^2 + 3*((2*a^2*b - b^3)*c^2*x^2 - 4*a^2*b + 2* b^3)*arccsc(c*x) - 3*(2*b^3*arccsc(c*x)^2 + 4*a*b^2*arccsc(c*x) + 2*a^2*b - b^3)*sqrt(c^2*x^2 - 1))/x^2
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \]
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
3/4*a^2*b*((c^4*x*sqrt(-1/(c^2*x^2) + 1)/(c^2*x^2*(1/(c^2*x^2) - 1) - 1) - c^3*arctan(c*x*sqrt(-1/(c^2*x^2) + 1)))/c - 2*arccsc(c*x)/x^2) - 1/2*a^3/ x^2 - 1/8*(4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*b^3*arctan2 (1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 + 12*(a*b^2*c^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log(c)^2 + 16*b^3*c^2*integrate(1/8*x^2*arc tan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^5 - x^3), x)*log(c)^2 - 16*b^3 *c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2 )/(c^2*x^5 - x^3), x)*log(c) + 32*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt (c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^5 - x^3), x)*log(c) - 16*a*b^2*c^2 *integrate(1/8*x^2*log(c^2*x^2)/(c^2*x^5 - x^3), x)*log(c) + 32*a*b^2*c^2* integrate(1/8*x^2*log(x)/(c^2*x^5 - x^3), x)*log(c) - 16*b^3*c^2*integrate (1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(c^2* x^5 - x^3), x) + 16*b^3*c^2*integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt (c*x - 1)))*log(x)^2/(c^2*x^5 - x^3), x) - 16*a*b^2*c^2*integrate(1/8*x^2* arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^5 - x^3), x) + 8*b^3*c^2* integrate(1/8*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^ 2*x^5 - x^3), x) + 4*a*b^2*c^2*integrate(1/8*x^2*log(c^2*x^2)^2/(c^2*x^5 - x^3), x) - 16*a*b^2*c^2*integrate(1/8*x^2*log(c^2*x^2)*log(x)/(c^2*x^5 - x^3), x) + 16*a*b^2*c^2*integrate(1/8*x^2*log(x)^2/(c^2*x^5 - x^3), x) - ( c^2*log(c*x + 1) + c^2*log(c*x - 1) - 2*c^2*log(x) + 1/x^2)*a*b^2*log(c...
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (109) = 218\).
Time = 0.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.42 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=-\frac {1}{8} \, {\left (4 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3} + 12 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2} + 2 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )^{3} + 12 \, a^{2} b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) - 6 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 6 \, a b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2} + 4 \, a^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 6 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 6 \, a^{2} b c \arcsin \left (\frac {1}{c x}\right ) - 3 \, b^{3} c \arcsin \left (\frac {1}{c x}\right ) + \frac {6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - 3 \, a b^{2} c + \frac {12 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {6 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} - \frac {3 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x}\right )} c \]
-1/8*(4*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^3 + 12*a*b^2*c*(1/(c^2*x^2 ) - 1)*arcsin(1/(c*x))^2 + 2*b^3*c*arcsin(1/(c*x))^3 + 12*a^2*b*c*(1/(c^2* x^2) - 1)*arcsin(1/(c*x)) - 6*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) + 6* a*b^2*c*arcsin(1/(c*x))^2 + 4*a^3*c*(1/(c^2*x^2) - 1) - 6*a*b^2*c*(1/(c^2* x^2) - 1) + 6*a^2*b*c*arcsin(1/(c*x)) - 3*b^3*c*arcsin(1/(c*x)) + 6*b^3*sq rt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))^2/x - 3*a*b^2*c + 12*a*b^2*sqrt(-1/(c ^2*x^2) + 1)*arcsin(1/(c*x))/x + 6*a^2*b*sqrt(-1/(c^2*x^2) + 1)/x - 3*b^3* sqrt(-1/(c^2*x^2) + 1)/x)*c
Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x^3} \,d x \]