Integrand size = 14, antiderivative size = 170 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3} \]
-2/27*b^3*c^3*(1-1/c^2/x^2)^(3/2)+2/9*b^2*(a+b*arccsc(c*x))/x^3+4/3*b^2*c^ 2*(a+b*arccsc(c*x))/x-1/3*(a+b*arccsc(c*x))^3/x^3+14/9*b^3*c^3*(1-1/c^2/x^ 2)^(1/2)-2/3*b*c^3*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)-1/3*b*c*(a+b*ar ccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/x^2
Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {-9 a^3-9 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+6 a b^2 \left (1+6 c^2 x^2\right )+2 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+20 c^2 x^2\right )+3 b \left (-9 a^2-6 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+2 b^2 \left (1+6 c^2 x^2\right )\right ) \csc ^{-1}(c x)-9 b^2 \left (3 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )\right ) \csc ^{-1}(c x)^2-9 b^3 \csc ^{-1}(c x)^3}{27 x^3} \]
(-9*a^3 - 9*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 6*a*b^2*(1 + 6*c^2*x^2) + 2*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 20*c^2*x^2) + 3*b*(-9*a ^2 - 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 2*b^2*(1 + 6*c^2*x^ 2))*ArcCsc[c*x] - 9*b^2*(3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) )*ArcCsc[c*x]^2 - 9*b^3*ArcCsc[c*x]^3)/(27*x^3)
Time = 0.72 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5746, 4904, 3042, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 3118}
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.
| \(\displaystyle \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx\) |
| \(\Big \downarrow \) 5746 |
| \(\displaystyle -c^3 \int \frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^3}{c^2 x^2}d\csc ^{-1}(c x)\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{c^3 x^3}d\csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{c x}d\csc ^{-1}(c x)-\frac {2}{9} b^2 \int \frac {1}{c^3 x^3}d\csc ^{-1}(c x)+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\frac {2}{9} b^2 \int \sin \left (\csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)+\frac {2}{9} b^2 \int \frac {1}{c^2 x^2}d\sqrt {1-\frac {1}{c^2 x^2}}+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (2 b \int \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (2 b \int \left (a+b \csc ^{-1}(c x)\right ) \sin \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (2 b \left (b \int -\frac {1}{c x}d\csc ^{-1}(c x)+\frac {a+b \csc ^{-1}(c x)}{c x}\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (2 b \left (\frac {a+b \csc ^{-1}(c x)}{c x}-b \int \frac {1}{c x}d\csc ^{-1}(c x)\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2}{3} \left (2 b \left (\frac {a+b \csc ^{-1}(c x)}{c x}-b \int \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )+\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -c^3 \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 c^3 x^3}-b \left (\frac {2 b \left (a+b \csc ^{-1}(c x)\right )}{9 c^3 x^3}-\frac {\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 c^2 x^2}+\frac {2}{3} \left (2 b \left (\frac {a+b \csc ^{-1}(c x)}{c x}+b \sqrt {1-\frac {1}{c^2 x^2}}\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )+\frac {2}{9} b^2 \left (\sqrt {1-\frac {1}{c^2 x^2}}-\frac {1}{3} \left (1-\frac {1}{c^2 x^2}\right )^{3/2}\right )\right )\right )\) |
-(c^3*((a + b*ArcCsc[c*x])^3/(3*c^3*x^3) - b*((2*b^2*(Sqrt[1 - 1/(c^2*x^2) ] - (1 - 1/(c^2*x^2))^(3/2)/3))/9 + (2*b*(a + b*ArcCsc[c*x]))/(9*c^3*x^3) - (Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(3*c^2*x^2) + (2*(-(Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2) + 2*b*(b*Sqrt[1 - 1/(c^2*x^2)] + (a + b*ArcCsc[c*x])/(c*x))))/3)))
3.1.31.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
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. \(298\) vs. \(2(148)=296\).
Time = 1.51 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.76
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{3 c^{3} x^{3}}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}+\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arccsc}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arccsc}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(299\) |
| default | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{3 c^{3} x^{3}}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}+\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arccsc}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arccsc}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(299\) |
| parts | \(-\frac {a^{3}}{3 x^{3}}+b^{3} c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{3 c^{3} x^{3}}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}+\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\operatorname {arccsc}\left (c x \right )}{3 c x}+\frac {2 \,\operatorname {arccsc}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \,c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\) | \(301\) |
c^3*(-1/3*a^3/c^3/x^3+b^3*(-1/3/c^3/x^3*arccsc(c*x)^3-1/3*arccsc(c*x)^2*(2 *c^2*x^2+1)/c^2/x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)+4/3*((c^2*x^2-1)/c^2/x^2)^ (1/2)+4/3/c/x*arccsc(c*x)+2/9/c^3/x^3*arccsc(c*x)+2/27*(2*c^2*x^2+1)/c^2/x ^2*((c^2*x^2-1)/c^2/x^2)^(1/2))+3*a*b^2*(-1/3/c^3/x^3*arccsc(c*x)^2-2/9*ar ccsc(c*x)*(2*c^2*x^2+1)/c^2/x^2*((c^2*x^2-1)/c^2/x^2)^(1/2)+2/27/c^3/x^3+4 /9/c/x)+3*a^2*b*(-1/3/c^3/x^3*arccsc(c*x)-1/9*(c^2*x^2-1)*(2*c^2*x^2+1)/(( c^2*x^2-1)/c^2/x^2)^(1/2)/c^4/x^4))
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \operatorname {arccsc}\left (c x\right )^{3} - 27 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2} - 9 \, a^{3} + 6 \, a b^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (2 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} c^{2} x^{2} + 9 \, a^{2} b - 2 \, b^{3} + 9 \, {\left (2 \, b^{3} c^{2} x^{2} + b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 18 \, {\left (2 \, a b^{2} c^{2} x^{2} + a b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, x^{3}} \]
1/27*(36*a*b^2*c^2*x^2 - 9*b^3*arccsc(c*x)^3 - 27*a*b^2*arccsc(c*x)^2 - 9* a^3 + 6*a*b^2 + 3*(12*b^3*c^2*x^2 - 9*a^2*b + 2*b^3)*arccsc(c*x) - (2*(9*a ^2*b - 20*b^3)*c^2*x^2 + 9*a^2*b - 2*b^3 + 9*(2*b^3*c^2*x^2 + b^3)*arccsc( c*x)^2 + 18*(2*a*b^2*c^2*x^2 + a*b^2)*arccsc(c*x))*sqrt(c^2*x^2 - 1))/x^3
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \]
1/3*a^2*b*((c^4*(-1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))/c - 3*arccsc(c*x)/x^3) - a*b^2*arccsc(c*x)^2/x^3 + 1/12*(12*x^3*integrate(- 1/4*(12*c^2*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 - 12*arct an2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 + 12*(c^2*x^2*arctan2(1, sqrt (c*x + 1)*sqrt(c*x - 1)) - arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x) ^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*(4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1) )^2 - log(c^2*x^2)^2) - 4*((3*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))* log(c) - c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^2 - 3*arctan2(1, s qrt(c*x + 1)*sqrt(c*x - 1))*log(c) + 3*(c^2*x^2*arctan2(1, sqrt(c*x + 1)*s qrt(c*x - 1)) - arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x) + arctan2( 1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2) + 24*(c^2*x^2*arctan2(1, sqr t(c*x + 1)*sqrt(c*x - 1))*log(c) - arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) *log(c))*log(x))/(c^2*x^6 - x^4), x) - 4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2)*b^3/x ^3 - 1/3*a^3/x^3 - 2/9*(6*c^5*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - 3*c^3*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - (6*c^3*x^2 + c)*sqrt (c*x + 1)*sqrt(c*x - 1) - 3*c*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*a*b ^2/(sqrt(c*x + 1)*sqrt(c*x - 1)*c*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (148) = 296\).
Time = 0.31 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.52 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\frac {1}{27} \, {\left (9 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )^{2} + 18 \, a b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right ) - 27 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2} - \frac {9 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3}}{x} + 9 \, a^{2} b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 2 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 54 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) - \frac {27 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - \frac {9 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )^{3}}{x} - 27 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 42 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {27 \, a^{2} b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {6 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {27 \, a b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2}}{x} + \frac {6 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{x} - \frac {27 \, a^{2} b c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {42 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {42 \, a b^{2} c}{x} - \frac {9 \, a^{3}}{c x^{3}}\right )} c \]
1/27*(9*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x))^2 + 18*a*b^2*c^2* (-1/(c^2*x^2) + 1)^(3/2)*arcsin(1/(c*x)) - 27*b^3*c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))^2 - 9*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))^3/x + 9*a ^2*b*c^2*(-1/(c^2*x^2) + 1)^(3/2) - 2*b^3*c^2*(-1/(c^2*x^2) + 1)^(3/2) - 5 4*a*b^2*c^2*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x)) - 27*a*b^2*c*(1/(c^2*x^ 2) - 1)*arcsin(1/(c*x))^2/x - 9*b^3*c*arcsin(1/(c*x))^3/x - 27*a^2*b*c^2*s qrt(-1/(c^2*x^2) + 1) + 42*b^3*c^2*sqrt(-1/(c^2*x^2) + 1) - 27*a^2*b*c*(1/ (c^2*x^2) - 1)*arcsin(1/(c*x))/x + 6*b^3*c*(1/(c^2*x^2) - 1)*arcsin(1/(c*x ))/x - 27*a*b^2*c*arcsin(1/(c*x))^2/x + 6*a*b^2*c*(1/(c^2*x^2) - 1)/x - 27 *a^2*b*c*arcsin(1/(c*x))/x + 42*b^3*c*arcsin(1/(c*x))/x + 42*a*b^2*c/x - 9 *a^3/(c*x^3))*c
Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]