Integrand size = 14, antiderivative size = 80 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}+\frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \]
6*b^2*(a+b*arccsc(c*x))/x-(a+b*arccsc(c*x))^3/x+6*b^3*c*(1-1/c^2/x^2)^(1/2 )-3*b*c*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)
Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {a^3-6 a b^2+3 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x-6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+3 b \left (a^2-2 b^2+2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)+3 b^2 \left (a+b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)^2+b^3 \csc ^{-1}(c x)^3}{x} \]
-((a^3 - 6*a*b^2 + 3*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 - 1/ (c^2*x^2)]*x + 3*b*(a^2 - 2*b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[ c*x] + 3*b^2*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x]^2 + b^3*ArcCsc[ c*x]^3)/x)
Time = 0.47 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5746, 3042, 3777, 25, 3042, 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.
\(\displaystyle \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx\) |
\(\Big \downarrow \) 5746 |
\(\displaystyle -c \int \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \int \left (a+b \csc ^{-1}(c x)\right )^3 \sin \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -c \left (3 b \int -\frac {\left (a+b \csc ^{-1}(c x)\right )^2}{c x}d\csc ^{-1}(c x)+\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{c x}d\csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \int \left (a+b \csc ^{-1}(c x)\right )^2 \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \left (2 b \int \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \left (2 b \int \left (a+b \csc ^{-1}(c x)\right ) \sin \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \left (2 b \left (b \int -\frac {1}{c x}d\csc ^{-1}(c x)+\frac {a+b \csc ^{-1}(c x)}{c x}\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \left (2 b \left (\frac {a+b \csc ^{-1}(c x)}{c x}-b \int \frac {1}{c x}d\csc ^{-1}(c x)\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \left (2 b \left (\frac {a+b \csc ^{-1}(c x)}{c x}-b \int \sin \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -c \left (\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{c x}-3 b \left (2 b \left (\frac {a+b \csc ^{-1}(c x)}{c x}+b \sqrt {1-\frac {1}{c^2 x^2}}\right )-\sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )\right )\) |
-(c*((a + b*ArcCsc[c*x])^3/(c*x) - 3*b*(-(Sqrt[1 - 1/(c^2*x^2)]*(a + b*Arc Csc[c*x])^2) + 2*b*(b*Sqrt[1 - 1/(c^2*x^2)] + (a + b*ArcCsc[c*x])/(c*x)))) )
3.1.29.3.1 Defintions of rubi rules used
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[((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. \(196\) vs. \(2(76)=152\).
Time = 0.89 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.46
method | result | size |
parts | \(-\frac {a^{3}}{x}+b^{3} c \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{c x}-3 \operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+\frac {6 \,\operatorname {arccsc}\left (c x \right )}{c x}\right )+3 a \,b^{2} c \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{c x}+\frac {2}{c x}-2 \,\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+3 a^{2} b c \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\) | \(197\) |
derivativedivides | \(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{c x}-3 \operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+\frac {6 \,\operatorname {arccsc}\left (c x \right )}{c x}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{c x}+\frac {2}{c x}-2 \,\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(199\) |
default | \(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{c x}-3 \operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+\frac {6 \,\operatorname {arccsc}\left (c x \right )}{c x}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{c x}+\frac {2}{c x}-2 \,\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\) | \(199\) |
-a^3/x+b^3*c*(-1/c/x*arccsc(c*x)^3-3*arccsc(c*x)^2*((c^2*x^2-1)/c^2/x^2)^( 1/2)+6*((c^2*x^2-1)/c^2/x^2)^(1/2)+6/c/x*arccsc(c*x))+3*a*b^2*c*(-1/c/x*ar ccsc(c*x)^2+2/c/x-2*arccsc(c*x)*((c^2*x^2-1)/c^2/x^2)^(1/2))+3*a^2*b*c*(-1 /c/x*arccsc(c*x)-1/((c^2*x^2-1)/c^2/x^2)^(1/2)/c^2/x^2*(c^2*x^2-1))
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \operatorname {arccsc}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2} + a^{3} - 6 \, a b^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) + 3 \, {\left (b^{3} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b^{2} \operatorname {arccsc}\left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{x} \]
-(b^3*arccsc(c*x)^3 + 3*a*b^2*arccsc(c*x)^2 + a^3 - 6*a*b^2 + 3*(a^2*b - 2 *b^3)*arccsc(c*x) + 3*(b^3*arccsc(c*x)^2 + 2*a*b^2*arccsc(c*x) + a^2*b - 2 *b^3)*sqrt(c^2*x^2 - 1))/x
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \operatorname {arccsc}\left (c x\right )^{3}}{x} - 3 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} a^{2} b - 6 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} - 3 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right )^{2} - 2 \, c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {2 \, \operatorname {arccsc}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \]
-b^3*arccsc(c*x)^3/x - 3*(c*sqrt(-1/(c^2*x^2) + 1) + arccsc(c*x)/x)*a^2*b - 6*(c*sqrt(-1/(c^2*x^2) + 1)*arccsc(c*x) - 1/x)*a*b^2 - 3*(c*sqrt(-1/(c^2 *x^2) + 1)*arccsc(c*x)^2 - 2*c*sqrt(-1/(c^2*x^2) + 1) - 2*arccsc(c*x)/x)*b ^3 - 3*a*b^2*arccsc(c*x)^2/x - a^3/x
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (76) = 152\).
Time = 0.32 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-{\left (3 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2} + 6 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) + \frac {b^{3} \arcsin \left (\frac {1}{c x}\right )^{3}}{c x} + 3 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {3 \, a b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c x} + \frac {3 \, a^{2} b \arcsin \left (\frac {1}{c x}\right )}{c x} - \frac {6 \, b^{3} \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a^{3}}{c x} - \frac {6 \, a b^{2}}{c x}\right )} c \]
-(3*b^3*sqrt(-1/(c^2*x^2) + 1)*arcsin(1/(c*x))^2 + 6*a*b^2*sqrt(-1/(c^2*x^ 2) + 1)*arcsin(1/(c*x)) + b^3*arcsin(1/(c*x))^3/(c*x) + 3*a^2*b*sqrt(-1/(c ^2*x^2) + 1) - 6*b^3*sqrt(-1/(c^2*x^2) + 1) + 3*a*b^2*arcsin(1/(c*x))^2/(c *x) + 3*a^2*b*arcsin(1/(c*x))/(c*x) - 6*b^3*arcsin(1/(c*x))/(c*x) + a^3/(c *x) - 6*a*b^2/(c*x))*c
Time = 1.02 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=\frac {b^3\,\left (6\,\mathrm {asin}\left (\frac {1}{c\,x}\right )-{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^3\right )}{x}-\frac {a^3}{x}-3\,a^2\,b\,c\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {\mathrm {asin}\left (\frac {1}{c\,x}\right )}{c\,x}\right )-b^3\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}\,\left (3\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-6\right )-3\,a\,b^2\,c\,\left (2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-2}{c\,x}\right ) \]