\(\int \frac {(a+b \csc ^{-1}(c x))^3}{x^2} \, dx\) [29]

Optimal result

Integrand size = 14, antiderivative size = 80 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}}+\frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5331, 3377, 2718} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=\frac {6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} \]

[In]

Int[(a + b*ArcCsc[c*x])^3/x^2,x]

[Out]

6*b^3*c*Sqrt[1 - 1/(c^2*x^2)] + (6*b^2*(a + b*ArcCsc[c*x]))/x - 3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]
)^2 - (a + b*ArcCsc[c*x])^3/x

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5331

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

Rubi steps \begin{align*} \text {integral}& = -\left (c \text {Subst}\left (\int (a+b x)^3 \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\right ) \\ & = -\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x}+(3 b c) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = -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}+\left (6 b^2 c\right ) \text {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \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}-\left (6 b^3 c\right ) \text {Subst}\left (\int \sin (x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {a^3-6 a b^2+3 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x-6 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+3 b \left (a^2-2 b^2+2 a b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)+3 b^2 \left (a+b c \sqrt {1-\frac {1}{c^2 x^2}} x\right ) \csc ^{-1}(c x)^2+b^3 \csc ^{-1}(c x)^3}{x} \]

[In]

Integrate[(a + b*ArcCsc[c*x])^3/x^2,x]

[Out]

-((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*ArcCs
c[c*x]^3)/x)

Maple [B] (verified)

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	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{3}}{c x}-3 \operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+\frac {6 \,\operatorname {arccsc}\left (c x \right )}{c x}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )^{2}}{c x}+\frac {2}{c x}-2 \,\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )+3 a^{2} b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{c x}-\frac {c^{2} x^{2}-1}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}\right )\right )\)	\(199\)

[In]

int((a+b*arccsc(c*x))^3/x^2,x,method=_RETURNVERBOSE)

[Out]

-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*arccsc(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))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \operatorname {arccsc}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2} + a^{3} - 6 \, a b^{2} + 3 \, {\left (a^{2} b - 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) + 3 \, {\left (b^{3} \operatorname {arccsc}\left (c x\right )^{2} + 2 \, a b^{2} \operatorname {arccsc}\left (c x\right ) + a^{2} b - 2 \, b^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{x} \]

[In]

integrate((a+b*arccsc(c*x))^3/x^2,x, algorithm="fricas")

[Out]

-(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

Sympy [F]

\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]

[In]

integrate((a+b*acsc(c*x))**3/x**2,x)

[Out]

Integral((a + b*acsc(c*x))**3/x**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \operatorname {arccsc}\left (c x\right )^{3}}{x} - 3 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} a^{2} b - 6 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} - 3 \, {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right )^{2} - 2 \, c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {2 \, \operatorname {arccsc}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \]

[In]

integrate((a+b*arccsc(c*x))^3/x^2,x, algorithm="maxima")

[Out]

-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

Giac [B] (verification not implemented)

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 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=-{\left (3 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2} + 6 \, a b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) + \frac {b^{3} \arcsin \left (\frac {1}{c x}\right )^{3}}{c x} + 3 \, a^{2} b \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 6 \, b^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {3 \, a b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c x} + \frac {3 \, a^{2} b \arcsin \left (\frac {1}{c x}\right )}{c x} - \frac {6 \, b^{3} \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {a^{3}}{c x} - \frac {6 \, a b^{2}}{c x}\right )} c \]

[In]

integrate((a+b*arccsc(c*x))^3/x^2,x, algorithm="giac")

[Out]

-(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

Mupad [B] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx=\frac {b^3\,\left (6\,\mathrm {asin}\left (\frac {1}{c\,x}\right )-{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^3\right )}{x}-\frac {a^3}{x}-3\,a^2\,b\,c\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {\mathrm {asin}\left (\frac {1}{c\,x}\right )}{c\,x}\right )-b^3\,c\,\sqrt {1-\frac {1}{c^2\,x^2}}\,\left (3\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-6\right )-3\,a\,b^2\,c\,\left (2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\sqrt {1-\frac {1}{c^2\,x^2}}+\frac {{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2-2}{c\,x}\right ) \]

[In]

int((a + b*asin(1/(c*x)))^3/x^2,x)

[Out]

(b^3*(6*asin(1/(c*x)) - asin(1/(c*x))^3))/x - a^3/x - 3*a^2*b*c*((1 - 1/(c^2*x^2))^(1/2) + asin(1/(c*x))/(c*x)
) - b^3*c*(1 - 1/(c^2*x^2))^(1/2)*(3*asin(1/(c*x))^2 - 6) - 3*a*b^2*c*(2*asin(1/(c*x))*(1 - 1/(c^2*x^2))^(1/2)
+ (asin(1/(c*x))^2 - 2)/(c*x))