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

Optimal. Leaf size=80 \[ \frac{6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac{\left (a+b \csc ^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt{1-\frac{1}{c^2 x^2}} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0857982, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5223, 3296, 2638} \[ \frac{6 b^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac{\left (a+b \csc ^{-1}(c x)\right )^3}{x}+6 b^3 c \sqrt{1-\frac{1}{c^2 x^2}} \]

Antiderivative was successfully verified.

[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 5223

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])

Rule 3296

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 2638

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \csc ^{-1}(c x)\right )^3}{x^2} \, dx &=-\left (c \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.200749, size = 135, normalized size = 1.69 \[ -\frac{3 b \csc ^{-1}(c x) \left (a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 b^2\right )+3 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+a^3+3 b^2 \csc ^{-1}(c x)^2 \left (a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right )-6 a b^2-6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+b^3 \csc ^{-1}(c x)^3}{x} \]

Antiderivative was successfully verified.

[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]  time = 0.25, size = 199, normalized size = 2.5 \begin{align*} c \left ( -{\frac{{a}^{3}}{cx}}+{b}^{3} \left ( -{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}}{cx}}-3\, \left ({\rm arccsc} \left (cx\right ) \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}}}}+6\,{\frac{{\rm arccsc} \left (cx\right )}{cx}} \right ) +3\,a{b}^{2} \left ( -{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{cx}}+2\,{\frac{1}{cx}}-2\,\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}{\rm arccsc} \left (cx\right ) \right ) +3\,{a}^{2}b \left ( -{\frac{{\rm arccsc} \left (cx\right )}{cx}}-{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02653, size = 198, normalized size = 2.48 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 2.37818, size = 238, normalized size = 2.98 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsc}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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