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

Optimal. Leaf size=50 \[ -2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x}+\frac{2 b^2}{x} \]

[Out]

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

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Rubi [A]  time = 0.0608282, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5223, 3296, 2637} \[ -2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x}+\frac{2 b^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^2/x^2,x]

[Out]

(2*b^2)/x - 2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]) - (a + b*ArcCsc[c*x])^2/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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x^2} \, dx &=-\left (c \operatorname{Subst}\left (\int (a+b x)^2 \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x}+(2 b c) \operatorname{Subst}\left (\int (a+b x) \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x}+\left (2 b^2 c\right ) \operatorname{Subst}\left (\int \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{2 b^2}{x}-2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x}\\ \end{align*}

Mathematica [A]  time = 0.146587, size = 71, normalized size = 1.42 \[ -\frac{a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b \csc ^{-1}(c x) \left (a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right )+b^2 \csc ^{-1}(c x)^2-2 b^2}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.219, size = 118, normalized size = 2.4 \begin{align*} c \left ( -{\frac{{a}^{2}}{cx}}+{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 ) +2\,ab \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))^2/x^2,x)

[Out]

c*(-a^2/c/x+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))+2*a*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)))

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Maxima [A]  time = 1.00159, size = 107, normalized size = 2.14 \begin{align*} -2 \,{\left (c \sqrt{-\frac{1}{c^{2} x^{2}} + 1} + \frac{\operatorname{arccsc}\left (c x\right )}{x}\right )} a b - 2 \,{\left (c \sqrt{-\frac{1}{c^{2} x^{2}} + 1} \operatorname{arccsc}\left (c x\right ) - \frac{1}{x}\right )} b^{2} - \frac{b^{2} \operatorname{arccsc}\left (c x\right )^{2}}{x} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.8743, size = 140, normalized size = 2.8 \begin{align*} -\frac{b^{2} \operatorname{arccsc}\left (c x\right )^{2} + 2 \, a b \operatorname{arccsc}\left (c x\right ) + a^{2} - 2 \, b^{2} + 2 \, \sqrt{c^{2} x^{2} - 1}{\left (b^{2} \operatorname{arccsc}\left (c x\right ) + a b\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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