3.22 \(\int \frac{(a+b \csc ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=102 \[ -\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{2 b^2}{27 x^3} \]

[Out]

(2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) - (4*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/9 - (2*b*c*Sqrt[1 -
1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(9*x^2) - (a + b*ArcCsc[c*x])^2/(3*x^3)

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Rubi [A]  time = 0.0963057, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5223, 4404, 3310, 3296, 2637} \[ -\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{2 b^2}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^2/x^4,x]

[Out]

(2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) - (4*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/9 - (2*b*c*Sqrt[1 -
1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(9*x^2) - (a + b*ArcCsc[c*x])^2/(3*x^3)

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 4404

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] - Dist[(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]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 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^4} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int (a+b x)^2 \cos (x) \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sin ^3(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{2 b^2}{27 x^3}-\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{9} \left (4 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{2 b^2}{27 x^3}-\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{9} \left (4 b^2 c^3\right ) \operatorname{Subst}\left (\int \cos (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{2 b^2}{27 x^3}+\frac{4 b^2 c^2}{9 x}-\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )-\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \csc ^{-1}(c x)\right )^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.225241, size = 108, normalized size = 1.06 \[ -\frac{9 a^2+6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )+6 b \csc ^{-1}(c x) \left (3 a+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 x^2+1\right )\right )-2 b^2 \left (6 c^2 x^2+1\right )+9 b^2 \csc ^{-1}(c x)^2}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])^2/x^4,x]

[Out]

-(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) + 6*b*(3*a + b*c*Sqrt[1 - 1/
(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcCsc[c*x] + 9*b^2*ArcCsc[c*x]^2)/(27*x^3)

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Maple [A]  time = 0.219, size = 154, normalized size = 1.5 \begin{align*}{c}^{3} \left ( -{\frac{{a}^{2}}{3\,{c}^{3}{x}^{3}}}+{b}^{2} \left ( -{\frac{ \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{3\,{c}^{3}{x}^{3}}}-{\frac{2\,{\rm arccsc} \left (cx\right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{9\,{c}^{2}{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{2}{27\,{c}^{3}{x}^{3}}}+{\frac{4}{9\,cx}} \right ) +2\,ab \left ( -1/3\,{\frac{{\rm arccsc} \left (cx\right )}{{c}^{3}{x}^{3}}}-1/9\,{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{{c}^{4}{x}^{4}}{\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^4,x)

[Out]

c^3*(-1/3*a^2/c^3/x^3+b^2*(-1/3/c^3/x^3*arccsc(c*x)^2-2/9*arccsc(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)+2*a*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))

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Maxima [B]  time = 2.10965, size = 266, normalized size = 2.61 \begin{align*} \frac{2}{9} \, a b{\left (\frac{c^{4}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, c^{4} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{3 \, \operatorname{arccsc}\left (c x\right )}{x^{3}}\right )} - \frac{b^{2} \operatorname{arccsc}\left (c x\right )^{2}}{3 \, x^{3}} - \frac{a^{2}}{3 \, x^{3}} - \frac{2 \,{\left (6 \, c^{5} x^{4} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) - 3 \, c^{3} x^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) -{\left (6 \, c^{3} x^{2} + c\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 3 \, c \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} b^{2}}{27 \, \sqrt{c x + 1} \sqrt{c x - 1} c x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^2/x^4,x, algorithm="maxima")

[Out]

2/9*a*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) - 1/3*b^2*arccsc
(c*x)^2/x^3 - 1/3*a^2/x^3 - 2/27*(6*c^5*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - 3*c^3*x^2*arctan2(1, sqr
t(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)))*b^2/(sqrt(c*x + 1)*sqrt(c*x - 1)*c*x^3)

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Fricas [A]  time = 2.01381, size = 224, normalized size = 2.2 \begin{align*} \frac{12 \, b^{2} c^{2} x^{2} - 9 \, b^{2} \operatorname{arccsc}\left (c x\right )^{2} - 18 \, a b \operatorname{arccsc}\left (c x\right ) - 9 \, a^{2} + 2 \, b^{2} - 6 \,{\left (2 \, a b c^{2} x^{2} + a b +{\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname{arccsc}\left (c x\right )\right )} \sqrt{c^{2} x^{2} - 1}}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^2/x^4,x, algorithm="fricas")

[Out]

1/27*(12*b^2*c^2*x^2 - 9*b^2*arccsc(c*x)^2 - 18*a*b*arccsc(c*x) - 9*a^2 + 2*b^2 - 6*(2*a*b*c^2*x^2 + a*b + (2*
b^2*c^2*x^2 + b^2)*arccsc(c*x))*sqrt(c^2*x^2 - 1))/x^3

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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^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))**2/x**4,x)

[Out]

Integral((a + b*acsc(c*x))**2/x**4, 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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^2/x^4,x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)^2/x^4, x)