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

Optimal. Leaf size=100 \[ -\frac{4}{9} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2 b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}-\frac{2 b^2}{27 x^3} \]

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

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Rubi [A]  time = 0.10826, antiderivative size = 100, 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 = {6286, 5446, 3310, 3296, 2637} \[ -\frac{4}{9} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2 b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}-\frac{2 b^2}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[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*ArcCsch[c*x]))/9 + (2*b*c*Sqrt[1
+ 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(9*x^2) - (a + b*ArcCsch[c*x])^2/(3*x^3)

Rule 6286

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

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[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 \text{csch}^{-1}(c x)\right )^2}{x^4} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int (a+b x)^2 \cosh (x) \sinh ^2(x) \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sinh ^3(x) \, dx,x,\text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^3}-\frac{1}{9} \left (4 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text{csch}^{-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 \text{csch}^{-1}(c x)\right )+\frac{2 b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{9} \left (4 b^2 c^3\right ) \operatorname{Subst}\left (\int \cosh (x) \, dx,x,\text{csch}^{-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 \text{csch}^{-1}(c x)\right )+\frac{2 b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{9 x^2}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.183789, size = 106, normalized size = 1.06 \[ \frac{-9 a^2+6 a b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (1-2 c^2 x^2\right )-6 b \text{csch}^{-1}(c x) \left (3 a+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (2 c^2 x^2-1\right )\right )+2 b^2 \left (6 c^2 x^2-1\right )-9 b^2 \text{csch}^{-1}(c x)^2}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[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))*ArcCsch[c*x] - 9*b^2*ArcCsch[c*x]^2)/(27*x^3)

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Maple [F]  time = 0.186, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))^2/x^4,x)

[Out]

int((a+b*arccsch(c*x))^2/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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{arcsch}\left (c x\right )}{x^{3}}\right )} - \frac{1}{3} \, b^{2}{\left (\frac{\log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )^{2}}{x^{3}} + 3 \, \int -\frac{3 \, c^{2} x^{2} \log \left (c\right )^{2} + 3 \,{\left (c^{2} x^{2} + 1\right )} \log \left (x\right )^{2} + 3 \, \log \left (c\right )^{2} + 6 \,{\left (c^{2} x^{2} \log \left (c\right ) + \log \left (c\right )\right )} \log \left (x\right ) - 2 \,{\left (3 \, c^{2} x^{2} \log \left (c\right ) + 3 \,{\left (c^{2} x^{2} + 1\right )} \log \left (x\right ) +{\left (c^{2} x^{2}{\left (3 \, \log \left (c\right ) - 1\right )} + 3 \,{\left (c^{2} x^{2} + 1\right )} \log \left (x\right ) + 3 \, \log \left (c\right )\right )} \sqrt{c^{2} x^{2} + 1} + 3 \, \log \left (c\right )\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + 3 \,{\left (c^{2} x^{2} \log \left (c\right )^{2} +{\left (c^{2} x^{2} + 1\right )} \log \left (x\right )^{2} + \log \left (c\right )^{2} + 2 \,{\left (c^{2} x^{2} \log \left (c\right ) + \log \left (c\right )\right )} \log \left (x\right )\right )} \sqrt{c^{2} x^{2} + 1}}{3 \,{\left (c^{2} x^{6} + x^{4} +{\left (c^{2} x^{6} + x^{4}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac{a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(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*arccsch(c*x)/x^3) - 1/3*b^2*(log(sq
rt(c^2*x^2 + 1) + 1)^2/x^3 + 3*integrate(-1/3*(3*c^2*x^2*log(c)^2 + 3*(c^2*x^2 + 1)*log(x)^2 + 3*log(c)^2 + 6*
(c^2*x^2*log(c) + log(c))*log(x) - 2*(3*c^2*x^2*log(c) + 3*(c^2*x^2 + 1)*log(x) + (c^2*x^2*(3*log(c) - 1) + 3*
(c^2*x^2 + 1)*log(x) + 3*log(c))*sqrt(c^2*x^2 + 1) + 3*log(c))*log(sqrt(c^2*x^2 + 1) + 1) + 3*(c^2*x^2*log(c)^
2 + (c^2*x^2 + 1)*log(x)^2 + log(c)^2 + 2*(c^2*x^2*log(c) + log(c))*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^6 + x^4
+ (c^2*x^6 + x^4)*sqrt(c^2*x^2 + 1)), x)) - 1/3*a^2/x^3

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Fricas [B]  time = 2.20302, size = 385, normalized size = 3.85 \begin{align*} \frac{12 \, b^{2} c^{2} x^{2} - 9 \, b^{2} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 9 \, a^{2} - 2 \, b^{2} - 6 \,{\left (3 \, a b +{\left (2 \, b^{2} c^{3} x^{3} - b^{2} c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 6 \,{\left (2 \, a b c^{3} x^{3} - a b c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsch}{\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*acsch(c*x))**2/x**4,x)

[Out]

Integral((a + b*acsch(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{arcsch}\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*arccsch(c*x))^2/x^4,x, algorithm="giac")

[Out]

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