3.23 \(\int \frac{(a+b \text{csch}^{-1}(c x))^2}{x^5} \, dx\)

Optimal. Leaf size=132 \[ -\frac{3 b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{16 x}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{8 x^3}+\frac{3}{16} a b c^4 \text{csch}^{-1}(c x)-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{4 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{3}{32} b^2 c^4 \text{csch}^{-1}(c x)^2-\frac{b^2}{32 x^4} \]

[Out]

-b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3*a*b*c^4*ArcCsch[c*x])/16 + (3*b^2*c^4*ArcCsch[c*x]^2)/32 + (b*c*Sqrt
[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(8*x^3) - (3*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(16*x)
- (a + b*ArcCsch[c*x])^2/(4*x^4)

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Rubi [A]  time = 0.116934, antiderivative size = 132, 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 = {6286, 5446, 3310} \[ -\frac{3 b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{16 x}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{8 x^3}+\frac{3}{16} a b c^4 \text{csch}^{-1}(c x)-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{4 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{3}{32} b^2 c^4 \text{csch}^{-1}(c x)^2-\frac{b^2}{32 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])^2/x^5,x]

[Out]

-b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3*a*b*c^4*ArcCsch[c*x])/16 + (3*b^2*c^4*ArcCsch[c*x]^2)/32 + (b*c*Sqrt
[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(8*x^3) - (3*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(16*x)
- (a + b*ArcCsch[c*x])^2/(4*x^4)

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{x^5} \, dx &=-\left (c^4 \operatorname{Subst}\left (\int (a+b x)^2 \cosh (x) \sinh ^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )\right )\\ &=-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{4 x^4}+\frac{1}{2} \left (b c^4\right ) \operatorname{Subst}\left (\int (a+b x) \sinh ^4(x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{b^2}{32 x^4}+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{8 x^3}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{4 x^4}-\frac{1}{8} \left (3 b c^4\right ) \operatorname{Subst}\left (\int (a+b x) \sinh ^2(x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{b^2}{32 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{8 x^3}-\frac{3 b c^3 \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{16 x}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{4 x^4}+\frac{1}{16} \left (3 b c^4\right ) \operatorname{Subst}\left (\int (a+b x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=-\frac{b^2}{32 x^4}+\frac{3 b^2 c^2}{32 x^2}+\frac{3}{16} a b c^4 \text{csch}^{-1}(c x)+\frac{3}{32} b^2 c^4 \text{csch}^{-1}(c x)^2+\frac{b c \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{8 x^3}-\frac{3 b c^3 \sqrt{1+\frac{1}{c^2 x^2}} \left (a+b \text{csch}^{-1}(c x)\right )}{16 x}-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.173547, size = 147, normalized size = 1.11 \[ \frac{-8 a^2-6 a b c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+4 a b c x \sqrt{\frac{1}{c^2 x^2}+1}+6 a b c^4 x^4 \sinh ^{-1}\left (\frac{1}{c x}\right )-2 b \text{csch}^{-1}(c x) \left (8 a+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (3 c^2 x^2-2\right )\right )+3 b^2 c^2 x^2+b^2 \left (3 c^4 x^4-8\right ) \text{csch}^{-1}(c x)^2-b^2}{32 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])^2/x^5,x]

[Out]

(-8*a^2 - b^2 + 4*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + 3*b^2*c^2*x^2 - 6*a*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3 - 2*b*(8
*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-2 + 3*c^2*x^2))*ArcCsch[c*x] + b^2*(-8 + 3*c^4*x^4)*ArcCsch[c*x]^2 + 6*a*b*
c^4*x^4*ArcSinh[1/(c*x)])/(32*x^4)

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Maple [F]  time = 0.204, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}}{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))^2/x^5,x)

[Out]

int((a+b*arccsch(c*x))^2/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{32} \, a b{\left (\frac{3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - 3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right ) - \frac{2 \,{\left (3 \, c^{8} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, c^{6} x \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} + 1}}{c} - \frac{16 \, \operatorname{arcsch}\left (c x\right )}{x^{4}}\right )} - \frac{1}{4} \, b^{2}{\left (\frac{\log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )^{2}}{x^{4}} + 4 \, \int -\frac{2 \, c^{2} x^{2} \log \left (c\right )^{2} + 2 \,{\left (c^{2} x^{2} + 1\right )} \log \left (x\right )^{2} + 2 \, \log \left (c\right )^{2} + 4 \,{\left (c^{2} x^{2} \log \left (c\right ) + \log \left (c\right )\right )} \log \left (x\right ) -{\left (4 \, c^{2} x^{2} \log \left (c\right ) + 4 \,{\left (c^{2} x^{2} + 1\right )} \log \left (x\right ) +{\left (c^{2} x^{2}{\left (4 \, \log \left (c\right ) - 1\right )} + 4 \,{\left (c^{2} x^{2} + 1\right )} \log \left (x\right ) + 4 \, \log \left (c\right )\right )} \sqrt{c^{2} x^{2} + 1} + 4 \, \log \left (c\right )\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + 2 \,{\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}}{2 \,{\left (c^{2} x^{7} + x^{5} +{\left (c^{2} x^{7} + x^{5}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac{a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2/x^5,x, algorithm="maxima")

[Out]

1/32*a*b*((3*c^5*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) - 3*c^5*log(c*x*sqrt(1/(c^2*x^2) + 1) - 1) - 2*(3*c^8*x^3*
(1/(c^2*x^2) + 1)^(3/2) - 5*c^6*x*sqrt(1/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) + 1)^2 - 2*c^2*x^2*(1/(c^2*x^2)
+ 1) + 1))/c - 16*arccsch(c*x)/x^4) - 1/4*b^2*(log(sqrt(c^2*x^2 + 1) + 1)^2/x^4 + 4*integrate(-1/2*(2*c^2*x^2
*log(c)^2 + 2*(c^2*x^2 + 1)*log(x)^2 + 2*log(c)^2 + 4*(c^2*x^2*log(c) + log(c))*log(x) - (4*c^2*x^2*log(c) + 4
*(c^2*x^2 + 1)*log(x) + (c^2*x^2*(4*log(c) - 1) + 4*(c^2*x^2 + 1)*log(x) + 4*log(c))*sqrt(c^2*x^2 + 1) + 4*log
(c))*log(sqrt(c^2*x^2 + 1) + 1) + 2*(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^7 + x^5 + (c^2*x^7 + x^5)*sqrt(c^2*x^2 + 1)), x)) - 1/4*a^2/x^4

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Fricas [A]  time = 2.22867, size = 432, normalized size = 3.27 \begin{align*} \frac{3 \, b^{2} c^{2} x^{2} +{\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 8 \, a^{2} - b^{2} + 2 \,{\left (3 \, a b c^{4} x^{4} - 8 \, a b -{\left (3 \, b^{2} c^{3} x^{3} - 2 \, 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 ) - 2 \,{\left (3 \, a b c^{3} x^{3} - 2 \, a b c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{32 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2/x^5,x, algorithm="fricas")

[Out]

1/32*(3*b^2*c^2*x^2 + (3*b^2*c^4*x^4 - 8*b^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))^2 - 8*a^2 - b
^2 + 2*(3*a*b*c^4*x^4 - 8*a*b - (3*b^2*c^3*x^3 - 2*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)) - 2*(3*a*b*c^3*x^3 - 2*a*b*c*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/x^4

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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^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))**2/x**5,x)

[Out]

Integral((a + b*acsch(c*x))**2/x**5, 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2/x^5,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)^2/x^5, x)