Integrand size = 14, antiderivative size = 87 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {b^2}{4 x^2}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {csch}^{-1}(c x)^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2} \]
-1/4*b^2/x^2-1/2*a*b*c^2*arccsch(c*x)-1/4*b^2*c^2*arccsch(c*x)^2-1/2*(a+b* arccsch(c*x))^2/x^2+1/2*b*c*(a+b*arccsch(c*x))*(1+1/c^2/x^2)^(1/2)/x
Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=-\frac {2 a^2+b^2-2 a b c \sqrt {1+\frac {1}{c^2 x^2}} x-2 b \left (-2 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {csch}^{-1}(c x)+b^2 \left (2+c^2 x^2\right ) \text {csch}^{-1}(c x)^2+2 a b c^2 x^2 \text {arcsinh}\left (\frac {1}{c x}\right )}{4 x^2} \]
-1/4*(2*a^2 + b^2 - 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x - 2*b*(-2*a + b*c*Sqrt [1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x] + b^2*(2 + c^2*x^2)*ArcCsch[c*x]^2 + 2*a *b*c^2*x^2*ArcSinh[1/(c*x)])/x^2
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6840, 5969, 3042, 25, 3791, 17}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx\) |
\(\Big \downarrow \) 6840 |
\(\displaystyle -c^2 \int \frac {\sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{c x}d\text {csch}^{-1}(c x)\) |
\(\Big \downarrow \) 5969 |
\(\displaystyle -c^2 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c^2 x^2}-b \int \frac {a+b \text {csch}^{-1}(c x)}{c^2 x^2}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^2 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c^2 x^2}-b \int -\left (\left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^2\right )d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c^2 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c^2 x^2}+b \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle -c^2 \left (b \left (\frac {1}{2} \int \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c x}+\frac {b}{4 c^2 x^2}\right )+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c^2 x^2}\right )\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -c^2 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c^2 x^2}+b \left (-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c x}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 b}+\frac {b}{4 c^2 x^2}\right )\right )\) |
-(c^2*((a + b*ArcCsch[c*x])^2/(2*c^2*x^2) + b*(b/(4*c^2*x^2) - (Sqrt[1 + 1 /(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(2*c*x) + (a + b*ArcCsch[c*x])^2/(4*b))) )
3.1.21.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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] - Simp[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]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, A rcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{2}}{x^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (75) = 150\).
Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=\frac {2 \, a b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, {\left (a b c^{2} x^{2} - b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, a b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, x^{2}} \]
1/4*(2*a*b*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - (b^2*c^2*x^2 + 2*b^2)*log(( c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))^2 - 2*a^2 - b^2 - 2*(a*b*c^2 *x^2 - b^2*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*a*b)*log((c*x*sqrt((c^2*x ^2 + 1)/(c^2*x^2)) + 1)/(c*x)))/x^2
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
1/4*a*b*((2*c^4*x*sqrt(1/(c^2*x^2) + 1)/(c^2*x^2*(1/(c^2*x^2) + 1) - 1) - c^3*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) + c^3*log(c*x*sqrt(1/(c^2*x^2) + 1) - 1))/c - 4*arccsch(c*x)/x^2) - 1/2*b^2*(log(sqrt(c^2*x^2 + 1) + 1)^2/x^2 + 2*integrate(-(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) - (2*c^2*x^2*log(c) + 2*(c^2*x^2 + 1)*log (x) + (c^2*x^2*(2*log(c) - 1) + 2*(c^2*x^2 + 1)*log(x) + 2*log(c))*sqrt(c^ 2*x^2 + 1) + 2*log(c))*log(sqrt(c^2*x^2 + 1) + 1) + (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^5 + x^3 + (c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)), x)) - 1/2*a^2/x^2
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \]