Integrand size = 14, antiderivative size = 132 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=-\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{16} a b c^4 \text {csch}^{-1}(c x)+\frac {3}{32} b^2 c^4 \text {csch}^{-1}(c x)^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4} \]
-1/32*b^2/x^4+3/32*b^2*c^2/x^2+3/16*a*b*c^4*arccsch(c*x)+3/32*b^2*c^4*arcc sch(c*x)^2-1/4*(a+b*arccsch(c*x))^2/x^4+1/8*b*c*(a+b*arccsch(c*x))*(1+1/c^ 2/x^2)^(1/2)/x^3-3/16*b*c^3*(a+b*arccsch(c*x))*(1+1/c^2/x^2)^(1/2)/x
Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\frac {-8 a^2-b^2+4 a b c \sqrt {1+\frac {1}{c^2 x^2}} x+3 b^2 c^2 x^2-6 a b c^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3-2 b \left (8 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-2+3 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)+b^2 \left (-8+3 c^4 x^4\right ) \text {csch}^{-1}(c x)^2+6 a b c^4 x^4 \text {arcsinh}\left (\frac {1}{c x}\right )}{32 x^4} \]
(-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)
Time = 0.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6840, 5969, 3042, 3791, 25, 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.
\(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx\) |
\(\Big \downarrow \) 6840 |
\(\displaystyle -c^4 \int \frac {\sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^3 x^3}d\text {csch}^{-1}(c x)\) |
\(\Big \downarrow \) 5969 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \int \frac {a+b \text {csch}^{-1}(c x)}{c^4 x^4}d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^4d\text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \int -\frac {a+b \text {csch}^{-1}(c x)}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {a+b \text {csch}^{-1}(c x)}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (-\frac {3}{4} \int -\left (\left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^2\right )d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \left (\frac {1}{2} \int \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c x}+\frac {b}{4 c^2 x^2}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^4 x^4}-\frac {1}{2} b \left (\frac {3}{4} \left (-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 c x}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 b}+\frac {b}{4 c^2 x^2}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{4 c^3 x^3}-\frac {b}{16 c^4 x^4}\right )\right )\) |
-(c^4*((a + b*ArcCsch[c*x])^2/(4*c^4*x^4) - (b*(-1/16*b/(c^4*x^4) + (Sqrt[ 1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(4*c^3*x^3) + (3*(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)))/4))/2))
3.1.23.3.1 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])
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{2}}{x^{5}}d x\]
Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\frac {3 \, b^{2} c^{2} x^{2} + {\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 8 \, a^{2} - b^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} - 8 \, a b - {\left (3 \, b^{2} c^{3} x^{3} - 2 \, b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (3 \, a b c^{3} x^{3} - 2 \, a b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{32 \, x^{4}} \]
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
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{5}}\, dx \]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
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*l og(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
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^5} \,d x \]