3.1.22 \(\int \frac {(a+b \text {csch}^{-1}(c x))^2}{x^4} \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 14, antiderivative size = 100 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=-\frac {2 b^2}{27 x^3}+\frac {4 b^2 c^2}{9 x}-\frac {4}{9} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3} \]

output

-2/27*b^2/x^3+4/9*b^2*c^2/x-1/3*(a+b*arccsch(c*x))^2/x^3-4/9*b*c^3*(a+b*ar 
ccsch(c*x))*(1+1/c^2/x^2)^(1/2)+2/9*b*c*(a+b*arccsch(c*x))*(1+1/c^2/x^2)^( 
1/2)/x^2
 

3.1.22.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=\frac {-9 a^2+6 a b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-2 c^2 x^2\right )+2 b^2 \left (-1+6 c^2 x^2\right )-6 b \left (3 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-1+2 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)-9 b^2 \text {csch}^{-1}(c x)^2}{27 x^3} \]

input

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

output

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

3.1.22.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6840, 5969, 3042, 26, 3791, 26, 3042, 26, 3777, 3042, 3117}

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.

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

\(\Big \downarrow \) 6840

\(\displaystyle -c^3 \int \frac {\sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^2 x^2}d\text {csch}^{-1}(c x)\)

\(\Big \downarrow \) 5969

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \int \frac {a+b \text {csch}^{-1}(c x)}{c^3 x^3}d\text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} b \int i \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^3d\text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )^3d\text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (\frac {2}{3} \int \frac {i \left (a+b \text {csch}^{-1}(c x)\right )}{c x}d\text {csch}^{-1}(c x)-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {i b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (\frac {2}{3} i \int \frac {a+b \text {csch}^{-1}(c x)}{c x}d\text {csch}^{-1}(c x)-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {i b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (\frac {2}{3} i \int -i \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {i b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (\frac {2}{3} \int \left (a+b \text {csch}^{-1}(c x)\right ) \sin \left (i \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {i b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-i b \int \sqrt {1+\frac {1}{c^2 x^2}}d\text {csch}^{-1}(c x)\right )-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {i b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-i b \int \sin \left (i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(c x)\right )-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {i b}{9 c^3 x^3}\right )\right )\)

\(\Big \downarrow \) 3117

\(\displaystyle -c^3 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 c^3 x^3}-\frac {2}{3} i b \left (-\frac {i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^2 x^2}+\frac {2}{3} \left (i \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {i b}{c x}\right )+\frac {i b}{9 c^3 x^3}\right )\right )\)

input

Int[(a + b*ArcCsch[c*x])^2/x^4,x]
 

output

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

3.1.22.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

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]
 

rule 5969

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]
 

rule 6840

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])
 

3.1.22.4 Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{2}}{x^{4}}d x\]

input

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

output

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

3.1.22.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (86) = 172\).

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=\frac {12 \, b^{2} c^{2} x^{2} - 9 \, b^{2} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 9 \, a^{2} - 2 \, b^{2} - 6 \, {\left (3 \, a b + {\left (2 \, b^{2} c^{3} x^{3} - b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 6 \, {\left (2 \, a b c^{3} x^{3} - a b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]

input

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

output

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
 

3.1.22.6 Sympy [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]

input

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

output

Integral((a + b*acsch(c*x))**2/x**4, x)
 

3.1.22.7 Maxima [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]

input

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

output

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(sqrt(c^2*x^2 + 1) + 1)^2/x^3 + 3*integra 
te(-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))*sq 
rt(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
 

3.1.22.8 Giac [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]

input

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

output

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

3.1.22.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^4} \,d x \]

input

int((a + b*asinh(1/(c*x)))^2/x^4,x)
 

output

int((a + b*asinh(1/(c*x)))^2/x^4, x)