3.1.47 \(\int \frac {(a+b \text {sech}^{-1}(c x))^3}{x^2} \, dx\) [47]

3.1.47.1 Optimal result

Integrand size = 14, antiderivative size = 102 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=\frac {6 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x}-\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x} \]

output

-6*b^2*(a+b*arcsech(c*x))/x-(a+b*arcsech(c*x))^3/x+6*b^3*(c*x+1)*((-c*x+1) 
/(c*x+1))^(1/2)/x+3*b*(c*x+1)*(a+b*arcsech(c*x))^2*((-c*x+1)/(c*x+1))^(1/2 
)/x
 

3.1.47.2 Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {a^3+6 a b^2-3 a^2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x)-6 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)+3 b \left (a^2+2 b^2-2 a b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)-3 b^2 \left (-a+b \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right ) \text {sech}^{-1}(c x)^2+b^3 \text {sech}^{-1}(c x)^3}{x} \]

input

Integrate[(a + b*ArcSech[c*x])^3/x^2,x]
 

output

-((a^3 + 6*a*b^2 - 3*a^2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 6*b^3*Sqr 
t[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 3*b*(a^2 + 2*b^2 - 2*a*b*Sqrt[(1 - c*x) 
/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] - 3*b^2*(-a + b*Sqrt[(1 - c*x)/(1 + c* 
x)]*(1 + c*x))*ArcSech[c*x]^2 + b^3*ArcSech[c*x]^3)/x)
 

3.1.47.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6839, 3042, 26, 3777, 3042, 3777, 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.

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

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

input

Int[(a + b*ArcSech[c*x])^3/x^2,x]
 

output

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

3.1.47.3.1 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 6839

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A 
rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

3.1.47.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(224\) vs. \(2(98)=196\).

Time = 0.54 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.21

method	result	size
parts	\(-\frac {a^{3}}{x}+b^{3} c \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{c x}+3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}-\frac {6 \,\operatorname {arcsech}\left (c x \right )}{c x}+6 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )+3 a \,b^{2} c \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{c x}+2 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+3 b \,a^{2} c \left (-\frac {\operatorname {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\)	\(225\)
derivativedivides	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{c x}+3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}-\frac {6 \,\operatorname {arcsech}\left (c x \right )}{c x}+6 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{c x}+2 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+3 b \,a^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\)	\(227\)
default	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{3}}{c x}+3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \operatorname {arcsech}\left (c x \right )^{2}-\frac {6 \,\operatorname {arcsech}\left (c x \right )}{c x}+6 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )+3 a \,b^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )^{2}}{c x}+2 \,\operatorname {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+3 b \,a^{2} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right )\)	\(227\)

input

int((a+b*arcsech(c*x))^3/x^2,x,method=_RETURNVERBOSE)
 

output

-a^3/x+b^3*c*(-1/c/x*arcsech(c*x)^3+3*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^( 
1/2)*arcsech(c*x)^2-6/c/x*arcsech(c*x)+6*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x 
)^(1/2))+3*a*b^2*c*(-1/c/x*arcsech(c*x)^2+2*arcsech(c*x)*(-(c*x-1)/c/x)^(1 
/2)*((c*x+1)/c/x)^(1/2)-2/c/x)+3*b*a^2*c*(-1/c/x*arcsech(c*x)+(-(c*x-1)/c/ 
x)^(1/2)*((c*x+1)/c/x)^(1/2))
 

3.1.47.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (98) = 196\).

Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \, {\left (b^{3} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \, {\left (2 \, a b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \]

input

integrate((a+b*arcsech(c*x))^3/x^2,x, algorithm="fricas")
 

output

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

3.1.47.6 Sympy [F]

\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]

input

integrate((a+b*asech(c*x))**3/x**2,x)
 

output

Integral((a + b*asech(c*x))**3/x**2, x)
 

3.1.47.7 Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=-\frac {b^{3} \operatorname {arsech}\left (c x\right )^{3}}{x} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} a^{2} b + 6 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right )^{2} + 2 \, c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {2 \, \operatorname {arsech}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arsech}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \]

input

integrate((a+b*arcsech(c*x))^3/x^2,x, algorithm="maxima")
 

output

-b^3*arcsech(c*x)^3/x + 3*(c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*a^2*b 
 + 6*(c*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x) - 1/x)*a*b^2 + 3*(c*sqrt(1/(c^2 
*x^2) - 1)*arcsech(c*x)^2 + 2*c*sqrt(1/(c^2*x^2) - 1) - 2*arcsech(c*x)/x)* 
b^3 - 3*a*b^2*arcsech(c*x)^2/x - a^3/x
 

3.1.47.8 Giac [F]

\[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*arcsech(c*x))^3/x^2,x, algorithm="giac")
 

output

integrate((b*arcsech(c*x) + a)^3/x^2, x)
 

3.1.47.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^2} \,d x \]

input

int((a + b*acosh(1/(c*x)))^3/x^2,x)
 

output

int((a + b*acosh(1/(c*x)))^3/x^2, x)