3.1.61 \(\int \frac {1}{x^3 (a+b \text {sech}^{-1}(c x))^2} \, dx\) [61]

3.1.61.1 Optimal result

Integrand size = 14, antiderivative size = 85 \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=-\frac {c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^2}+\frac {c^2 \sinh \left (2 \text {sech}^{-1}(c x)\right )}{2 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b^2} \]

output

-c^2*Chi(2*a/b+2*arcsech(c*x))*cosh(2*a/b)/b^2+c^2*Shi(2*a/b+2*arcsech(c*x 
))*sinh(2*a/b)/b^2+1/2*c^2*sinh(2*arcsech(c*x))/b/(a+b*arcsech(c*x))
 

3.1.61.2 Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\frac {\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )}-c^2 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+c^2 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )}{b^2} \]

input

Integrate[1/(x^3*(a + b*ArcSech[c*x])^2),x]
 

output

((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x^2*(a + b*ArcSech[c*x])) - c^2* 
Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSech[c*x])] + c^2*Sinh[(2*a)/b]*Sin 
hIntegral[2*(a/b + ArcSech[c*x])])/b^2
 

3.1.61.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6839, 5971, 27, 3042, 26, 3778, 3042, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 6839

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

\(\displaystyle \frac {1}{2} i c^2 \left (\frac {2 i \left (\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \text {sech}^{-1}(c x)\right )}{b}\right )}{b}-\frac {i \sinh \left (2 \text {sech}^{-1}(c x)\right )}{b \left (a+b \text {sech}^{-1}(c x)\right )}\right )\)

input

Int[1/(x^3*(a + b*ArcSech[c*x])^2),x]
 

output

(I/2)*c^2*(((-I)*Sinh[2*ArcSech[c*x]])/(b*(a + b*ArcSech[c*x])) + ((2*I)*( 
(Cosh[(2*a)/b]*CoshIntegral[(2*a)/b + 2*ArcSech[c*x]])/b - (Sinh[(2*a)/b]* 
SinhIntegral[(2*a)/b + 2*ArcSech[c*x]])/b))/b)
 

3.1.61.3.1 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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3778

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 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])
 

3.1.61.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(83)=166\).

Time = 0.76 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.19

method	result	size
derivativedivides	\(c^{2} \left (\frac {2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+c^{2} x^{2}-2}{4 c^{2} x^{2} b \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{2 b^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{4 b \,c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{2}}\right )\)	\(186\)
default	\(c^{2} \left (\frac {2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+c^{2} x^{2}-2}{4 c^{2} x^{2} b \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (\frac {2 a}{b}+2 \,\operatorname {arcsech}\left (c x \right )\right )}{2 b^{2}}-\frac {c^{2} x^{2}-2-2 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{4 b \,c^{2} x^{2} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsech}\left (c x \right )-\frac {2 a}{b}\right )}{2 b^{2}}\right )\)	\(186\)

input

int(1/x^3/(a+b*arcsech(c*x))^2,x,method=_RETURNVERBOSE)
 

output

c^2*(1/4*(2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)+c^2*x^2-2)/c^2/x^ 
2/b/(a+b*arcsech(c*x))+1/2/b^2*exp(2*a/b)*Ei(1,2*a/b+2*arcsech(c*x))-1/4/b 
*(c^2*x^2-2-2*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2))/c^2/x^2/(a+b*a 
rcsech(c*x))+1/2/b^2*exp(-2*a/b)*Ei(1,-2*arcsech(c*x)-2*a/b))
 

3.1.61.5 Fricas [F]

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

input

integrate(1/x^3/(a+b*arcsech(c*x))^2,x, algorithm="fricas")
 

output

integral(1/(b^2*x^3*arcsech(c*x)^2 + 2*a*b*x^3*arcsech(c*x) + a^2*x^3), x)
 

3.1.61.6 Sympy [F]

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

input

integrate(1/x**3/(a+b*asech(c*x))**2,x)
 

output

Integral(1/(x**3*(a + b*asech(c*x))**2), x)
 

3.1.61.7 Maxima [F]

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

input

integrate(1/x^3/(a+b*arcsech(c*x))^2,x, algorithm="maxima")
 

output

-(c^2*x^3 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - x)/((b^2*c^2*x^2 
- b^2)*x^3*log(x) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b)*x^ 
3 - (b^2*x^3*log(x) + (b^2*log(c) - a*b)*x^3)*sqrt(c*x + 1)*sqrt(-c*x + 1) 
 + (sqrt(c*x + 1)*sqrt(-c*x + 1)*b^2*x^3 - (b^2*c^2*x^2 - b^2)*x^3)*log(sq 
rt(c*x + 1)*sqrt(-c*x + 1) + 1)) + integrate(-(2*c^4*x^4 - 4*c^2*x^2 - 2*( 
c*x + 1)*(c*x - 1) + (c^4*x^4 - 4*c^2*x^2 + 4)*sqrt(c*x + 1)*sqrt(-c*x + 1 
) + 2)/((b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*x^3*log(x) + ((b^2*c^4*log(c) 
- a*b*c^4)*x^4 - 2*(b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b)*x^3 
- (b^2*x^3*log(x) + (b^2*log(c) - a*b)*x^3)*(c*x + 1)*(c*x - 1) - 2*((b^2* 
c^2*x^2 - b^2)*x^3*log(x) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + 
 a*b)*x^3)*sqrt(c*x + 1)*sqrt(-c*x + 1) + ((c*x + 1)*(c*x - 1)*b^2*x^3 + 2 
*(b^2*c^2*x^2 - b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^3 - (b^2*c^4*x^4 - 2*b 
^2*c^2*x^2 + b^2)*x^3)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)), x)
 

3.1.61.8 Giac [F]

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

input

integrate(1/x^3/(a+b*arcsech(c*x))^2,x, algorithm="giac")
 

output

integrate(1/((b*arcsech(c*x) + a)^2*x^3), x)
 

3.1.61.9 Mupad [F(-1)]

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

input

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

output

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