3.2.45 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx\) [145]

3.2.45.1 Optimal result

Integrand size = 23, antiderivative size = 294 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\frac {b c^3 x^2 \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{d \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {b e x \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]

output

-(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/d/x+b*c^3*x^2*(e*x^2+d)^(1/2)/d/(-c^2* 
x^2)^(1/2)/(-c^2*x^2-1)^(1/2)+b*c*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/(-c 
^2*x^2)^(1/2)-b*c^2*x*(1/(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)*EllipticE(c* 
x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))*(e*x^2+d)^(1/2)/d/(-c^2*x^2)^(1/2)/ 
(-c^2*x^2-1)^(1/2)/((e*x^2+d)/d/(c^2*x^2+1))^(1/2)+b*e*x*(1/(c^2*x^2+1))^( 
1/2)*(c^2*x^2+1)^(1/2)*EllipticF(c*x/(c^2*x^2+1)^(1/2),(1-e/c^2/d)^(1/2))* 
(e*x^2+d)^(1/2)/d^2/(-c^2*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)/((e*x^2+d)/d/(c^2* 
x^2+1))^(1/2)
 

3.2.45.2 Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.47 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (-a+b c \sqrt {1+\frac {1}{c^2 x^2}} x-b \text {csch}^{-1}(c x)\right )}{d x}-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} E\left (\arcsin \left (\sqrt {-\frac {e}{d}} x\right )|\frac {c^2 d}{e}\right )}{d \sqrt {-\frac {e}{d}} \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]

input

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

output

(Sqrt[d + e*x^2]*(-a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x - b*ArcCsch[c*x]))/(d*x 
) - (b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sq 
rt[-(e/d)]*x], (c^2*d)/e])/(d*Sqrt[-(e/d)]*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^ 
2])
 

3.2.45.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6856, 25, 27, 377, 27, 324, 320, 388, 313}

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.

\(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 6856

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {b c x \left (\frac {\sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{x}-e \left (c^2 \left (-\int \frac {x^2}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx\right )-\frac {\sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )\right )}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {b c x \left (\frac {\sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{x}-e \left (-\left (c^2 \left (\frac {\int \frac {\sqrt {e x^2+d}}{\left (-c^2 x^2-1\right )^{3/2}}dx}{e}+\frac {x \sqrt {d+e x^2}}{e \sqrt {-c^2 x^2-1}}\right )\right )-\frac {\sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )\right )}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {b c x \left (\frac {\sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{x}-e \left (-\frac {\sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{c d \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\left (c^2 \left (\frac {x \sqrt {d+e x^2}}{e \sqrt {-c^2 x^2-1}}-\frac {\sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{c e \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}\right )\right )\right )\right )}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}\)

input

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

output

-((Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/(d*x)) + (b*c*x*((Sqrt[-1 - c^2*x 
^2]*Sqrt[d + e*x^2])/x - e*(-(c^2*((x*Sqrt[d + e*x^2])/(e*Sqrt[-1 - c^2*x^ 
2]) - (Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(c*e*Sqrt[-1 
 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]))) - (Sqrt[d + e*x^2]*Elli 
pticF[ArcTan[c*x], 1 - e/(c^2*d)])/(c*d*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2 
)/(d*(1 + c^2*x^2))]))))/(d*Sqrt[-(c^2*x^2)])
 

3.2.45.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 324

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b   Int[x^2/(Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c 
] && PosQ[b/a]
 

rule 377

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( 
m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) 
+ 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b 
*c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m 
, 2, p, q, x]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 6856

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( 
x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si 
mp[(a + b*ArcCsch[c*x])   u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[Simpl 
ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, 
 f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 
0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I 
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

3.2.45.4 Maple [F]

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

input

int((a+b*arccsch(c*x))/x^2/(e*x^2+d)^(1/2),x)
 

output

int((a+b*arccsch(c*x))/x^2/(e*x^2+d)^(1/2),x)
 

3.2.45.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.61 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {-c^{2}} b c^{4} d^{\frac {3}{2}} x E(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) + \sqrt {e x^{2} + d} b c^{2} d \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{4} d + b e\right )} \sqrt {-c^{2}} \sqrt {d} x F(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - {\left (b c^{3} d x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - a c^{2} d\right )} \sqrt {e x^{2} + d}}{c^{2} d^{2} x} \]

input

integrate((a+b*arccsch(c*x))/x^2/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

output

-(sqrt(-c^2)*b*c^4*d^(3/2)*x*elliptic_e(arcsin(sqrt(-c^2)*x), e/(c^2*d)) + 
 sqrt(e*x^2 + d)*b*c^2*d*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x) 
) - (b*c^4*d + b*e)*sqrt(-c^2)*sqrt(d)*x*elliptic_f(arcsin(sqrt(-c^2)*x), 
e/(c^2*d)) - (b*c^3*d*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - a*c^2*d)*sqrt(e*x^ 
2 + d))/(c^2*d^2*x)
 

3.2.45.6 Sympy [F]

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

input

integrate((a+b*acsch(c*x))/x**2/(e*x**2+d)**(1/2),x)
 

output

Integral((a + b*acsch(c*x))/(x**2*sqrt(d + e*x**2)), x)
 

3.2.45.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]

input

integrate((a+b*arccsch(c*x))/x^2/(e*x^2+d)^(1/2),x, algorithm="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.2.45.8 Giac [F]

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

input

integrate((a+b*arccsch(c*x))/x^2/(e*x^2+d)^(1/2),x, algorithm="giac")
 

output

integrate((b*arccsch(c*x) + a)/(sqrt(e*x^2 + d)*x^2), x)
 

3.2.45.9 Mupad [F(-1)]

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

input

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

output

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