3.2.5 \(\int \frac {x (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^2} \, dx\) [105]

3.2.5.1 Optimal result

Integrand size = 19, antiderivative size = 139 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \arctan \left (\sqrt {-1-c^2 x^2}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {c^2 d-e} \sqrt {e} \sqrt {-c^2 x^2}} \]

output

1/2*(-a-b*arccsch(c*x))/e/(e*x^2+d)+1/2*b*c*x*arctan((-c^2*x^2-1)^(1/2))/d 
/e/(-c^2*x^2)^(1/2)+1/2*b*c*x*arctanh(e^(1/2)*(-c^2*x^2-1)^(1/2)/(c^2*d-e) 
^(1/2))/d/(c^2*d-e)^(1/2)/e^(1/2)/(-c^2*x^2)^(1/2)
 

3.2.5.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.95 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a}{d+e x^2}+\frac {2 b \text {csch}^{-1}(c x)}{d+e x^2}-\frac {2 b \text {arcsinh}\left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \log \left (-\frac {4 \left (i d e+c d \sqrt {e} \left (c \sqrt {d}+i \sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \sqrt {-c^2 d+e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d+e}}+\frac {b \sqrt {e} \log \left (\frac {4 i \left (d e+c d \sqrt {e} \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \sqrt {-c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d+e}}}{4 e} \]

input

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

output

-1/4*((2*a)/(d + e*x^2) + (2*b*ArcCsch[c*x])/(d + e*x^2) - (2*b*ArcSinh[1/ 
(c*x)])/d + (b*Sqrt[e]*Log[(-4*(I*d*e + c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-( 
c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*Sqrt[-(c^2*d) + e]*(Sqrt[d] - I* 
Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) + e]) + (b*Sqrt[e]*Log[((4*I)*(d*e + c*d*Sq 
rt[e]*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*Sqrt 
[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) + e]))/e
 

3.2.5.3 Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6854, 354, 97, 73, 218, 221}

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.

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

\(\Big \downarrow \) 6854

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {b c x \left (\frac {2 e \int \frac {1}{-\frac {e x^4}{c^2}+d-\frac {e}{c^2}}d\sqrt {-c^2 x^2-1}}{c^2 d}+\frac {2 \arctan \left (\sqrt {-c^2 x^2-1}\right )}{d}\right )}{4 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b c x \left (\frac {2 \arctan \left (\sqrt {-c^2 x^2-1}\right )}{d}+\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{d \sqrt {c^2 d-e}}\right )}{4 e \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}\)

input

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

output

-1/2*(a + b*ArcCsch[c*x])/(e*(d + e*x^2)) + (b*c*x*((2*ArcTan[Sqrt[-1 - c^ 
2*x^2]])/d + (2*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/Sqrt[c^2*d - 
e]])/(d*Sqrt[c^2*d - e])))/(4*e*Sqrt[-(c^2*x^2)])
 

3.2.5.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 354

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

rule 6854

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

3.2.5.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs. \(2(120)=240\).

Time = 6.22 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.95

method	result	size
parts	\(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arccsch}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {c \sqrt {c^{2} x^{2}+1}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d -e}{e}}}\right )}{c^{2}}\)	\(271\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d -e}{e}}}\right )}{c^{2}}\)	\(282\)
default	\(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsch}\left (c x \right )}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right ) \sqrt {-\frac {c^{2} d -e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x +2 e}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d -e}{e}}}\right )}{c^{2}}\)	\(282\)

input

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

output

-1/2*a/e/(e*x^2+d)+b/c^2*(-1/2*c^4/e/(c^2*e*x^2+c^2*d)*arccsch(c*x)+1/4*c/ 
e*(c^2*x^2+1)^(1/2)*(2*arctanh(1/(c^2*x^2+1)^(1/2))*(-(c^2*d-e)/e)^(1/2)-l 
n(-2*((c^2*x^2+1)^(1/2)*(-(c^2*d-e)/e)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c 
*e*x+(-c^2*d*e)^(1/2)))-ln(-2*(-(c^2*x^2+1)^(1/2)*(-(c^2*d-e)/e)^(1/2)*e+( 
-c^2*d*e)^(1/2)*c*x-e)/(c*e*x+(-c^2*d*e)^(1/2))))/((c^2*x^2+1)/c^2/x^2)^(1 
/2)/x/d/(-(c^2*d-e)/e)^(1/2))
 

3.2.5.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (117) = 234\).

Time = 0.29 (sec) , antiderivative size = 615, normalized size of antiderivative = 4.42 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} - 2 \, a d e + \sqrt {-c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e + e^{2}} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, e}{e x^{2} + d}\right ) - 2 \, {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 2 \, {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} - a d e + \sqrt {c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (-\frac {\sqrt {c^{2} d e - e^{2}} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{c^{2} d - e}\right ) - {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}\right ] \]

input

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

output

[-1/4*(2*a*c^2*d^2 - 2*a*d*e + sqrt(-c^2*d*e + e^2)*(b*e*x^2 + b*d)*log((c 
^2*e*x^2 - c^2*d - 2*sqrt(-c^2*d*e + e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2) 
) + 2*e)/(e*x^2 + d)) - 2*(b*c^2*d^2 - b*d*e + (b*c^2*d*e - b*e^2)*x^2)*lo 
g(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + 2*(b*c^2*d^2 - b*d*e + (b 
*c^2*d*e - b*e^2)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 
2*(b*c^2*d^2 - b*d*e)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)))/ 
(c^2*d^3*e - d^2*e^2 + (c^2*d^2*e^2 - d*e^3)*x^2), -1/2*(a*c^2*d^2 - a*d*e 
 + sqrt(c^2*d*e - e^2)*(b*e*x^2 + b*d)*arctan(-sqrt(c^2*d*e - e^2)*c*x*sqr 
t((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d - e)) - (b*c^2*d^2 - b*d*e + (b*c^2*d*e 
- b*e^2)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + (b*c^2*d^ 
2 - b*d*e + (b*c^2*d*e - b*e^2)*x^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) 
 - c*x - 1) + (b*c^2*d^2 - b*d*e)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 
 1)/(c*x)))/(c^2*d^3*e - d^2*e^2 + (c^2*d^2*e^2 - d*e^3)*x^2)]
 

3.2.5.6 Sympy [F]

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

input

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

output

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

3.2.5.7 Maxima [F]

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

input

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

output

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

3.2.5.8 Giac [F]

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

input

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

output

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

3.2.5.9 Mupad [F(-1)]

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

input

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

output

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