\(\int x (d+e x^2) (a+b \text {csch}^{-1}(c x)) \, dx\) [86]

Optimal result

Integrand size = 17, antiderivative size = 146 \[ \int x \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b \left (2 c^2 d-e\right ) x \sqrt {-1-c^2 x^2}}{4 c^3 \sqrt {-c^2 x^2}}-\frac {b e x \left (-1-c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b c d^2 x \arctan \left (\sqrt {-1-c^2 x^2}\right )}{4 e \sqrt {-c^2 x^2}} \] Output:

1/4*b*(2*c^2*d-e)*x*(-c^2*x^2-1)^(1/2)/c^3/(-c^2*x^2)^(1/2)-1/12*b*e*x*(-c 
^2*x^2-1)^(3/2)/c^3/(-c^2*x^2)^(1/2)+1/4*(e*x^2+d)^2*(a+b*arccsch(c*x))/e- 
1/4*b*c*d^2*x*arctan((-c^2*x^2-1)^(1/2))/e/(-c^2*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53 \[ \int x \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {x \left (3 a c^3 x \left (2 d+e x^2\right )+b \sqrt {1+\frac {1}{c^2 x^2}} \left (-2 e+c^2 \left (6 d+e x^2\right )\right )+3 b c^3 x \left (2 d+e x^2\right ) \text {csch}^{-1}(c x)\right )}{12 c^3} \] Input:

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

Output:

(x*(3*a*c^3*x*(2*d + e*x^2) + b*Sqrt[1 + 1/(c^2*x^2)]*(-2*e + c^2*(6*d + e 
*x^2)) + 3*b*c^3*x*(2*d + e*x^2)*ArcCsch[c*x]))/(12*c^3)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6854, 354, 99, 2009}

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.

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

\(\Big \downarrow \) 6854

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

\(\Big \downarrow \) 354

\(\displaystyle \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b c x \int \frac {\left (e x^2+d\right )^2}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{8 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b c x \int \left (\frac {d^2}{x^2 \sqrt {-c^2 x^2-1}}-\frac {e^2 \sqrt {-c^2 x^2-1}}{c^2}-\frac {e \left (e-2 c^2 d\right )}{c^2 \sqrt {-c^2 x^2-1}}\right )dx^2}{8 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b c x \left (2 d^2 \arctan \left (\sqrt {-c^2 x^2-1}\right )-\frac {2 e \sqrt {-c^2 x^2-1} \left (2 c^2 d-e\right )}{c^4}+\frac {2 e^2 \left (-c^2 x^2-1\right )^{3/2}}{3 c^4}\right )}{8 e \sqrt {-c^2 x^2}}\)

Input:

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

Output:

((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/(4*e) - (b*c*x*((-2*(2*c^2*d - e)*e*S 
qrt[-1 - c^2*x^2])/c^4 + (2*e^2*(-1 - c^2*x^2)^(3/2))/(3*c^4) + 2*d^2*ArcT 
an[Sqrt[-1 - c^2*x^2]]))/(8*e*Sqrt[-(c^2*x^2)])
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.24

method	result	size
parts	\(\frac {a \left (x^{2} e +d \right )^{2}}{4 e}+\frac {b \left (\frac {c^{2} e \,\operatorname {arccsch}\left (c x \right ) x^{4}}{4}+\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{2} x^{2}}{2}+\frac {c^{2} \operatorname {arccsch}\left (c x \right ) d^{2}}{4 e}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (3 c^{4} d^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-6 c^{2} d e \sqrt {c^{2} x^{2}+1}-e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 e^{2} \sqrt {c^{2} x^{2}+1}\right )}{12 c^{3} e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c^{2}}\)	\(181\)
derivativedivides	\(\frac {\frac {a \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) c^{4} d^{2}}{4 e}+\frac {\operatorname {arccsch}\left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {e \,\operatorname {arccsch}\left (c x \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (3 c^{4} d^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-6 c^{2} d e \sqrt {c^{2} x^{2}+1}-e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 e^{2} \sqrt {c^{2} x^{2}+1}\right )}{12 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{2}}\)	\(195\)
default	\(\frac {\frac {a \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) c^{4} d^{2}}{4 e}+\frac {\operatorname {arccsch}\left (c x \right ) c^{4} d \,x^{2}}{2}+\frac {e \,\operatorname {arccsch}\left (c x \right ) c^{4} x^{4}}{4}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (3 c^{4} d^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-6 c^{2} d e \sqrt {c^{2} x^{2}+1}-e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 e^{2} \sqrt {c^{2} x^{2}+1}\right )}{12 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{2}}\)	\(195\)

Input:

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

Output:

1/4*a*(e*x^2+d)^2/e+b/c^2*(1/4*c^2*e*arccsch(c*x)*x^4+1/2*arccsch(c*x)*d*c 
^2*x^2+1/4*c^2/e*arccsch(c*x)*d^2-1/12/c^3/e*(c^2*x^2+1)^(1/2)*(3*c^4*d^2* 
arctanh(1/(c^2*x^2+1)^(1/2))-6*c^2*d*e*(c^2*x^2+1)^(1/2)-e^2*c^2*x^2*(c^2* 
x^2+1)^(1/2)+2*e^2*(c^2*x^2+1)^(1/2))/((c^2*x^2+1)/c^2/x^2)^(1/2)/x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.84 \[ \int x \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {3 \, a c^{3} e x^{4} + 6 \, a c^{3} d x^{2} + 3 \, {\left (b c^{3} e x^{4} + 2 \, b c^{3} d x^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} e x^{3} + 2 \, {\left (3 \, b c^{2} d - b e\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{3}} \] Input:

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

Output:

1/12*(3*a*c^3*e*x^4 + 6*a*c^3*d*x^2 + 3*(b*c^3*e*x^4 + 2*b*c^3*d*x^2)*log( 
(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (b*c^2*e*x^3 + 2*(3*b*c^2 
*d - b*e)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^3
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int x \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e \] Input:

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

Output:

1/4*a*e*x^4 + 1/2*a*d*x^2 + 1/2*(x^2*arccsch(c*x) + x*sqrt(1/(c^2*x^2) + 1 
)/c)*b*d + 1/12*(3*x^4*arccsch(c*x) + (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 3 
*x*sqrt(1/(c^2*x^2) + 1))/c^3)*b*e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsch} \left (c x \right ) x^{3}d x \right ) b e +\left (\int \mathit {acsch} \left (c x \right ) x d x \right ) b d +\frac {a d \,x^{2}}{2}+\frac {a e \,x^{4}}{4} \] Input:

int(x*(e*x^2+d)*(a+b*acsch(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(4*int(acsch(c*x)*x**3,x)*b*e + 4*int(acsch(c*x)*x,x)*b*d + 2*a*d*x**2 + a 
*e*x**4)/4