\(\int x^2 (d+e x^2) (a+b \text {csch}^{-1}(c x)) \, dx\) [78]

Optimal result

Integrand size = 19, antiderivative size = 167 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b \left (20 c^2 d-9 e\right ) x^2 \sqrt {-1-c^2 x^2}}{120 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^4 \sqrt {-1-c^2 x^2}}{20 c \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \left (20 c^2 d-9 e\right ) x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{120 c^4 \sqrt {-c^2 x^2}} \] Output:

1/120*b*(20*c^2*d-9*e)*x^2*(-c^2*x^2-1)^(1/2)/c^3/(-c^2*x^2)^(1/2)+1/20*b* 
e*x^4*(-c^2*x^2-1)^(1/2)/c/(-c^2*x^2)^(1/2)+1/3*d*x^3*(a+b*arccsch(c*x))+1 
/5*e*x^5*(a+b*arccsch(c*x))+1/120*b*(20*c^2*d-9*e)*x*arctan(c*x/(-c^2*x^2- 
1)^(1/2))/c^4/(-c^2*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.71 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {c^2 x^2 \left (8 a c^3 x \left (5 d+3 e x^2\right )+b \sqrt {1+\frac {1}{c^2 x^2}} \left (-9 e+c^2 \left (20 d+6 e x^2\right )\right )\right )+8 b c^5 x^3 \left (5 d+3 e x^2\right ) \text {csch}^{-1}(c x)+b \left (-20 c^2 d+9 e\right ) \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{120 c^5} \] Input:

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

Output:

(c^2*x^2*(8*a*c^3*x*(5*d + 3*e*x^2) + b*Sqrt[1 + 1/(c^2*x^2)]*(-9*e + c^2* 
(20*d + 6*e*x^2))) + 8*b*c^5*x^3*(5*d + 3*e*x^2)*ArcCsch[c*x] + b*(-20*c^2 
*d + 9*e)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(120*c^5)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6856, 27, 363, 262, 224, 216}

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.

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

\(\Big \downarrow \) 6856

\(\displaystyle -\frac {b c x \int \frac {x^2 \left (3 e x^2+5 d\right )}{15 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {x^2 \left (3 e x^2+5 d\right )}{\sqrt {-c^2 x^2-1}}dx}{15 \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c x \left (\frac {1}{4} \left (20 d-\frac {9 e}{c^2}\right ) \int \frac {x^2}{\sqrt {-c^2 x^2-1}}dx-\frac {3 e x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {b c x \left (\frac {1}{4} \left (20 d-\frac {9 e}{c^2}\right ) \left (-\frac {\int \frac {1}{\sqrt {-c^2 x^2-1}}dx}{2 c^2}-\frac {x \sqrt {-c^2 x^2-1}}{2 c^2}\right )-\frac {3 e x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {b c x \left (\frac {1}{4} \left (20 d-\frac {9 e}{c^2}\right ) \left (-\frac {\int \frac {1}{\frac {c^2 x^2}{-c^2 x^2-1}+1}d\frac {x}{\sqrt {-c^2 x^2-1}}}{2 c^2}-\frac {x \sqrt {-c^2 x^2-1}}{2 c^2}\right )-\frac {3 e x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b c x \left (\frac {1}{4} \left (-\frac {\arctan \left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{2 c^3}-\frac {x \sqrt {-c^2 x^2-1}}{2 c^2}\right ) \left (20 d-\frac {9 e}{c^2}\right )-\frac {3 e x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}\)

Input:

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

Output:

(d*x^3*(a + b*ArcCsch[c*x]))/3 + (e*x^5*(a + b*ArcCsch[c*x]))/5 - (b*c*x*( 
(-3*e*x^3*Sqrt[-1 - c^2*x^2])/(4*c^2) + ((20*d - (9*e)/c^2)*(-1/2*(x*Sqrt[ 
-1 - c^2*x^2])/c^2 - ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]]/(2*c^3)))/4))/(15*Sq 
rt[-(c^2*x^2)])
 

Defintions of rubi rules used

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 262

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

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

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]))
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.95

method	result	size
parts	\(a \left (\frac {1}{5} e \,x^{5}+\frac {1}{3} d \,x^{3}\right )+\frac {b \left (\frac {c^{3} \operatorname {arccsch}\left (c x \right ) e \,x^{5}}{5}+\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{3} x^{3}}{3}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-20 d \,c^{3} x \sqrt {c^{2} x^{2}+1}-6 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+20 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+9 e c x \sqrt {c^{2} x^{2}+1}-9 e \,\operatorname {arcsinh}\left (c x \right )\right )}{120 c^{3} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c^{3}}\)	\(158\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-20 d \,c^{3} x \sqrt {c^{2} x^{2}+1}-6 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+20 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+9 e c x \sqrt {c^{2} x^{2}+1}-9 e \,\operatorname {arcsinh}\left (c x \right )\right )}{120 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{3}}\)	\(171\)
default	\(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{5} x^{3}}{3}+\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{5} x^{5}}{5}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-20 d \,c^{3} x \sqrt {c^{2} x^{2}+1}-6 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+20 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+9 e c x \sqrt {c^{2} x^{2}+1}-9 e \,\operatorname {arcsinh}\left (c x \right )\right )}{120 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{3}}\)	\(171\)

Input:

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

Output:

a*(1/5*e*x^5+1/3*d*x^3)+b/c^3*(1/5*c^3*arccsch(c*x)*e*x^5+1/3*arccsch(c*x) 
*d*c^3*x^3-1/120/c^3*(c^2*x^2+1)^(1/2)*(-20*d*c^3*x*(c^2*x^2+1)^(1/2)-6*e* 
c^3*x^3*(c^2*x^2+1)^(1/2)+20*d*c^2*arcsinh(c*x)+9*e*c*x*(c^2*x^2+1)^(1/2)- 
9*e*arcsinh(c*x))/((c^2*x^2+1)/c^2/x^2)^(1/2)/x)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.63 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e x^{5} + 40 \, a c^{5} d x^{3} + 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (20 \, b c^{2} d - 9 \, b e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 8 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3} - 5 \, b c^{5} d - 3 \, b c^{5} e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} e x^{4} + {\left (20 \, b c^{4} d - 9 \, b c^{2} e\right )} x^{2}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{120 \, c^{5}} \] Input:

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

Output:

1/120*(24*a*c^5*e*x^5 + 40*a*c^5*d*x^3 + 8*(5*b*c^5*d + 3*b*c^5*e)*log(c*x 
*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + (20*b*c^2*d - 9*b*e)*log(c*x*s 
qrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) - 8*(5*b*c^5*d + 3*b*c^5*e)*log(c*x*sq 
rt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 8*(3*b*c^5*e*x^5 + 5*b*c^5*d*x^3 
- 5*b*c^5*d - 3*b*c^5*e)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x) 
) + (6*b*c^4*e*x^4 + (20*b*c^4*d - 9*b*c^2*e)*x^2)*sqrt((c^2*x^2 + 1)/(c^2 
*x^2)))/c^5
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.36 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arcsch}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e \] Input:

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

Output:

1/5*a*e*x^5 + 1/3*a*d*x^3 + 1/12*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) 
 + 1)/(c^2*(1/(c^2*x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^2 + 
 log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d + 1/80*(16*x^5*arccsch(c*x) - 
(2*(3*(1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(1/(c^2*x^2) + 1))/(c^4*(1/(c^2*x^2) 
 + 1)^2 - 2*c^4*(1/(c^2*x^2) + 1) + c^4) - 3*log(sqrt(1/(c^2*x^2) + 1) + 1 
)/c^4 + 3*log(sqrt(1/(c^2*x^2) + 1) - 1)/c^4)/c)*b*e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsch} \left (c x \right ) x^{4}d x \right ) b e +\left (\int \mathit {acsch} \left (c x \right ) x^{2}d x \right ) b d +\frac {a d \,x^{3}}{3}+\frac {a e \,x^{5}}{5} \] Input:

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

Output:

(15*int(acsch(c*x)*x**4,x)*b*e + 15*int(acsch(c*x)*x**2,x)*b*d + 5*a*d*x** 
3 + 3*a*e*x**5)/15