Integrand size = 21, antiderivative size = 1276 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:
1/16*b*c*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/(-d)^(1/2)/e^(1/2)/(c^2*d+e)/((- d)^(1/2)*e^(1/2)-d/x)+1/16*b*c*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/(-d)^(1/2) /e^(1/2)/(c^2*d+e)/((-d)^(1/2)*e^(1/2)+d/x)+1/16*(a+b*arcsech(c*x))/(-d)^( 1/2)/e^(1/2)/((-d)^(1/2)*e^(1/2)-d/x)^2+1/16*(a+b*arcsech(c*x))/d/e/((-d)^ (1/2)*e^(1/2)-d/x)-1/16*(a+b*arcsech(c*x))/(-d)^(1/2)/e^(1/2)/((-d)^(1/2)* e^(1/2)+d/x)^2-1/16*(a+b*arcsech(c*x))/d/e/((-d)^(1/2)*e^(1/2)+d/x)+1/8*b* arctan((c*d-(-d)^(1/2)*e^(1/2))^(1/2)*(1+1/c/x)^(1/2)/(c*d+(-d)^(1/2)*e^(1 /2))^(1/2)/(-1+1/c/x)^(1/2))/(c*d-(-d)^(1/2)*e^(1/2))^(3/2)/(c*d+(-d)^(1/2 )*e^(1/2))^(3/2)-1/8*b*arctan((c*d-(-d)^(1/2)*e^(1/2))^(1/2)*(1+1/c/x)^(1/ 2)/(c*d+(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x)^(1/2))/d/(c*d-(-d)^(1/2)*e^(1 /2))^(1/2)/(c*d+(-d)^(1/2)*e^(1/2))^(1/2)/e+1/8*b*arctan((c*d+(-d)^(1/2)*e ^(1/2))^(1/2)*(1+1/c/x)^(1/2)/(c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x)^(1 /2))/(c*d-(-d)^(1/2)*e^(1/2))^(3/2)/(c*d+(-d)^(1/2)*e^(1/2))^(3/2)-1/8*b*a rctan((c*d+(-d)^(1/2)*e^(1/2))^(1/2)*(1+1/c/x)^(1/2)/(c*d-(-d)^(1/2)*e^(1/ 2))^(1/2)/(-1+1/c/x)^(1/2))/d/(c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(c*d+(-d)^(1/ 2)*e^(1/2))^(1/2)/e-1/16*(a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1 /c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(3/2) +1/16*(a+b*arcsech(c*x))*ln(1+c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/ x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(3/2)-1/16*(a+b*arcsech( c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/...
Result contains complex when optimal does not.
Time = 6.07 (sec) , antiderivative size = 2030, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
Output:
-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt [e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*(((-1/16*I)*(((-I)*Sqrt[e]*Sqrt[( 1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e ]*x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sq rt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)] ]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2 *d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[ e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x) ) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sq rt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2 *Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e] *x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/( d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) - (-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[ (1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x) /(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I *Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d])/(16*d*e) - (-(ArcSech...
Time = 3.61 (sec) , antiderivative size = 1340, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 6374, 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.
\(\displaystyle \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6857 |
\(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3 x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 6374 |
\(\displaystyle -\int \left (\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d \left (\frac {d}{x^2}+e\right )^2}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )^3}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 \sqrt {-d} \sqrt {e} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{16 d e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{16 d e \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{16 \sqrt {-d} \sqrt {e} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{16 \sqrt {-d} \sqrt {e} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2}}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 d \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2}}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}\) |
Input:
Int[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
Output:
(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e )*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/( 16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcCosh [1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcCo sh[1/(c*x)])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCosh[1/(c*x)])/ (16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcCosh[1/(c*x)] )/(16*d*e*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[ e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])]) /(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) - (b*Ar cTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d] *Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) + (b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - S qrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) - (b*ArcTan[(Sqrt[c *d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sq rt[-1 + 1/(c*x)])])/(8*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]* Sqrt[e]]*e) - ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c *x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcC osh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2* d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCosh[1/(c*x)])*Log[1 - (...
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCosh[x/c])^n/x ^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 ] && IntegersQ[m, p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 72.88 (sec) , antiderivative size = 1398, normalized size of antiderivative = 1.10
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1398\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1417\) |
default | \(\text {Expression too large to display}\) | \(1417\) |
Input:
int(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1 /2)))+b/c^3*(1/8*c^5*x*(c^4*d*e*arcsech(c*x)*x^2-d^2*c^4*arcsech(c*x)-(-(c *x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*e^2*c^3*x^3-(-(c*x-1)/c/x)^(1/2)*((c* x+1)/c/x)^(1/2)*c^3*d*e*x+e^2*arcsech(c*x)*c^2*x^2-c^2*d*e*arcsech(c*x))/d /e/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2-1/16/d/(c^2*d+e)*c^4*sum(_R1/(_R1^2*c^2*d +c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/ _R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c ^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))-1/16/(c^2*d+e)/e*c^6*sum(_R1/(_R1^2*c ^2*d+c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/ 2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=Root Of(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))+1/16/d/(c^2*d+e)*c^4*sum(1/_R1/(_ R1^2*c^2*d+c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/ x)^(1/2))/_R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R 1=RootOf(c^2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))+1/8*(-(c^2*d-2*(e*(c^2*d+e) )^(1/2)+2*e)*d)^(1/2)*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*c*arctanh(c*d*(1/c /x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d )^(1/2))/(c^2*d+e)/e/d^3-1/8*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)* (c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^2+2*(e*(c^2*d+e))^(1/2)*e)*c*arct anh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^ (1/2)-2*e)*d)^(1/2))/(c^2*d+e)^2/e/d^3+1/8*((c^2*d+2*(e*(c^2*d+e))^(1/2...
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b*x^2*arcsech(c*x) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
Timed out. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**2*(a+b*asech(c*x))/(e*x**2+d)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,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
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^2*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*arcsech(c*x) + a)*x^2/(e*x^2 + d)^3, x)
Timed out. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int((x^2*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3,x)
Output:
int((x^2*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3, x)
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+2 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{2}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{4} e^{2}+16 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{2}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3} x^{2}+8 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{2}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e^{4} x^{4}-a \,d^{2} e x +a d \,e^{2} x^{3}}{8 d^{2} e^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int(x^2*(a+b*asech(c*x))/(e*x^2+d)^3,x)
Output:
(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 2*sqrt(e)*sqrt(d)* atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + sqrt(e)*sqrt(d)*atan((e*x)/(sqr t(e)*sqrt(d)))*a*e**2*x**4 + 8*int((asech(c*x)*x**2)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**4*e**2 + 16*int((asech(c*x)*x**2)/(d **3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3*x**2 + 8*i nt((asech(c*x)*x**2)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x) *b*d**2*e**4*x**4 - a*d**2*e*x + a*d*e**2*x**3)/(8*d**2*e**2*(d**2 + 2*d*e *x**2 + e**2*x**4))