Integrand size = 21, antiderivative size = 1272 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:
1/16*b*c*(-d)^(1/2)*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/e^(3/2)/(c^2*d+e)/((- d)^(1/2)*e^(1/2)-d/x)+1/16*b*c*(-d)^(1/2)*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2) /e^(3/2)/(c^2*d+e)/((-d)^(1/2)*e^(1/2)+d/x)+1/16*(-d)^(1/2)*(a+b*arcsech(c *x))/e^(3/2)/((-d)^(1/2)*e^(1/2)-d/x)^2+3/16*(a+b*arcsech(c*x))/e^2/((-d)^ (1/2)*e^(1/2)-d/x)-1/16*(-d)^(1/2)*(a+b*arcsech(c*x))/e^(3/2)/((-d)^(1/2)* e^(1/2)+d/x)^2-3/16*(a+b*arcsech(c*x))/e^2/((-d)^(1/2)*e^(1/2)+d/x)-3/8*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))^(1/2)/(c*d+(-d)^(1/2 )*e^(1/2))^(1/2)/e^2-1/8*b*d*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)/e-3/8*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))^(1/2)/(c*d+(-d)^(1/2)*e^(1/2))^(1/2)/e^2 -1/8*b*d*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)/e+3/16*(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)^(1/2) /e^(5/2)-3/16*(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)^(1/2)/e^(5/2)+3/16*(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)...
Result contains complex when optimal does not.
Time = 5.56 (sec) , antiderivative size = 1823, normalized size of antiderivative = 1.43 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
Output:
((b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/((c^2*d + e)*((-I)*Sqrt[d] + Sq rt[e]*x)) + (b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/((c^2*d + e)*(I*Sqrt [d] + Sqrt[e]*x)) + (4*a*d*Sqrt[e]*x)/(d + e*x^2)^2 - (10*a*Sqrt[e]*x)/(d + e*x^2) + (5*b*ArcSech[c*x])/(I*Sqrt[d] - Sqrt[e]*x) + (I*b*Sqrt[d]*ArcSe ch[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (I*b*Sqrt[d]*ArcSech[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 - (5*b*ArcSech[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + (6*a*ArcTan [(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (I*b*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))])/(Sqrt[d]*(c^2*d + e)^(3/2)) - (I*b*S qrt[e]*(2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sq rt[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))])/(Sqr t[d]*(c^2*d + e)^(3/2)) + ((5*I)*b*Sqrt[e]*(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]))/S qrt[d] - ((5*I)*b*Sqrt[e]*(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*Sqrt[e]*(I*Sqrt[d]*S qrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt...
Time = 2.52 (sec) , antiderivative size = 1336, 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, 6324, 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.
| \(\displaystyle \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle -\int \left (-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d^3}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^3}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d^3}{8 (-d)^{3/2} e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^3}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d}{8 e^2 \left (-\frac {d^2}{x^2}-e d\right )}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \sqrt {-d} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 e^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b \sqrt {-d} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 e^{3/2} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {-d} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {-d} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}-\frac {b d \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} e}-\frac {3 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 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e^2}-\frac {b d \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} e}-\frac {3 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 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e^2}+\frac {3 \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 \sqrt {-d} e^{5/2}}-\frac {3 \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 \sqrt {-d} e^{5/2}}+\frac {3 \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 \sqrt {-d} e^{5/2}}-\frac {3 \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 \sqrt {-d} e^{5/2}}-\frac {3 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 \sqrt {-d} e^{5/2}}+\frac {3 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 \sqrt {-d} e^{5/2}}-\frac {3 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 \sqrt {-d} e^{5/2}}+\frac {3 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 \sqrt {-d} e^{5/2}}\) |
Input:
Int[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
Output:
(b*c*Sqrt[-d]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*e^(3/2)*(c^2*d + e )*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[-d]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/ (c*x)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcCosh[1/(c*x)]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b* ArcCosh[1/(c*x)]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*Ar cCosh[1/(c*x)]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcCo sh[1/(c*x)]))/(16*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (3*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*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e] ]*e^2) - (b*d*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqr t[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)*e) - (3*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*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e^2) - (b*d*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)*e) + (3*(a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d] *e^(5/2)) - (3*(a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c *x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*A...
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
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 = 75.24 (sec) , antiderivative size = 1960, normalized size of antiderivative = 1.54
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1960\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1983\) |
| default | \(\text {Expression too large to display}\) | \(1983\) |
Input:
int(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
a*((-5/8/e*x^3-3/8/e^2*d*x)/(e*x^2+d)^2+3/8/e^2/(d*e)^(1/2)*arctan(x*e/(d* e)^(1/2)))+b/c^5*(-1/8*c^7*x*(3*d^2*c^4*arcsech(c*x)+5*c^4*d*e*arcsech(c*x )*x^2-(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c^3*d*e*x-(-(c*x-1)/c/x)^(1 /2)*((c*x+1)/c/x)^(1/2)*e^2*c^3*x^3+3*c^2*d*e*arcsech(c*x)+5*e^2*arcsech(c *x)*c^2*x^2)/e^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)-3/8*(-(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^3*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)^2/e^2/d^2-3/8*((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^3*arctan(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 ^2/d^2+1/2*(-(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^3-1/2*(-(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*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)^2/e/d^3+1 /2*((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*arctan(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^3-1/2*((c^2*d+2*(e*(c^2*d...
\[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b*x^4*arcsech(c*x) + a*x^4)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
Timed out. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**4*(a+b*asech(c*x))/(e*x**2+d)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4*(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
\[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*arcsech(c*x) + a)*x^4/(e*x^2 + d)^3, x)
Timed out. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int((x^4*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3,x)
Output:
int((x^4*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3, x)
\[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3}+16 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e^{4} x^{2}+8 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{4}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{5} x^{4}-3 a \,d^{2} e x -5 a d \,e^{2} x^{3}}{8 d \,e^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int(x^4*(a+b*asech(c*x))/(e*x^2+d)^3,x)
Output:
(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 6*sqrt(e)*sqrt(d )*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/ (sqrt(e)*sqrt(d)))*a*e**2*x**4 + 8*int((asech(c*x)*x**4)/(d**3 + 3*d**2*e* x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3 + 16*int((asech(c*x)*x**4 )/(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**2 + 8*int((asech(c*x)*x**4)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6 ),x)*b*d*e**5*x**4 - 3*a*d**2*e*x - 5*a*d*e**2*x**3)/(8*d*e**3*(d**2 + 2*d *e*x**2 + e**2*x**4))