3.125 \(\int \frac {x (a+b \text {sech}^{-1}(c x))}{(d+e x^2)^3} \, dx\)
Optimal. Leaf size=217 \[ -\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 d^2 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )} \]
[Out]
1/4*(-a-b*arcsech(c*x))/e/(e*x^2+d)^2+1/4*b*arctanh((-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/d^2/e-
1/8*b*(3*c^2*d+2*e)*arctanh(e^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*d+e)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/d^2/(c
^2*d+e)^(3/2)/e^(1/2)-1/8*b*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)
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Rubi [A] time = 0.29, antiderivative size = 217, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used =
{6299, 517, 446, 103, 156, 63, 208} \[ -\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 d^2 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (3 c^2 d+2 e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
[Out]
-(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(8*d*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSech[c*x])
/(4*e*(d + e*x^2)^2) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(4*d^2*e) - (b*(3*c^2
*d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt
[e]*(c^2*d + e)^(3/2))
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 156
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 517
Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))
Rule 6299
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcSech[c*x]))/(2*e*(p + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(2*e*(p + 1)), Int[(d +
e*x^2)^(p + 1)/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 e}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {c^2 d+e-\frac {1}{2} c^2 e x}{x \sqrt {1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{8 d e \left (c^2 d+e\right )}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{8 d^2 e}+\frac {\left (b \left (\frac {1}{2} c^2 d e+e \left (c^2 d+e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{8 d^2 e \left (c^2 d+e\right )}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{4 c^2 d^2 e}-\frac {\left (b \left (\frac {1}{2} c^2 d e+e \left (c^2 d+e\right )\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{d+\frac {e}{c^2}-\frac {e x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{4 c^2 d^2 e \left (c^2 d+e\right )}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.10, size = 486, normalized size = 2.24 \[ \frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {c^2 d+e} \left (c x \sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 d+e}+\sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 d+e}-i c^2 \sqrt {d} x+\sqrt {e}\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {e} x-i \sqrt {d}\right )}\right )}{d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {c^2 d+e} \left (c x \sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 d+e}+\sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 d+e}+i c^2 \sqrt {d} x+\sqrt {e}\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}-\frac {2 \sqrt {\frac {1-c x}{c x+1}} (b c x+b)}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac {4 b \log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )}{d^2 e}-\frac {4 b \text {sech}^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac {4 b \log (x)}{d^2 e}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
[Out]
((-4*a)/(e*(d + e*x^2)^2) - (2*Sqrt[(1 - c*x)/(1 + c*x)]*(b + b*c*x))/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcSe
ch[c*x])/(e*(d + e*x^2)^2) - (4*b*Log[x])/(d^2*e) + (4*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x
)/(1 + c*x)]])/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + S
qrt[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)]))/(b*(3*c^2*d + 2*e)*
((-I)*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*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)]))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)))/16
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fricas [B] time = 0.72, size = 1232, normalized size = 5.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
[Out]
[-1/16*(4*a*c^4*d^4 + 2*(4*a + b)*c^2*d^3*e + 2*(2*a + b)*d^2*e^2 + 2*(b*c^2*d*e^3 + b*e^4)*x^4 + 4*(b*c^2*d^2
*e^2 + b*d*e^3)*x^2 - (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)
*x^2)*sqrt(c^2*d*e + e^2)*log((c^4*d^2 + 4*c^2*d*e - (c^4*d*e + 2*c^2*e^2)*x^2 + 4*(c^3*d*e + c*e^2)*x*sqrt(-(
c^2*x^2 - 1)/(c^2*x^2)) + 4*e^2 + 2*(c^2*e*x^2 - c^2*d - (c^3*d + 2*c*e)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 2*
e)*sqrt(c^2*d*e + e^2))/(e*x^2 + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*d*e
^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1
)/x) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*((b*c
^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^6*e + 2*c^
2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)*
x^2), -1/8*(2*a*c^4*d^4 + (4*a + b)*c^2*d^3*e + (2*a + b)*d^2*e^2 + (b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^2*d^2*e
^2 + b*d*e^3)*x^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)*x
^2)*sqrt(-c^2*d*e - e^2)*arctan((sqrt(-c^2*d*e - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - sqrt(-c^2*d*e - e
^2)*(e*x^2 + d))/((c^2*d*e + e^2)*x^2)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2 + (b*c^4*d^2*e^2 + 2*b*c^2*
d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))
- 1)/x) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + ((b*
c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^6*e + 2*c
^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)
*x^2)]
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")
[Out]
integrate((b*arcsech(c*x) + a)*x/(e*x^2 + d)^3, x)
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maple [B] time = 0.10, size = 3289, normalized size = 15.16 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arcsech(c*x))/(e*x^2+d)^3,x)
[Out]
-1/4*c^4*a/e/(c^2*e*x^2+c^2*d)^2-1/4*c^4*b/e/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)-1/4*c^7*b*(-(c*x-1)/c/x)^(1/2)*x
^3*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*
e)^(1/2)+e)^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-1/4*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^
(1/2)*e^2/(-c*x*e+(-c^2*d*e)^(1/2))*d/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(
-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))+3/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^3/(-
c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+
e)^2/(-c^2*x^2+1)^(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*x*e+(-c^2*d*
e)^(1/2)))+3/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^2/(-c*x*e+(-c^2*d*e)^(1/2))*d/((c^2*d+e)/e)
^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))+3/16*c^7*b*(-(c*x-1)/c/x)^
(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^
2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))+3/16*c^7*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^2/(-c
*x*e+(-c^2*d*e)^(1/2))*d/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)
+e)^2/(-c^2*x^2+1)^(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*x*e+(-c^2
*d*e)^(1/2)))+1/8*c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*
d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2-1/2*c^5*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/
2)*e^4/(-c*x*e+(-c^2*d*e)^(1/2))/d/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^
2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-1/2*c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+
(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(1/2)*ar
ctanh(1/(-c^2*x^2+1)^(1/2))+5/16*c^5*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^4/(-c*x*e+(-c^2*d*e)^(1/
2))/d/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^
(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*x*e+(-c^2*d*e)^(1/2)))+5/16*c^
5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*
d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))+5/16*c^5*b*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c
/x)^(1/2)*e^4/(-c*x*e+(-c^2*d*e)^(1/2))/d/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/
((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))+5/16*c^5*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(-c*x*e+(-c^2*d*e)^(1
/2))/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*x^2+1)^(
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*x*e+(-c^2*d*e)^(1/2)))+1/8*c^
3*b*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d
*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d-1/4*c^3*b*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(-c*x*e+(-c
^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d^2/(-c^2*x^2+1)^(1/2)*a
rctanh(1/(-c^2*x^2+1)^(1/2))-1/4*c^3*b*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2)
)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*
x^2+1)^(1/2))+1/8*c^3*b*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/
e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d^2/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))+1/8*c^3*b*e^4*(-(c*x
-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((
-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))+1/8*c^3*b*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)
/(-c*x*e+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/
2)+e)^2/d^2/(-c^2*x^2+1)^(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*x*e
+(-c^2*d*e)^(1/2)))+1/8*c^3*b*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c*x*e+(-c^2*d*e)^(1/2))/((c^2*d
+e)/e)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/((-c^2*d*e)^(1/2)-e)^2/((-c^2*d*e)^(1/2)+e)^2/d/(-c^2*x^2+1)^(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*x*e+(-c^2*d*e)^(1/2)))
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3,x)
[Out]
int((x*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3, x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*asech(c*x))/(e*x**2+d)**3,x)
[Out]
Timed out
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