3.2.25 \(\int \frac {x (a+b \text {sech}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [125]
Optimal. Leaf size=217 \[ -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}} \]
[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 = 0.22, 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 = {6434, 531, 457,
105, 162, 65, 214} \begin {gather*} -\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 )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
[Out]
-1/8*(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(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*Sq
rt[e]*(c^2*d + e)^(3/2))
Rule 65
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 105
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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Rule 162
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 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 457
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 531
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 6434
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]/(2*e*(p + 1)))*Sqrt[1/(1 + c*x)], 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 0.75, size = 486, normalized size = 2.24 \begin {gather*} \frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}-\frac {2 \sqrt {\frac {1-c x}{1+c x}} (b+b c x)}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \text {sech}^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac {4 b \log (x)}{d^2 e}+\frac {4 b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d+e} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (3 c^2 d+2 e\right ) \left (-i \sqrt {d}+\sqrt {e} x\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 (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d+e} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (3 c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2}}\right ) \end {gather*}
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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3304\) vs.
\(2(186)=372\).
time = 4.54, size = 3305, normalized size = 15.23
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| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(3305\) |
| default |
\(\text {Expression too large to display}\) |
\(3305\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arcsech(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
[Out]
1/c^2*(-1/4*a*c^6/e/(c^2*e*x^2+c^2*d)^2-1/4*b*c^6/e/(c^2*e*x^2+c^2*d)^2*arcsech(c*x)-1/4*b*c^9*(-(c*x-1)/c/x)^
(1/2)*x*((c*x+1)/c/x)^(1/2)*e^2*d/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(
e+(-c^2*d*e)^(1/2))^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-1/4*b*c^9*(-(c*x-1)/c/x)^(1/2)*x^3*((c*
x+1)/c/x)^(1/2)*e^3/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(
1/2))^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))+3/16*b*c^9*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)
*e^2*d/(e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2
*d*e)^(1/2))^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)/(e*c*
x+(-c^2*d*e)^(1/2)))+3/16*b*c^9*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^3/(e*c*x+(-c^2*d*e)^(1/2))/((c^
2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(e*c*x+(-c^2*d*e)^(1/2)))+3/16*b*c^9*(-(c
*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^2*d/(e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(
1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(-e*c*x+(-c^2*d*e)^(1/2)))+3/16*b*c^9*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)
^(1/2)*e^3/(e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(
-c^2*d*e)^(1/2))^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)/
(-e*c*x+(-c^2*d*e)^(1/2)))-1/2*b*c^7*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(e*c*x+(-c^2*d*e)^(1/2))/(
-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2
+1)^(1/2))-1/2*b*c^7*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^4/d/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2
*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))+1
/8*b*c^7*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/(-e
+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^2+5/16*b*c^7*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(e*c*x+(
-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(e*c*x+(-c^2*d*e)^(1
/2)))+5/16*b*c^7*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^4/d/(e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/
2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(e*c*x+(-c^2*d*e)^(1/2)))+5/16*b*c^7*(-(c*x-1)/c/x)^(1
/2)*x*((c*x+1)/c/x)^(1/2)*e^3/(e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2
*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(-e*c*x+(-c^2*d*e)^(1/2)))+5/16*b*c^7*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)*e^4/d/(e
*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/(-e*c*x+(-c^2*d*e)^(1/2))/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/
2))^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)/(-e*c*x+(-c^2
*d*e)^(1/2)))-1/4*b*c^5*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*
d*e)^(1/2))/d/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))-
1/4*b*c^5*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/
d^2/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^2/(-c^2*x^2+1)^(1/2)*arctanh(1/(-c^2*x^2+1)^(1/2))+1/8*b*c^5*
e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/d/(-e+(-c^2*
d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^2+1/8*b*c^5*e^4*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(e*c*x+(-c^2*d*e
)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/d/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(e*c*x+(-c^2*d*e)^(1/2)))+
1/8*b*c^5*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/
((c^2*d+e)/e)^(1/2)/d^2/(-e+(-c^2*d*e)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(e*c*x+(-c^2*d*e)^(1/2)))+1/8*b*c^5*e^4*(-(c*x-1)/c/x)^(1/2
)*x*((c*x+1)/c/x)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/((c^2*d+e)/e)^(1/2)/d/(-e+(-c^2*d*e
)^(1/2))^2/(e+(-c^2*d*e)^(1/2))^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)/(-e*c*x+(-c^2*d*e)^(1/2)))+1/8*b*c^5*e^5*(-(c*x-1)/c/x)^(1/2)*x^3*((c*x+1)/c/x)^(1/2)/(e*c*x+(-
c^2*d*e)^(1/2))/(-e*c*x+(-c^2*d*e)^(1/2))/((c^2...
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1526 vs.
\(2 (137) = 274\).
time = 0.56, size = 3218, normalized size = 14.83 \begin {gather*} \text {Too large to display} \end {gather*}
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*b*x^4*cosh(1)^4 + 2*b*x^4*sinh(1)^4 + 2*(4*a + b)*c^2*d^3*cosh(1) + 2*(b*c^2*d*x^4 + 2
*b*d*x^2)*cosh(1)^3 + 2*(b*c^2*d*x^4 + 4*b*x^4*cosh(1) + 2*b*d*x^2)*sinh(1)^3 + 2*(2*b*c^2*d^2*x^2 + (2*a + b)
*d^2)*cosh(1)^2 + 2*(2*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + (2*a + b)*d^2 + 3*(b*c^2*d*x^4 + 2*b*d*x^2)*cosh(1)
)*sinh(1)^2 - (2*b*x^4*cosh(1)^3 + 2*b*x^4*sinh(1)^3 + 3*b*c^2*d^3 + (3*b*c^2*d*x^4 + 4*b*d*x^2)*cosh(1)^2 + (
3*b*c^2*d*x^4 + 6*b*x^4*cosh(1) + 4*b*d*x^2)*sinh(1)^2 + 2*(3*b*c^2*d^2*x^2 + b*d^2)*cosh(1) + 2*(3*b*c^2*d^2*
x^2 + 3*b*x^4*cosh(1)^2 + b*d^2 + (3*b*c^2*d*x^4 + 4*b*d*x^2)*cosh(1))*sinh(1))*sqrt((c^2*d + cosh(1) + sinh(1
))/(cosh(1) - sinh(1)))*log((c^4*d^2 - 2*(c^2*x^2 - 2)*cosh(1)^2 - 2*(c^2*x^2 - 2)*sinh(1)^2 - (c^4*d*x^2 - 4*
c^2*d)*cosh(1) - (c^4*d*x^2 - 4*c^2*d + 4*(c^2*x^2 - 2)*cosh(1))*sinh(1) - 2*(c^2*d - (c^2*x^2 - 2)*cosh(1) -
(c^2*x^2 - 2)*sinh(1))*sqrt((c^2*d + cosh(1) + sinh(1))/(cosh(1) - sinh(1))) + 2*(2*c^3*d*x*cosh(1) + 2*c*x*co
sh(1)^2 + 2*c*x*sinh(1)^2 + 2*(c^3*d*x + 2*c*x*cosh(1))*sinh(1) - (c^3*d*x + 2*c*x*cosh(1) + 2*c*x*sinh(1))*sq
rt((c^2*d + cosh(1) + sinh(1))/(cosh(1) - sinh(1))))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(x^2*cosh(1) + x^2*sinh(1
) + d)) + 4*(b*c^4*d^4 + b*x^4*cosh(1)^4 + b*x^4*sinh(1)^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^3 + 2*(b*c^2*d*
x^4 + 2*b*x^4*cosh(1) + b*d*x^2)*sinh(1)^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^2 + (b*c^4*d^2*
x^4 + 4*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + b*d^2 + 6*(b*c^2*d*x^4 + b*d*x^2)*cosh(1))*sinh(1)^2 + 2*(b*c^4*d^
3*x^2 + b*c^2*d^3)*cosh(1) + 2*(b*c^4*d^3*x^2 + 2*b*x^4*cosh(1)^3 + b*c^2*d^3 + 3*(b*c^2*d*x^4 + b*d*x^2)*cosh
(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*cosh(1))*sinh(1))*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*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 + 2*(b*c^2*d^3 + b*d^2*cosh(1))
*sinh(1))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(4*b*x^4*cosh(1)^3 + (4*a + b)*c^2*d^3 + 3*(
b*c^2*d*x^4 + 2*b*d*x^2)*cosh(1)^2 + 2*(2*b*c^2*d^2*x^2 + (2*a + b)*d^2)*cosh(1))*sinh(1) + 2*(b*c^3*d^3*x*cos
h(1) + b*c*d*x^3*cosh(1)^3 + b*c*d*x^3*sinh(1)^3 + (b*c^3*d^2*x^3 + b*c*d^2*x)*cosh(1)^2 + (b*c^3*d^2*x^3 + 3*
b*c*d*x^3*cosh(1) + b*c*d^2*x)*sinh(1)^2 + (b*c^3*d^3*x + 3*b*c*d*x^3*cosh(1)^2 + 2*(b*c^3*d^2*x^3 + b*c*d^2*x
)*cosh(1))*sinh(1))*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^6*cosh(1) + d^2*x^4*cosh(1)^5 + d^2*x^4*sinh(1)^5 +
2*(c^2*d^3*x^4 + d^3*x^2)*cosh(1)^4 + (2*c^2*d^3*x^4 + 5*d^2*x^4*cosh(1) + 2*d^3*x^2)*sinh(1)^4 + (c^4*d^4*x^
4 + 4*c^2*d^4*x^2 + d^4)*cosh(1)^3 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 + 10*d^2*x^4*cosh(1)^2 + d^4 + 8*(c^2*d^3*x^
4 + d^3*x^2)*cosh(1))*sinh(1)^3 + 2*(c^4*d^5*x^2 + c^2*d^5)*cosh(1)^2 + (2*c^4*d^5*x^2 + 10*d^2*x^4*cosh(1)^3
+ 2*c^2*d^5 + 12*(c^2*d^3*x^4 + d^3*x^2)*cosh(1)^2 + 3*(c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)*cosh(1))*sinh(1)^2
+ (c^4*d^6 + 5*d^2*x^4*cosh(1)^4 + 8*(c^2*d^3*x^4 + d^3*x^2)*cosh(1)^3 + 3*(c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)
*cosh(1)^2 + 4*(c^4*d^5*x^2 + c^2*d^5)*cosh(1))*sinh(1)), -1/8*(2*a*c^4*d^4 + b*x^4*cosh(1)^4 + b*x^4*sinh(1)^
4 + (4*a + b)*c^2*d^3*cosh(1) + (b*c^2*d*x^4 + 2*b*d*x^2)*cosh(1)^3 + (b*c^2*d*x^4 + 4*b*x^4*cosh(1) + 2*b*d*x
^2)*sinh(1)^3 + (2*b*c^2*d^2*x^2 + (2*a + b)*d^2)*cosh(1)^2 + (2*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + (2*a + b)
*d^2 + 3*(b*c^2*d*x^4 + 2*b*d*x^2)*cosh(1))*sinh(1)^2 + (2*b*x^4*cosh(1)^3 + 2*b*x^4*sinh(1)^3 + 3*b*c^2*d^3 +
(3*b*c^2*d*x^4 + 4*b*d*x^2)*cosh(1)^2 + (3*b*c^2*d*x^4 + 6*b*x^4*cosh(1) + 4*b*d*x^2)*sinh(1)^2 + 2*(3*b*c^2*
d^2*x^2 + b*d^2)*cosh(1) + 2*(3*b*c^2*d^2*x^2 + 3*b*x^4*cosh(1)^2 + b*d^2 + (3*b*c^2*d*x^4 + 4*b*d*x^2)*cosh(1
))*sinh(1))*sqrt(-(c^2*d + cosh(1) + sinh(1))/(cosh(1) - sinh(1)))*arctan((c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)
) - x^2*cosh(1) - x^2*sinh(1) - d)*sqrt(-(c^2*d + cosh(1) + sinh(1))/(cosh(1) - sinh(1)))/(c^2*d*x^2*cosh(1) +
x^2*cosh(1)^2 + x^2*sinh(1)^2 + (c^2*d*x^2 + 2*x^2*cosh(1))*sinh(1))) + 2*(b*c^4*d^4 + b*x^4*cosh(1)^4 + b*x^
4*sinh(1)^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^3 + 2*(b*c^2*d*x^4 + 2*b*x^4*cosh(1) + b*d*x^2)*sinh(1)^3 + (b
*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + b*d
^2 + 6*(b*c^2*d*x^4 + b*d*x^2)*cosh(1))*sinh(1)^2 + 2*(b*c^4*d^3*x^2 + b*c^2*d^3)*cosh(1) + 2*(b*c^4*d^3*x^2 +
2*b*x^4*cosh(1)^3 + b*c^2*d^3 + 3*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^
2)*cosh(1))*sinh(1))*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*cosh(1) + b*
d^2*cosh(1)^2 + b*d^2*sinh(1)^2 + 2*(b*c^2*d^3 + b*d^2*cosh(1))*sinh(1))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2
)) + 1)/(c*x)) + (4*b*x^4*cosh(1)^3 + (4*a + b)*c^2*d^3 + 3*(b*c^2*d*x^4 + 2*b*d*x^2)*cosh(1)^2 + 2*(2*b*c^2*d
^2*x^2 + (2*a + b)*d^2)*cosh(1))*sinh(1) + (b*c^3*d^3*x*cosh(1) + b*c*d*x^3*cosh(1)^3 + b*c*d*x^3*sinh(1)^3 +
(b*c^3*d^2*x^3 + b*c*d^2*x)*cosh(1)^2 + (b*c^3*d^2*x^3 + 3*b*c*d*x^3*cosh(1) + b*c*d^2*x)*sinh(1)^2 + (b*c^3*d
^3*x + 3*b*c*d*x^3*cosh(1)^2 + 2*(b*c^3*d^2*x^3...
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}
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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