3.117 \(\int \frac {x (a+b \text {sech}^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=147 \[ -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e}} \]

[Out]

1/2*(-a-b*arcsech(c*x))/e/(e*x^2+d)+1/2*b*arctanh((-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/d/e-1/2*
b*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/e^(1/2)/(c^2*d+e)^(1/2
)

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Rubi [A]  time = 0.24, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6299, 517, 446, 86, 63, 208} \[ -\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*ArcSech[c*x])/(2*e*(d + e*x^2)) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(2
*d*e) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(2*d*Sqrt[
e]*Sqrt[c^2*d + e])

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 86

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

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 )^2} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\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 )} \, dx}{2 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\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 )} \, dx}{2 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\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)} \, dx,x,x^2\right )}{4 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {\left (b \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 )}{4 d}-\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 )}{4 d e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {\left (b \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 )}{2 c^2 d}+\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 )}{2 c^2 d e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d e}-\frac {b \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 )}{2 d \sqrt {e} \sqrt {c^2 d+e}}\\ \end {align*}

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Mathematica [C]  time = 1.05, size = 345, normalized size = 2.35 \[ -\frac {\frac {2 a}{d+e x^2}+\frac {b \sqrt {e} \log \left (\frac {4 \left (\frac {c^2 d^{3/2} \sqrt {e} x+i d e}{\sqrt {c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}+\frac {d e \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{e x-i \sqrt {d} \sqrt {e}}\right )}{b}\right )}{d \sqrt {c^2 d+e}}+\frac {b \sqrt {e} \log \left (\frac {4 \left (\frac {d e+i c^2 d^{3/2} \sqrt {e} x}{\sqrt {c^2 d+e} \left (\sqrt {e} x+i \sqrt {d}\right )}+\frac {d e \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{e x+i \sqrt {d} \sqrt {e}}\right )}{b}\right )}{d \sqrt {c^2 d+e}}+\frac {2 b \text {sech}^{-1}(c x)}{d+e x^2}-\frac {2 b \log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )}{d}+\frac {2 b \log (x)}{d}}{4 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((2*a)/(d + e*x^2) + (2*b*ArcSech[c*x])/(d + e*x^2) + (2*b*Log[x])/d - (2*b*Log[1 + Sqrt[(1 - c*x)/(1 + c
*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/d + (b*Sqrt[e]*Log[(4*((I*d*e + c^2*d^(3/2)*Sqrt[e]*x)/(Sqrt[c^2*d + e]
*(Sqrt[d] + I*Sqrt[e]*x)) + (d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/((-I)*Sqrt[d]*Sqrt[e] + e*x)))/b])/(d*Sq
rt[c^2*d + e]) + (b*Sqrt[e]*Log[(4*((d*e + I*c^2*d^(3/2)*Sqrt[e]*x)/(Sqrt[c^2*d + e]*(I*Sqrt[d] + Sqrt[e]*x))
+ (d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(I*Sqrt[d]*Sqrt[e] + e*x)))/b])/(d*Sqrt[c^2*d + e]))/e

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fricas [B]  time = 0.66, size = 602, normalized size = 4.10 \[ \left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e - \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{4} d^{2} + 4 \, c^{2} d e - {\left (c^{4} d e + 2 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} d e + c e^{2}\right )} x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 4 \, e^{2} + 2 \, {\left (c^{2} e x^{2} - c^{2} d - {\left (c^{3} d + 2 \, c e\right )} x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, e\right )} \sqrt {c^{2} d e + e^{2}}}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {-c^{2} d e - e^{2}} c d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - \sqrt {-c^{2} d e - e^{2}} {\left (e x^{2} + d\right )}}{{\left (c^{2} d e + e^{2}\right )} x^{2}}\right ) + {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*c^2*d^2 + 2*a*d*e - sqrt(c^2*d*e + e^2)*(b*e*x^2 + b*d)*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)) + 2*(b*c^2*d^2 + b*d*e + (b*c^2*
d*e + b*e^2)*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + 2*(b*c^2*d^2 + b*d*e)*log((c*x*sqrt(-(c^2*
x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2), -1/2*(a*c^2*d^2 + a*d*e +
sqrt(-c^2*d*e - e^2)*(b*e*x^2 + b*d)*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)) + (b*c^2*d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*log((c*x*s
qrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c^2*d^2 + b*d*e)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)
))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.10, size = 840, normalized size = 5.71 \[ -\frac {c^{2} a}{2 e \left (c^{2} x^{2} e +c^{2} d \right )}-\frac {c^{2} b \,\mathrm {arcsech}\left (c x \right )}{2 e \left (c^{2} x^{2} e +c^{2} d \right )}-\frac {c^{3} b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2 \sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (\sqrt {-c^{2} d e}-e \right )}+\frac {c^{3} b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (-\frac {2 \left (\sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d +e}{e}}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c x e +\sqrt {-c^{2} d e}}\right )}{4 \sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (\sqrt {-c^{2} d e}-e \right ) \sqrt {\frac {c^{2} d +e}{e}}}+\frac {c^{3} b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d +e}{e}}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c x e +\sqrt {-c^{2} d e}}\right )}{4 \sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (\sqrt {-c^{2} d e}-e \right ) \sqrt {\frac {c^{2} d +e}{e}}}-\frac {c b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e}{2 \sqrt {-c^{2} x^{2}+1}\, d \left (\sqrt {-c^{2} d e}+e \right ) \left (\sqrt {-c^{2} d e}-e \right )}+\frac {c b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (-\frac {2 \left (\sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d +e}{e}}\, e -\sqrt {-c^{2} d e}\, c x +e \right )}{-c x e +\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {-c^{2} x^{2}+1}\, d \left (\sqrt {-c^{2} d e}+e \right ) \left (\sqrt {-c^{2} d e}-e \right ) \sqrt {\frac {c^{2} d +e}{e}}}+\frac {c b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d +e}{e}}\, e +2 \sqrt {-c^{2} d e}\, c x +2 e}{c x e +\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {-c^{2} x^{2}+1}\, d \left (\sqrt {-c^{2} d e}+e \right ) \left (\sqrt {-c^{2} d e}-e \right ) \sqrt {\frac {c^{2} d +e}{e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^2*a/e/(c^2*e*x^2+c^2*d)-1/2*c^2*b/e/(c^2*e*x^2+c^2*d)*arcsech(c*x)-1/2*c^3*b*(-(c*x-1)/c/x)^(1/2)*x*((c
*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/((-c^2*d*e)^(1/2)+e)/((-c^2*d*e)^(1/2)-e)*arctanh(1/(-c^2*x^2+1)^(1/2))+1/
4*c^3*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/((-c^2*d*e)^(1/2)+e)/((-c^2*d*e)^(1/2)-e
)/((c^2*d+e)/e)^(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)))+1/4*c^3*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/((-c^2*d*e)^(1/2)+e)/((-c^2
*d*e)^(1/2)-e)/((c^2*d+e)/e)^(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)))-1/2*c*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/d/((-c^2*d*e)^(1/2
)+e)/((-c^2*d*e)^(1/2)-e)*arctanh(1/(-c^2*x^2+1)^(1/2))*e+1/4*c*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)/(
-c^2*x^2+1)^(1/2)/d/((-c^2*d*e)^(1/2)+e)/((-c^2*d*e)^(1/2)-e)/((c^2*d+e)/e)^(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)))*e+1/4*c*b*(-(c*x-1)/c/x)^(1/2)*x*((c*x+
1)/c/x)^(1/2)/(-c^2*x^2+1)^(1/2)/d/((-c^2*d*e)^(1/2)+e)/((-c^2*d*e)^(1/2)-e)/((c^2*d+e)/e)^(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)))*e

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (2 \, c^{2} \int \frac {x^{3}}{2 \, {\left (c^{2} d^{2} x^{2} + {\left (c^{2} d e x^{2} - d e\right )} x^{2} + {\left (c^{2} d^{2} x^{2} + {\left (c^{2} d e x^{2} - d e\right )} x^{2} - d^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - d^{2}\right )}}\,{d x} + \frac {x^{2} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) - x^{2} \log \relax (c) - x^{2} \log \relax (x)}{d e x^{2} + d^{2}} - 2 \, \int \frac {x}{2 \, {\left (c^{2} d^{2} x^{2} + {\left (c^{2} d e x^{2} - d e\right )} x^{2} - d^{2}\right )}}\,{d x}\right )} b - \frac {a}{2 \, {\left (e^{2} x^{2} + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*c^2*integrate(1/2*x^3/(c^2*d^2*x^2 + (c^2*d*e*x^2 - d*e)*x^2 + (c^2*d^2*x^2 + (c^2*d*e*x^2 - d*e)*x^2 -
d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) - d^2), x) + (x^2*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) - x^2*log(c) - x^2*
log(x))/(d*e*x^2 + d^2) - 2*integrate(1/2*x/(c^2*d^2*x^2 + (c^2*d*e*x^2 - d*e)*x^2 - d^2), x))*b - 1/2*a/(e^2*
x^2 + d*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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