3.112 \(\int \frac {x^3 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=167 \[ \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c x \left (c^2 d-2 e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{8 d e^{3/2} \sqrt {-c^2 x^2} \left (c^2 d-e\right )^{3/2}}-\frac {b c x \sqrt {-c^2 x^2-1}}{8 e \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \left (d+e x^2\right )} \]

[Out]

1/4*x^4*(a+b*arccsch(c*x))/d/(e*x^2+d)^2+1/8*b*c*(c^2*d-2*e)*x*arctanh(e^(1/2)*(-c^2*x^2-1)^(1/2)/(c^2*d-e)^(1
/2))/d/(c^2*d-e)^(3/2)/e^(3/2)/(-c^2*x^2)^(1/2)-1/8*b*c*x*(-c^2*x^2-1)^(1/2)/(c^2*d-e)/e/(e*x^2+d)/(-c^2*x^2)^
(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {264, 6302, 12, 446, 78, 63, 208} \[ \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c x \left (c^2 d-2 e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{8 d e^{3/2} \sqrt {-c^2 x^2} \left (c^2 d-e\right )^{3/2}}-\frac {b c x \sqrt {-c^2 x^2-1}}{8 e \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

-(b*c*x*Sqrt[-1 - c^2*x^2])/(8*(c^2*d - e)*e*Sqrt[-(c^2*x^2)]*(d + e*x^2)) + (x^4*(a + b*ArcCsch[c*x]))/(4*d*(
d + e*x^2)^2) + (b*c*(c^2*d - 2*e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/Sqrt[c^2*d - e]])/(8*d*(c^2*d - e)^(
3/2)*e^(3/2)*Sqrt[-(c^2*x^2)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 6302

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

\begin {align*} \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c x) \int \frac {x^3}{4 d \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c x) \int \frac {x^3}{\sqrt {-1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d \sqrt {-c^2 x^2}}\\ &=\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1-c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 d \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{8 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {\left (b c \left (c^2 d-2 e\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{16 d \left (c^2 d-e\right ) e \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{8 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {\left (b \left (c^2 d-2 e\right ) 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 )}{8 c d \left (c^2 d-e\right ) e \sqrt {-c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1-c^2 x^2}}{8 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (c^2 d-2 e\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{8 d \left (c^2 d-e\right )^{3/2} e^{3/2} \sqrt {-c^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 1.48, size = 375, normalized size = 2.25 \[ -\frac {\frac {8 a}{d+e x^2}-\frac {4 a d}{\left (d+e x^2\right )^2}+\frac {b \sqrt {e} \left (2 e-c^2 d\right ) \log \left (\frac {16 d e^{3/2} \sqrt {e-c^2 d} \left (\sqrt {e}+c x \left (\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {e-c^2 d}-i c \sqrt {d}\right )\right )}{b \left (2 e-c^2 d\right ) \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{d \left (e-c^2 d\right )^{3/2}}+\frac {b \sqrt {e} \left (2 e-c^2 d\right ) \log \left (-\frac {16 i d e^{3/2} \sqrt {e-c^2 d} \left (\sqrt {e}+c x \left (\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {e-c^2 d}+i c \sqrt {d}\right )\right )}{b \left (c^2 d-2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \left (e-c^2 d\right )^{3/2}}-\frac {2 b c e x \sqrt {\frac {1}{c^2 x^2}+1}}{\left (e-c^2 d\right ) \left (d+e x^2\right )}+\frac {4 b \text {csch}^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}-\frac {4 b \sinh ^{-1}\left (\frac {1}{c x}\right )}{d}}{16 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

-1/16*((-4*a*d)/(d + e*x^2)^2 + (8*a)/(d + e*x^2) - (2*b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x)/((-(c^2*d) + e)*(d + e*x
^2)) + (4*b*(d + 2*e*x^2)*ArcCsch[c*x])/(d + e*x^2)^2 - (4*b*ArcSinh[1/(c*x)])/d + (b*Sqrt[e]*(-(c^2*d) + 2*e)
*Log[(16*d*e^(3/2)*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])
*x))/(b*(-(c^2*d) + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(-(c^2*d) + e)^(3/2)) + (b*Sqrt[e]*(-(c^2*d) + 2*e)*Log
[((-16*I)*d*e^(3/2)*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x
))/(b*(c^2*d - 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(-(c^2*d) + e)^(3/2)))/e^2

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fricas [B]  time = 3.05, size = 1381, normalized size = 8.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*a*c^4*d^4 - 8*a*c^2*d^3*e + 4*a*d^2*e^2 + 8*(a*c^4*d^3*e - 2*a*c^2*d^2*e^2 + a*d*e^3)*x^2 + (b*c^2*d
^3 + (b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(b*c^2*d^2*e - 2*b*d*e^2)*x^2)*sqrt(-c^2*d*e + e^2)*log((c^2*
e*x^2 - c^2*d - 2*sqrt(-c^2*d*e + e^2)*c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*e)/(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)) - c*x + 1) + 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)) - c*x - 1) + 4*(b*c^4*d^4 - 2*b*c^2*d^3*e + b*d^2*e^2 + 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)/(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^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4
- 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*x^2), -1/8*(2*a*c^4*d^4 - 4*a*c^2*d^
3*e + 2*a*d^2*e^2 + 4*(a*c^4*d^3*e - 2*a*c^2*d^2*e^2 + a*d*e^3)*x^2 + (b*c^2*d^3 + (b*c^2*d*e^2 - 2*b*e^3)*x^4
- 2*b*d^2*e + 2*(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*x*sqrt((c^2*
x^2 + 1)/(c^2*x^2))/(c^2*d - e)) - 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)) - c*x + 1
) + 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)) - c*x - 1) + 2*(b*c^4*d^4 - 2*b*c^2*d^3*
e + b*d^2*e^2 + 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)/(
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^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 - 2*c^2*d^3*e^4
+ d^2*e^5)*x^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x^3/(e*x^2 + d)^3, x)

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maple [B]  time = 0.09, size = 1922, normalized size = 11.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x)

[Out]

1/4*c^4*a/e^2*d/(c^2*e*x^2+c^2*d)^2-1/2*c^2*a/e^2/(c^2*e*x^2+c^2*d)+1/4*c^4*b*arccsch(c*x)/e^2*d/(c^2*e*x^2+c^
2*d)^2-1/2*c^2*b*arccsch(c*x)/e^2/(c^2*e*x^2+c^2*d)-1/4*c^3*b*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/
(c^2*d-e)/(c*x*e+(-c^2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))*arctanh(1/(c^2*x^2+1)^(1/2))-1/4*c^3*b*(c^2*x^2+1
)^(1/2)/e/((c^2*x^2+1)/c^2/x^2)^(1/2)/x*d/(c^2*d-e)/(c*x*e+(-c^2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))*arctanh
(1/(c^2*x^2+1)^(1/2))+1/16*c^3*b*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/(c^2*d-e)/(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))*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)))+1/16*c^3*b*(c^2*x^2+1)^(1/2)/e/((c^2*x^2+1)/c^2/x^2)^(1/2)/x*d/(c^2*
d-e)/(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))*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)))+1/16*c^3*b*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^
2/x^2)^(1/2)*x/(c^2*d-e)/(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))*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)))+1/16*c^3*b*(c^2*x^2+1)^(
1/2)/e/((c^2*x^2+1)/c^2/x^2)^(1/2)/x*d/(c^2*d-e)/(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))*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/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/(c^2*d-e)/(c*x*e+(-c^2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))+1/8*c*b/(
(c^2*x^2+1)/c^2/x^2)^(1/2)/x/(c^2*d-e)/(c*x*e+(-c^2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))+1/4*c*b*(c^2*x^2+1)^
(1/2)*e/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/d/(c^2*d-e)/(c*x*e+(-c^2*d*e)^(1/2))/(-c*x*e+(-c^2*d*e)^(1/2))*arctanh(1
/(c^2*x^2+1)^(1/2))+1/4*c*b*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/(c^2*d-e)/(c*x*e+(-c^2*d*e)^(1/2))
/(-c*x*e+(-c^2*d*e)^(1/2))*arctanh(1/(c^2*x^2+1)^(1/2))-1/8*c*b*(c^2*x^2+1)^(1/2)*e/((c^2*x^2+1)/c^2/x^2)^(1/2
)*x/d/(c^2*d-e)/(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))*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*b*(c^2*x^2+1)^(1/2)/((c^2*x
^2+1)/c^2/x^2)^(1/2)/x/(c^2*d-e)/(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))*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*b*(c^2*x^2
+1)^(1/2)*e/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/d/(c^2*d-e)/(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))*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*b*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/(c^2*d-e)/(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))*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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, b {\left (\frac {2 \, c^{4} d^{4} \log \relax (c) - 2 \, {\left (c^{4} d^{2} e^{2} - 2 \, c^{2} d e^{3} + e^{4}\right )} x^{4} \log \relax (x) + 2 \, d^{2} e^{2} \log \relax (c) + d^{2} e^{2} - {\left (4 \, d^{3} e \log \relax (c) + d^{3} e\right )} c^{2} + {\left (4 \, c^{4} d^{3} e \log \relax (c) + 4 \, d e^{3} \log \relax (c) + d e^{3} - {\left (8 \, d^{2} e^{2} \log \relax (c) + d^{2} e^{2}\right )} c^{2}\right )} x^{2} + {\left (c^{4} d^{4} - 2 \, c^{2} d^{3} e + {\left (c^{4} d^{2} e^{2} - 2 \, c^{2} d e^{3}\right )} x^{4} + 2 \, {\left (c^{4} d^{3} e - 2 \, c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (c^{4} d^{4} - 2 \, c^{2} d^{3} e + d^{2} e^{2} + 2 \, {\left (c^{4} d^{3} e - 2 \, c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}} + \frac {\log \left (e x^{2} + d\right )}{c^{4} d^{3} - 2 \, c^{2} d^{2} e + d e^{2}} - 8 \, \int \frac {2 \, c^{2} e x^{3} + c^{2} d x}{4 \, {\left (c^{2} e^{4} x^{6} + {\left (2 \, c^{2} d e^{3} + e^{4}\right )} x^{4} + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + 2 \, d e^{3}\right )} x^{2} + {\left (c^{2} e^{4} x^{6} + {\left (2 \, c^{2} d e^{3} + e^{4}\right )} x^{4} + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + 2 \, d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {{\left (2 \, e x^{2} + d\right )} a}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/8*b*((2*c^4*d^4*log(c) - 2*(c^4*d^2*e^2 - 2*c^2*d*e^3 + e^4)*x^4*log(x) + 2*d^2*e^2*log(c) + d^2*e^2 - (4*d^
3*e*log(c) + d^3*e)*c^2 + (4*c^4*d^3*e*log(c) + 4*d*e^3*log(c) + d*e^3 - (8*d^2*e^2*log(c) + d^2*e^2)*c^2)*x^2
+ (c^4*d^4 - 2*c^2*d^3*e + (c^4*d^2*e^2 - 2*c^2*d*e^3)*x^4 + 2*(c^4*d^3*e - 2*c^2*d^2*e^2)*x^2)*log(c^2*x^2 +
1) - 2*(c^4*d^4 - 2*c^2*d^3*e + d^2*e^2 + 2*(c^4*d^3*e - 2*c^2*d^2*e^2 + d*e^3)*x^2)*log(sqrt(c^2*x^2 + 1) +
1))/(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 - 2*c^
2*d^3*e^4 + d^2*e^5)*x^2) + log(e*x^2 + d)/(c^4*d^3 - 2*c^2*d^2*e + d*e^2) - 8*integrate(1/4*(2*c^2*e*x^3 + c^
2*d*x)/(c^2*e^4*x^6 + (2*c^2*d*e^3 + e^4)*x^4 + d^2*e^2 + (c^2*d^2*e^2 + 2*d*e^3)*x^2 + (c^2*e^4*x^6 + (2*c^2*
d*e^3 + e^4)*x^4 + d^2*e^2 + (c^2*d^2*e^2 + 2*d*e^3)*x^2)*sqrt(c^2*x^2 + 1)), x)) - 1/4*(2*e*x^2 + d)*a/(e^4*x
^4 + 2*d*e^3*x^2 + d^2*e^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3,x)

[Out]

int((x^3*(a + b*asinh(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**3*(a+b*acsch(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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