3.2.12 \(\int \frac {x^3 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [112]
Optimal. Leaf size=167 \[ -\frac {b c x \sqrt {-1-c^2 x^2}}{8 \left (c^2 d-e\right ) e \sqrt {-c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (c^2 d-2 e\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{8 d \left (c^2 d-e\right )^{3/2} e^{3/2} \sqrt {-c^2 x^2}} \]
[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 = 0.15, 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 = {270, 6437, 12,
457, 79, 65, 214} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]
[Out]
-1/8*(b*c*x*Sqrt[-1 - c^2*x^2])/((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 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 79
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] || L
tQ[p, n]))))
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 270
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 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 6437
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) \text {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 ) \text {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 ) \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 )}{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] Result contains complex when optimal does not.
time = 0.95, size = 375, normalized size = 2.25 \begin {gather*} -\frac {-\frac {4 a d}{\left (d+e x^2\right )^2}+\frac {8 a}{d+e x^2}-\frac {2 b c e \sqrt {1+\frac {1}{c^2 x^2}} x}{\left (-c^2 d+e\right ) \left (d+e x^2\right )}+\frac {4 b \left (d+2 e x^2\right ) \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2}-\frac {4 b \sinh ^{-1}\left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \left (-c^2 d+2 e\right ) \log \left (\frac {16 d e^{3/2} \sqrt {-c^2 d+e} \left (\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (-c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d \left (-c^2 d+e\right )^{3/2}}+\frac {b \sqrt {e} \left (-c^2 d+2 e\right ) \log \left (-\frac {16 i d e^{3/2} \sqrt {-c^2 d+e} \left (\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (c^2 d-2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \left (-c^2 d+e\right )^{3/2}}}{16 e^2} \end {gather*}
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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1880\) vs.
\(2(145)=290\).
time = 6.34, size = 1881, normalized size = 11.26
| | |
| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(1881\) |
| default |
\(\text {Expression too large to display}\) |
\(1881\) |
| | |
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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,method=_RETURNVERBOSE)
[Out]
1/c^4*(a*c^6*(-1/2/e^2/(c^2*e*x^2+c^2*d)+1/4*d*c^2/e^2/(c^2*e*x^2+c^2*d)^2)-1/2*b*c^6*arccsch(c*x)/e^2/(c^2*e*
x^2+c^2*d)+1/4*b*c^8*arccsch(c*x)*d/e^2/(c^2*e*x^2+c^2*d)^2-1/4*b*c^7*(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)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*arctanh(1/(c^2*x^2+1)^(1/2))-1/4*b*c^
7*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))
*arctanh(1/(c^2*x^2+1)^(1/2))+1/16*b*c^7*(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)/e)^(1
/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-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)/(-e*c*x+(-c^2*d*e)^(1/2)))+1/16*b*c^7*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)
*x/(-(c^2*d-e)/e)^(1/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-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)/(-e*c*x+(-c^2*d*e)^(1/2)))+1/16*b*c^7*(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)/e)^(1/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*l
n(-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)))+1/16*b*c^7*(
c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/(-(c^2*d-e)/e)^(1/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x
+(-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)/(e*c*x+(-c^2*d*e)^
(1/2)))+1/8*b*c^5*(c^2*x^2+1)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d
*e)^(1/2))+1/4*b*c^5*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*
x+(-c^2*d*e)^(1/2))*arctanh(1/(c^2*x^2+1)^(1/2))+1/4*b*c^5*(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)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-c^2*d*e)^(1/2))*arctanh(1/(c^2*x^2+1)^(1/2))-1/8*b*c^5*(c^2*x^2+
1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/(-(c^2*d-e)/e)^(1/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-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)/(-e*c*x+(-c^2*d*e)^(1/2)))-
1/8*b*c^5*(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)/e)^(1/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e
)^(1/2))/(e*c*x+(-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)/(-e*
c*x+(-c^2*d*e)^(1/2)))-1/8*b*c^5*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x/(-(c^2*d-e)/e)^(1/2)/(c^2*d-e
)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-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)/(e*c*x+(-c^2*d*e)^(1/2)))-1/8*b*c^5*(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)/e)^(1/2)/(c^2*d-e)/(-e*c*x+(-c^2*d*e)^(1/2))/(e*c*x+(-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)/(e*c*x+(-c^2*d*e)^(1/2))))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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) - (4*d^3*log(c) + d^3)*c^2*e + (4*c^
4*d^3*e*log(c) - (8*d^2*log(c) + d^2)*c^2*e^2 + (4*d*log(c) + d)*e^3)*x^2 + (2*d^2*log(c) + d^2)*e^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 + 2*(c^4*d^3*e - 2*c^2*d^2*e^2 + d*e^3)*x^2 + d^2*e^2)*log(sqrt(c^2*x^2 + 1) + 1))/(c^4*d
^5*e^2 - 2*c^2*d^4*e^3 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d*e^6)*x^4 + d^3*e^4 + 2*(c^4*d^4*e^3 - 2*c^2*d^3*e^4
+ d^2*e^5)*x^2) + log(x^2*e + d)/(c^4*d^3 - 2*c^2*d^2*e + d*e^2) - 8*integrate(1/4*(2*c^2*x^3*e + c^2*d*x)/(c^
2*x^6*e^4 + (2*c^2*d*e^3 + e^4)*x^4 + (c^2*d^2*e^2 + 2*d*e^3)*x^2 + d^2*e^2 + (c^2*x^6*e^4 + (2*c^2*d*e^3 + e^
4)*x^4 + (c^2*d^2*e^2 + 2*d*e^3)*x^2 + d^2*e^2)*sqrt(c^2*x^2 + 1)), x)) - 1/4*(2*x^2*e + d)*a/(x^4*e^4 + 2*d*x
^2*e^3 + d^2*e^2)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1953 vs.
\(2 (148) = 296\).
time = 0.66, size = 3944, normalized size = 23.62 \begin {gather*} \text {Too large to display} \end {gather*}
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*d*x^2*cosh(1)^3 + 8*a*d*x^2*sinh(1)^3 - 4*(4*a*c^2*d^2*x^2 - a*d^2)*cosh(1)^2 - 4*(4
*a*c^2*d^2*x^2 - 6*a*d*x^2*cosh(1) - a*d^2)*sinh(1)^2 - (2*b*x^4*cosh(1)^3 + 2*b*x^4*sinh(1)^3 - b*c^2*d^3 - (
b*c^2*d*x^4 - 4*b*d*x^2)*cosh(1)^2 - (b*c^2*d*x^4 - 6*b*x^4*cosh(1) - 4*b*d*x^2)*sinh(1)^2 - 2*(b*c^2*d^2*x^2
- b*d^2)*cosh(1) - 2*(b*c^2*d^2*x^2 - 3*b*x^4*cosh(1)^2 - b*d^2 + (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^2*d + 2*c*x*sqrt(-(c^2*d - cosh(1) - sinh(1))/(
cosh(1) - sinh(1)))*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - (c^2*x^2 + 2)*cosh(1) - (c^2*x^2 + 2)*sinh(1))/(x^2*cosh(1
) + x^2*sinh(1) + d)) + 8*(a*c^4*d^3*x^2 - a*c^2*d^3)*cosh(1) - 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)) - c*x + 1) + 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*c
osh(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)) - c*x - 1) + 4*(b*c^4*d^4 + 2*b*d*x^2*cosh(1)^3 + 2*b*d*x^2*s
inh(1)^3 - (4*b*c^2*d^2*x^2 - b*d^2)*cosh(1)^2 - (4*b*c^2*d^2*x^2 - 6*b*d*x^2*cosh(1) - b*d^2)*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 - b*c^2*d^3 + 3*b*d*x^2*cosh(1)^2 - (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)/(c*x)) + 8*(a*c^4*d^3*x^2 - a*c^2*d^3 + 3*
a*d*x^2*cosh(1)^2 - (4*a*c^2*d^2*x^2 - a*d^2)*cosh(1))*sinh(1) + 2*(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 - b*c*d^2*x)*cosh(1))*sinh(1))*sqrt((c
^2*x^2 + 1)/(c^2*x^2)))/(c^4*d^5*cosh(1)^2 + d*x^4*cosh(1)^6 + d*x^4*sinh(1)^6 - 2*(c^2*d^2*x^4 - d^2*x^2)*cos
h(1)^5 - 2*(c^2*d^2*x^4 - 3*d*x^4*cosh(1) - d^2*x^2)*sinh(1)^5 + (c^4*d^3*x^4 - 4*c^2*d^3*x^2 + d^3)*cosh(1)^4
+ (c^4*d^3*x^4 - 4*c^2*d^3*x^2 + 15*d*x^4*cosh(1)^2 + d^3 - 10*(c^2*d^2*x^4 - d^2*x^2)*cosh(1))*sinh(1)^4 + 2
*(c^4*d^4*x^2 - c^2*d^4)*cosh(1)^3 + 2*(c^4*d^4*x^2 + 10*d*x^4*cosh(1)^3 - c^2*d^4 - 10*(c^2*d^2*x^4 - d^2*x^2
)*cosh(1)^2 + 2*(c^4*d^3*x^4 - 4*c^2*d^3*x^2 + d^3)*cosh(1))*sinh(1)^3 + (c^4*d^5 + 15*d*x^4*cosh(1)^4 - 20*(c
^2*d^2*x^4 - d^2*x^2)*cosh(1)^3 + 6*(c^4*d^3*x^4 - 4*c^2*d^3*x^2 + d^3)*cosh(1)^2 + 6*(c^4*d^4*x^2 - c^2*d^4)*
cosh(1))*sinh(1)^2 + 2*(c^4*d^5*cosh(1) + 3*d*x^4*cosh(1)^5 - 5*(c^2*d^2*x^4 - d^2*x^2)*cosh(1)^4 + 2*(c^4*d^3
*x^4 - 4*c^2*d^3*x^2 + d^3)*cosh(1)^3 + 3*(c^4*d^4*x^2 - c^2*d^4)*cosh(1)^2)*sinh(1)), -1/8*(2*a*c^4*d^4 + 4*a
*d*x^2*cosh(1)^3 + 4*a*d*x^2*sinh(1)^3 - 2*(4*a*c^2*d^2*x^2 - a*d^2)*cosh(1)^2 - 2*(4*a*c^2*d^2*x^2 - 6*a*d*x^
2*cosh(1) - a*d^2)*sinh(1)^2 - (2*b*x^4*cosh(1)^3 + 2*b*x^4*sinh(1)^3 - b*c^2*d^3 - (b*c^2*d*x^4 - 4*b*d*x^2)*
cosh(1)^2 - (b*c^2*d*x^4 - 6*b*x^4*cosh(1) - 4*b*d*x^2)*sinh(1)^2 - 2*(b*c^2*d^2*x^2 - b*d^2)*cosh(1) - 2*(b*c
^2*d^2*x^2 - 3*b*x^4*cosh(1)^2 - b*d^2 + (b*c^2*d*x^4 - 4*b*d*x^2)*cosh(1))*sinh(1))*sqrt((c^2*d - cosh(1) - s
inh(1))/(cosh(1) - sinh(1)))*arctan(-c*x*sqrt((c^2*d - cosh(1) - sinh(1))/(cosh(1) - sinh(1)))*sqrt((c^2*x^2 +
1)/(c^2*x^2))/(c^2*d - cosh(1) - sinh(1))) + 4*(a*c^4*d^3*x^2 - a*c^2*d^3)*cosh(1) - 2*(b*c^4*d^4 + b*x^4*cos
h(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)*s
inh(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*co
sh(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)) - c*x + 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)*si
nh(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*cos
h(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*...
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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**3*(a+b*acsch(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^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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
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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