3.2.13 \(\int \frac {x (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [113]

Optimal. Leaf size=205 \[ \frac {b c x \sqrt {-1-c^2 x^2}}{8 d \left (c^2 d-e\right ) \sqrt {-c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \text {ArcTan}\left (\sqrt {-1-c^2 x^2}\right )}{4 d^2 e \sqrt {-c^2 x^2}}+\frac {b c \left (3 c^2 d-2 e\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{8 d^2 \left (c^2 d-e\right )^{3/2} \sqrt {e} \sqrt {-c^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6435, 457, 105, 162, 65, 211, 214} \begin {gather*} -\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \text {ArcTan}\left (\sqrt {-c^2 x^2-1}\right )}{4 d^2 e \sqrt {-c^2 x^2}}+\frac {b c x \left (3 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^2 \sqrt {e} \sqrt {-c^2 x^2} \left (c^2 d-e\right )^{3/2}}+\frac {b c x \sqrt {-c^2 x^2-1}}{8 d \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*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*x*Sqrt[-1 - c^2*x^2])/(8*d*(c^2*d - e)*Sqrt[-(c^2*x^2)]*(d + e*x^2)) - (a + b*ArcCsch[c*x])/(4*e*(d + e*x
^2)^2) + (b*c*x*ArcTan[Sqrt[-1 - c^2*x^2]])/(4*d^2*e*Sqrt[-(c^2*x^2)]) + (b*c*(3*c^2*d - 2*e)*x*ArcTanh[(Sqrt[
e]*Sqrt[-1 - c^2*x^2])/Sqrt[c^2*d - e]])/(8*d^2*(c^2*d - e)^(3/2)*Sqrt[e]*Sqrt[-(c^2*x^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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 6435

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p +
1)*((a + b*ArcCsch[c*x])/(2*e*(p + 1))), x] - Dist[b*c*(x/(2*e*(p + 1)*Sqrt[(-c^2)*x^2])), Int[(d + e*x^2)^(p
+ 1)/(x*Sqrt[-1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.62, size = 368, normalized size = 1.80 \begin {gather*} \frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} x}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {4 b \text {csch}^{-1}(c x)}{e \left (d+e x^2\right )^2}+\frac {4 b \sinh ^{-1}\left (\frac {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}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) 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 i d^2 \sqrt {e} \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 (3 c^2 d-2 e\right ) \left (\sqrt {d}+i \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*ArcCsch[c*x]))/(d + e*x^2)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1837\) vs. \(2(180)=360\).
time = 6.44, size = 1838, normalized size = 8.97

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1838\)
default	\(\text {Expression too large to display}\)	\(1838\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccsch(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*arccsch(c*x)-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))+3/16*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)))+3/16*b*c^5*(c^2*x^2+1)^(1/2)/((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)))*e+3/16*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)))+3/16*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^3*(c^2*x^2+1)/((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))*e+1/4*b*c^3*(c^2*x^2+1)^(1/2)/((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))*e+1/4*b*c^3*(c^2*x^2+1)^(1/2)/((c^
2*x^2+1)/c^2/x^2)^(1/2)*x/d^2/(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))*e^2-1/8*b*c^3*(c^2*x^2+1)^(1/2)/((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)))*e-1/8*b*c^3*(c^2*x^2+1)^(1/2)/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/d^2/(-(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)))*e^2-1/8*b*c^3*(c^2*x^2+1)^(1/2)/((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)))*e-1/8*b*c^3*(c^2*x^2+1)^(1/2)
/((c^2*x^2+1)/c^2/x^2)^(1/2)*x/d^2/(-(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)))*e^2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1635 vs. \(2 (179) = 358\).
time = 0.58, size = 3307, normalized size = 16.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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*cosh(1) + 4*a*d^2*cosh(1)^2 + 4*a*d^2*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^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)) - 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*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)) - 8*(a*c^2*d^3 - 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^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 - 4*a*c^2*d^3*cosh(1) + 2*a*d^2*cosh(1)^2 + 2*a*d^2*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*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) - s
inh(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)*cos
h(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)*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) + 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*(a*c^2*d^3 - a*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 -
b*c*d^2*x)*cosh(1))*sinh(1))*sqrt((c^2*x^2 + 1...

________________________________________________________________________________________

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*acsch(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\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*(a + b*asinh(1/(c*x))))/(d + e*x^2)^3,x)

[Out]

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

________________________________________________________________________________________