3.113 \(\int \frac{x (a+b \text{csch}^{-1}(c x))}{(d+e x^2)^3} \, dx\)
Optimal. Leaf size=205 \[ -\frac{a+b \text{csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac{b c x \tan ^{-1}\left (\sqrt{-c^2 x^2-1}\right )}{4 d^2 e \sqrt{-c^2 x^2}}+\frac{b c x \left (3 c^2 d-2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right )}{8 d^2 \sqrt{e} \sqrt{-c^2 x^2} \left (c^2 d-e\right )^{3/2}}+\frac{b c x \sqrt{-c^2 x^2-1}}{8 d \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \left (d+e x^2\right )} \]
[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)])
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Rubi [A] time = 0.204717, antiderivative size = 205, normalized size of antiderivative = 1.,
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 =
{6300, 446, 103, 156, 63, 205, 208} \[ -\frac{a+b \text{csch}^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac{b c x \tan ^{-1}\left (\sqrt{-c^2 x^2-1}\right )}{4 d^2 e \sqrt{-c^2 x^2}}+\frac{b c x \left (3 c^2 d-2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right )}{8 d^2 \sqrt{e} \sqrt{-c^2 x^2} \left (c^2 d-e\right )^{3/2}}+\frac{b c x \sqrt{-c^2 x^2-1}}{8 d \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \left (d+e x^2\right )} \]
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 6300
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]
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 103
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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 156
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 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 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
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]
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) \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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) \operatorname{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 ) \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 )}{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] time = 0.88306, size = 368, normalized size = 1.8 \[ \frac{1}{16} \left (-\frac{4 a}{e \left (d+e x^2\right )^2}+\frac{b \left (3 c^2 d-2 e\right ) \log \left (\frac{16 d^2 \sqrt{e} \sqrt{e-c^2 d} \left (\sqrt{e}+c x \left (\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}-i c \sqrt{d}\right )\right )}{b \left (2 e-3 c^2 d\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{d^2 \sqrt{e} \left (e-c^2 d\right )^{3/2}}+\frac{b \left (3 c^2 d-2 e\right ) \log \left (-\frac{16 i d^2 \sqrt{e} \sqrt{e-c^2 d} \left (\sqrt{e}+c x \left (\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}+i c \sqrt{d}\right )\right )}{b \left (3 c^2 d-2 e\right ) \left (\sqrt{d}+i \sqrt{e} x\right )}\right )}{d^2 \sqrt{e} \left (e-c^2 d\right )^{3/2}}+\frac{2 b c x \sqrt{\frac{1}{c^2 x^2}+1}}{d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{4 b \sinh ^{-1}\left (\frac{1}{c x}\right )}{d^2 e}-\frac{4 b \text{csch}^{-1}(c x)}{e \left (d+e x^2\right )^2}\right ) \]
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
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Maple [B] time = 0.279, size = 1884, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arccsch(c*x))/(e*x^2+d)^3,x)
[Out]
-1/4*c^4*a/e/(c^2*e*x^2+c^2*d)^2-1/4*c^4*b/e/(c^2*e*x^2+c^2*d)^2*arccsch(c*x)-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))-1/4*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))+3/16*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)/e)^(1/2)/(c^2*d-e)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))*ln(2*((-(c^2*d-e)/e)^(1/2)*
(c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)))+3/16*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)/e)^(1/2)/(c^2*d-e)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))*ln(2*((-(
c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)))+3/16*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)/e)^(1/2)/(c^2*d-e)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2
*d*e)^(1/2))*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*x*e+(-c^2*d*e)^(1/2))
)+3/16*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)/e)^(1/2)/(c^2*d-e)/(-c*x*e+(-c^2*d*e)
^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x+e)/(-c*x
*e+(-c^2*d*e)^(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/d^2/(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))*e^2+1/4*c*b*(c^2*x^2+1)^(1/2)/((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))*e-1/8*
c^3*b/((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))*e-1/8*c*b/(
(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))*e-1/8*c*b*(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)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-
c^2*d*e)^(1/2))*ln(2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(c*x*e+(-c^2*d*e)^(1/2)
))*e^2-1/8*c*b*(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)/(-c*x*e+(-c^2*
d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))*ln(2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x+e)/(c
*x*e+(-c^2*d*e)^(1/2)))*e-1/8*c*b*(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)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))*ln(-2*((-(c^2*d-e)/e)^(1/2)*(c^2*x^2+1)^(1/2)*e+(-c^
2*d*e)^(1/2)*c*x+e)/(-c*x*e+(-c^2*d*e)^(1/2)))*e^2-1/8*c*b*(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)/(-c*x*e+(-c^2*d*e)^(1/2))/(c*x*e+(-c^2*d*e)^(1/2))*ln(-2*((-(c^2*d-e)/e)^(1/2)*(
c^2*x^2+1)^(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., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \,{\left (8 \, c^{2} \int \frac{x}{4 \,{\left (c^{2} e^{3} x^{6} +{\left (2 \, c^{2} d e^{2} + e^{3}\right )} x^{4} + d^{2} e +{\left (c^{2} d^{2} e + 2 \, d e^{2}\right )} x^{2} +{\left (c^{2} e^{3} x^{6} +{\left (2 \, c^{2} d e^{2} + e^{3}\right )} x^{4} + d^{2} e +{\left (c^{2} d^{2} e + 2 \, d e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x} + \frac{{\left (2 \, c^{2} d - e\right )} \log \left (e x^{2} + d\right )}{c^{4} d^{4} - 2 \, c^{2} d^{3} e + d^{2} e^{2}} - \frac{2 \, c^{4} d^{4} \log \left (c\right ) + 2 \, d^{2} e^{2} \log \left (c\right ) - d^{2} e^{2} -{\left (4 \, d^{3} e \log \left (c\right ) - d^{3} e\right )} c^{2} +{\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2} +{\left (c^{4} d^{2} e^{2} x^{4} + 2 \, c^{4} d^{3} e x^{2} + c^{4} d^{4}\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \,{\left ({\left (c^{4} d^{2} e^{2} - 2 \, c^{2} d e^{3} + e^{4}\right )} x^{4} + 2 \,{\left (c^{4} d^{3} e - 2 \, c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )} \log \left (x\right ) - 2 \,{\left (c^{4} d^{4} - 2 \, c^{2} d^{3} e + d^{2} e^{2}\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} +{\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \,{\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}}\right )} b - \frac{a}{4 \,{\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} \end{align*}
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*e^3*x^6 + (2*c^2*d*e^2 + e^3)*x^4 + d^2*e + (c^2*d^2*e + 2*d*e^2)*x^2 + (c^2*
e^3*x^6 + (2*c^2*d*e^2 + e^3)*x^4 + d^2*e + (c^2*d^2*e + 2*d*e^2)*x^2)*sqrt(c^2*x^2 + 1)), x) + (2*c^2*d - e)*
log(e*x^2 + d)/(c^4*d^4 - 2*c^2*d^3*e + d^2*e^2) - (2*c^4*d^4*log(c) + 2*d^2*e^2*log(c) - d^2*e^2 - (4*d^3*e*l
og(c) - d^3*e)*c^2 + (c^2*d^2*e^2 - d*e^3)*x^2 + (c^4*d^2*e^2*x^4 + 2*c^4*d^3*e*x^2 + 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 + d^4*e^3 + (c^4*d^4*e^3 - 2*c
^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4)*x^2))*b - 1/4*a/(e^3*x^4 + 2*d*e^2*x^2 +
d^2*e)
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Fricas [B] time = 4.78139, size = 2596, normalized size = 12.66 \begin{align*} \text{result too large to display} \end{align*}
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*e + 4*a*d^2*e^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e +
2*(3*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*s
qrt((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)*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^6*e - 2*c^2*d^5*e^2 + d^4*e^3
+ (c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4)*x^2), -1/8*(2*a*c^4*
d^4 - 4*a*c^2*d^3*e + 2*a*d^2*e^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(3*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)*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^6*e - 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*x^4
+ 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4)*x^2)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
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)