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

Optimal. Leaf size=231 \[ \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 d e^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \sqrt{1-c^2 x^2} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{8 \sqrt{d} e^2 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^{3/2}}-\frac{b c x \left (1-c^2 x^2\right )}{8 e \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.361612, antiderivative size = 241, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {264, 5790, 12, 519, 470, 523, 217, 206, 377, 208} \[ \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{4 d e^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \sqrt{c^2 x^2-1} \left (2 c^2 d+3 e\right ) \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{8 \sqrt{d} e^2 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^{3/2}}-\frac{b c x \left (1-c^2 x^2\right )}{8 e \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5790

Int[((a_.) + ArcCosh[(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*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[
1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] &
& (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

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

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 377

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

Rubi steps

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

Mathematica [A]  time = 0.890434, size = 192, normalized size = 0.83 \[ \frac{\frac{\frac{b c e x \sqrt{c x-1} \sqrt{c x+1} \left (d+e x^2\right )}{c^2 d+e}-2 a \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 (-d)-e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{\sqrt{d} \sqrt{c^2 x^2-1} \left (c^2 (-d)-e\right )^{3/2}}-\frac{2 b \cosh ^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}}{8 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.038, size = 2499, normalized size = 10.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^2*a/e^2/(c^2*e*x^2+c^2*d)+1/4*c^4*a/e^2*d/(c^2*e*x^2+c^2*d)^2-1/2*c^2*b*arccosh(c*x)/e^2/(c^2*e*x^2+c^2
*d)+1/4*c^4*b*arccosh(c*x)/e^2*d/(c^2*e*x^2+c^2*d)^2+1/8*c^8*b*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*
e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1
/2)+e)^2/(c^2*x^2-1)^(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)))*x^2*d^2+1/8*c^8*b*e*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c
^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*d^3-1/8*c^8*b*e
^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e
)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2*d^2-1/8*c^8*b*e*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c
*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/(
(-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*d^3+1/8*c^5*b*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(c*x*e-(-c^2
*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2*x*d+5/16*c^6*b*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x
*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-
c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2*d+5/16*c^6*b*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e
)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1
)^(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)))*d^2
-5/16*c^6*b*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*
x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2*d-5/16*c^6*b*e^2*(c*x+1)^(1/2)*(c
*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d
*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*d^2+1/8*c^3*b*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2)
)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2*x+3/16*c^4*b*e^4*(c*x+1)^(1/2)*(c*x-1
)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^
(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2+3/16*c^4*b*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))
/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/
(c^2*x^2-1)^(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)))*d-3/16*c^4*b*e^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(
1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2-3/16*c^4*b*e^3*(c*x+1)^(1
/2)*(c*x-1)^(1/2)/(c*x*e+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(c*x*e-(-c^2*d*e)^(1/2))/(e-(
-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{8} \, b{\left (\frac{{\left (c^{4} d + 2 \, c^{2} e\right )} \log \left (e x^{2} + d\right )}{c^{4} d^{2} e^{2} + 2 \, c^{2} d e^{3} + e^{4}} + \frac{c^{4} d^{3} + c^{2} d^{2} e +{\left (c^{4} d^{2} e + c^{2} d e^{2}\right )} x^{2} + 2 \,{\left (c^{4} d^{3} + 2 \, c^{2} d^{2} e + d e^{2} + 2 \,{\left (c^{4} d^{2} e + 2 \, c^{2} d e^{2} + e^{3}\right )} x^{2}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) -{\left (c^{4} d^{3} + 2 \, c^{2} d^{2} e +{\left (c^{4} d e^{2} + 2 \, c^{2} e^{3}\right )} x^{4} + 2 \,{\left (c^{4} d^{2} e + 2 \, c^{2} d e^{2}\right )} x^{2}\right )} \log \left (c x + 1\right ) -{\left (c^{4} d^{3} + 2 \, c^{2} d^{2} e +{\left (c^{4} d e^{2} + 2 \, c^{2} e^{3}\right )} x^{4} + 2 \,{\left (c^{4} d^{2} e + 2 \, c^{2} d e^{2}\right )} x^{2}\right )} \log \left (c x - 1\right )}{c^{4} d^{4} e^{2} + 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4} +{\left (c^{4} d^{2} e^{4} + 2 \, c^{2} d e^{5} + e^{6}\right )} x^{4} + 2 \,{\left (c^{4} d^{3} e^{3} + 2 \, c^{2} d^{2} e^{4} + d e^{5}\right )} x^{2}} + 8 \, \int \frac{2 \, c e x^{2} + c d}{4 \,{\left (c^{3} e^{4} x^{7} - c d^{2} e^{2} x +{\left (2 \, c^{3} d e^{3} - c e^{4}\right )} x^{5} +{\left (c^{3} d^{2} e^{2} - 2 \, c d e^{3}\right )} x^{3} +{\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 )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b*((c^4*d + 2*c^2*e)*log(e*x^2 + d)/(c^4*d^2*e^2 + 2*c^2*d*e^3 + e^4) + (c^4*d^3 + c^2*d^2*e + (c^4*d^2*e
+ c^2*d*e^2)*x^2 + 2*(c^4*d^3 + 2*c^2*d^2*e + d*e^2 + 2*(c^4*d^2*e + 2*c^2*d*e^2 + e^3)*x^2)*log(c*x + sqrt(c
*x + 1)*sqrt(c*x - 1)) - (c^4*d^3 + 2*c^2*d^2*e + (c^4*d*e^2 + 2*c^2*e^3)*x^4 + 2*(c^4*d^2*e + 2*c^2*d*e^2)*x^
2)*log(c*x + 1) - (c^4*d^3 + 2*c^2*d^2*e + (c^4*d*e^2 + 2*c^2*e^3)*x^4 + 2*(c^4*d^2*e + 2*c^2*d*e^2)*x^2)*log(
c*x - 1))/(c^4*d^4*e^2 + 2*c^2*d^3*e^3 + d^2*e^4 + (c^4*d^2*e^4 + 2*c^2*d*e^5 + e^6)*x^4 + 2*(c^4*d^3*e^3 + 2*
c^2*d^2*e^4 + d*e^5)*x^2) + 8*integrate(1/4*(2*c*e*x^2 + c*d)/(c^3*e^4*x^7 - c*d^2*e^2*x + (2*c^3*d*e^3 - c*e^
4)*x^5 + (c^3*d^2*e^2 - 2*c*d*e^3)*x^3 + (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)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x)) - 1/4*(2*e*x^2 + d)*a/(e^4*x^4 + 2*d*e^3*x^2 + d^2*e^
2)

________________________________________________________________________________________

Fricas [B]  time = 3.33259, size = 2473, normalized size = 10.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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**3*(a+b*acosh(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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