3.506 \(\int \frac{x^5 (a+b \cosh ^{-1}(c x))}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=737 \[ \frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 e^3}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^3}+\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right ) \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{8 e^3 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^{3/2}}-\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{e^3 \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}} \]

[Out]

(b*c*d*x*(1 - c^2*x^2))/(8*e^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (d^2*(a + b*ArcCosh[c*x
]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcCosh[c*x]))/(e^3*(d + e*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*e^3) - (b
*c*Sqrt[d]*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e^3*Sqrt[c^2*d + e]*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d]*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqr
t[d]*Sqrt[-1 + c^2*x^2])])/(8*e^3*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[
1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt
[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^Arc
Cosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x]
)/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(
c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) +
(b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e
]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3)

________________________________________________________________________________________

Rubi [A]  time = 1.18224, antiderivative size = 737, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {5792, 5788, 519, 382, 377, 208, 5800, 5562, 2190, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^3}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 e^3}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^3}+\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right ) \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{8 e^3 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^{3/2}}-\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{e^3 \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*d*x*(1 - c^2*x^2))/(8*e^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (d^2*(a + b*ArcCosh[c*x
]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcCosh[c*x]))/(e^3*(d + e*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*e^3) - (b
*c*Sqrt[d]*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e^3*Sqrt[c^2*d + e]*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d]*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqr
t[d]*Sqrt[-1 + c^2*x^2])])/(8*e^3*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[
1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt
[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^Arc
Cosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x]
)/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(
c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) +
(b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e
]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3)

Rule 5792

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int
[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 5788

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcCosh[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*
Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]

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 382

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

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]

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (\frac{d^2 x \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^3}-\frac{2 d x \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^2}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx}{e^2}-\frac{(2 d) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac{d^2 \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx}{e^2}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{(b c d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )} \, dx}{e^3}+\frac{\left (b c d^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^2} \, dx}{4 e^3}+\frac{\int \left (-\frac{a+b \cosh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e^2}\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 e^{5/2}}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 e^{5/2}}-\frac{\left (b c d \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{e^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{5/2}}-\frac{\left (b c d \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 )}{e^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d \left (2 c^2 d+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^3 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \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 )}{e^3 \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{5/2}}+\frac{\left (b c d \left (2 c^2 d+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^3 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \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 )}{e^3 \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d} \left (2 c^2 d+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 e^3 \left (c^2 d+e\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^3}\\ &=\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \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 )}{e^3 \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d} \left (2 c^2 d+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 e^3 \left (c^2 d+e\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^3}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^3}\\ &=\frac{b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 e^3 \left (d+e x^2\right )^2}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^3 \left (d+e x^2\right )}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e^3}-\frac{b c \sqrt{d} \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 )}{e^3 \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d} \left (2 c^2 d+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 e^3 \left (c^2 d+e\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}+\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^3}\\ \end{align*}

Mathematica [C]  time = 7.1105, size = 1155, normalized size = 1.57 \[ -\frac{a d^2}{4 e^3 \left (e x^2+d\right )^2}+\frac{a d}{e^3 \left (e x^2+d\right )}+\frac{a \log \left (e x^2+d\right )}{2 e^3}+b \left (-\frac{7 i \sqrt{d} \left (\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left (\frac{2 e \left (\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c \sqrt{-d c^2-e} \left (i \sqrt{e} x+\sqrt{d}\right )}\right )}{\sqrt{-d c^2-e}}\right )}{16 e^3}-\frac{7 i \sqrt{d} \left (-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left (\frac{2 e \left (-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c \sqrt{-d c^2-e} \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{\sqrt{-d c^2-e}}\right )}{16 e^3}-\frac{d \left (\frac{\sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c^3 \left (d+i \sqrt{e} x \sqrt{d}\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left (d c^2+e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x-i \sqrt{d}\right )^2}\right )}{16 e^{5/2}}-\frac{d \left (-\frac{\sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c^3 \left (d-i \sqrt{d} \sqrt{e} x\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left (d c^2+e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x+i \sqrt{d}\right )^2}\right )}{16 e^{5/2}}+\frac{\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right )+\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right )\right )-\cosh ^{-1}(c x)\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )}{4 e^3}+\frac{\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right )+\log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )-\cosh ^{-1}(c x)\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )}{4 e^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]

[Out]

-(a*d^2)/(4*e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*((((-7*I)/16)*Sqrt[d
]*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/e^3 - (((7*I
)/16)*Sqrt[d]*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*
d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/e
^3 - (d*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*((-
I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqr
t[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(
16*e^(5/2)) - (d*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[
e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x +
Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2)))
)/(16*e^(5/2)) + (ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*
d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^A
rcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + S
qrt[-(c^2*d) - e]))])/(4*e^3) + (ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqr
t[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyL
og[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x
])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])/(4*e^3))

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Maple [C]  time = 0.79, size = 5196, normalized size = 7.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{5} \operatorname{arcosh}\left (c x\right ) + a x^{5}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*x^5*arccosh(c*x) + a*x^5)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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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**5*(a+b*acosh(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{arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*x^5/(e*x^2 + d)^3, x)