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

Optimal. Leaf size=177 \[ -\frac{a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac{b c \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right ) \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{8 d^{3/2} e \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^{3/2}}+\frac{b c x \left (1-c^2 x^2\right )}{8 d \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.135168, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5788, 519, 382, 377, 208} \[ -\frac{a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac{b c \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right ) \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{8 d^{3/2} e \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^{3/2}}+\frac{b c x \left (1-c^2 x^2\right )}{8 d \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.980429, size = 183, normalized size = 1.03 \[ \frac{1}{8} \left (-\frac{\frac{2 a}{e}+\frac{b c x \sqrt{c x-1} \sqrt{c x+1} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac{x \sqrt{c^2 (-d)-e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{d^{3/2} e \sqrt{c^2 x^2-1} \left (c^2 (-d)-e\right )^{3/2}}-\frac{2 b \cosh ^{-1}(c x)}{e \left (d+e x^2\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.03, size = 2443, normalized size = 13.8 \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*arccosh(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*arccosh(c*x)-1/8*c^8*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-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)))*d^2+1/8*c^8*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+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)))*d^2-1/8*c^5*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^6*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^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)))*d+3/16*c^6*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^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)))*d-1/8*c^3*b*e^4*(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/d*x-1/16*c^4*b*e^5*(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))/(-(c^2*d+e)/e)^(1/2)/(e-(-c^2*d*e)^
(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*d*e)^(1/2)/d/(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^2-1/16*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*x*e-(-c^2*d*e)^(1/2))/(-(c^2*d+e)/e)^(1/2)/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/
(-c^2*d*e)^(1/2)/(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)))+1/16*c^4*b*e^5*(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))/(-(c^2*d+e)/e)^(1/2)/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*d*e)^(1/2)/d/(c^2*x^2-1)^(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)/(c*x*e+(-c^2*d*e)^(1/2)))*x^2+1/16*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*x*e-(-c^2*d*e)^(1/2))/(-(c^2*d+e)/e)^(1/2)/(e-(
-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(-c^2*d*e)^(1/2)/(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)))

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

[Out]

-1/8*(c^4*log(e*x^2 + d)/(c^4*d^2*e + 2*c^2*d*e^2 + e^3) + 8*c*integrate(1/4/(c^3*e^3*x^7 + (2*c^3*d*e^2 - c*e
^3)*x^5 - c*d^2*e*x + (c^3*d^2*e - 2*c*d*e^2)*x^3 + (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)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x) - (c^4*d^2 + c^2*d*e + (c^4*d*e + c^2*e^2)*x^2
- 2*(c^4*d^2 + 2*c^2*d*e + e^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^4*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^
2)*log(c*x + 1) + (c^4*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^2)*log(c*x - 1))/(c^4*d^4*e + 2*c^2*d^3*e^2 + d^2*e^3 +
(c^4*d^2*e^3 + 2*c^2*d*e^4 + e^5)*x^4 + 2*(c^4*d^3*e^2 + 2*c^2*d^2*e^3 + d*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 = 3.39367, size = 2522, normalized size = 14.25 \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*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*a*d^2*e^2 + 2*(b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 4*(b
*c^4*d^3*e + b*c^2*d^2*e^2)*x^2 - (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e
+ 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^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)) - 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*s
qrt(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^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 + b)*c^4*d^4 + (4*a + b)*c^2*d^3*e + 2*a*d^2*e^2 + (b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 2*(b*c^4*d^3*e
+ b*c^2*d^2*e^2)*x^2 - (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + 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^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*(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)) + sqr
t(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^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*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}{{\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*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

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