3.5.85 \(\int \frac {(d+e x^2)^3 (a+b \cosh ^{-1}(c x))}{x^2} \, dx\) [485]

Optimal. Leaf size=265 \[ \frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^3 \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-d^3*(a+b*arccosh(c*x))/x+3*d^2*e*x*(a+b*arccosh(c*x))+d*e^2*x^3*(a+b*arccosh(c*x))+1/5*e^3*x^5*(a+b*arccosh(c
*x))+1/5*b*e*(15*c^4*d^2+5*c^2*d*e+e^2)*(-c^2*x^2+1)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/15*b*e^2*(5*c^2*d+2*e)*
(-c^2*x^2+1)^2/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/25*b*e^3*(-c^2*x^2+1)^3/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c*d
^3*arctan((c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5958, 1624, 1813, 1634, 65, 211} \begin {gather*} -\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^3 \sqrt {c^2 x^2-1} \text {ArcTan}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^2 \left (1-c^2 x^2\right )^2 \left (5 c^2 d+2 e\right )}{15 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \left (1-c^2 x^2\right ) \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{5 c^5 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*(15*c^4*d^2 + 5*c^2*d*e + e^2)*(1 - c^2*x^2))/(5*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(5*c^2*d + 2*
e)*(1 - c^2*x^2)^2)/(15*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(1 - c^2*x^2)^3)/(25*c^5*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/x + 3*d^2*e*x*(a + b*ArcCosh[c*x]) + d*e^2*x^3*(a + b*ArcCosh[c*x]) +
(e^3*x^5*(a + b*ArcCosh[c*x]))/5 + (b*c*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sq
rt[1 + c*x])

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 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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 5958

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]))

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {-d^3+3 d^2 e x^2+d e^2 x^4+\frac {e^3 x^6}{5}}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {-d^3+3 d^2 e x^2+d e^2 x^4+\frac {e^3 x^6}{5}}{x \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-d^3+3 d^2 e x+d e^2 x^2+\frac {e^3 x^3}{5}}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {e \left (15 c^4 d^2+5 c^2 d e+e^2\right )}{5 c^4 \sqrt {-1+c^2 x}}-\frac {d^3}{x \sqrt {-1+c^2 x}}+\frac {e^2 \left (5 c^2 d+2 e\right ) \sqrt {-1+c^2 x}}{5 c^4}+\frac {e^3 \left (-1+c^2 x\right )^{3/2}}{5 c^4}\right ) \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b e \left (15 c^4 d^2+5 c^2 d e+e^2\right ) \left (1-c^2 x^2\right )}{5 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (5 c^2 d+2 e\right ) \left (1-c^2 x^2\right )^2}{15 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )+d e^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c d^3 \sqrt {-1+c^2 x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.21, size = 182, normalized size = 0.69 \begin {gather*} -\frac {a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac {1}{5} a e^3 x^5-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \left (8 e^2+2 c^2 e \left (25 d+2 e x^2\right )+c^4 \left (225 d^2+25 d e x^2+3 e^2 x^4\right )\right )}{75 c^5}+\frac {b \left (-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right ) \cosh ^{-1}(c x)}{5 x}-b c d^3 \text {ArcTan}\left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d^3)/x) + 3*a*d^2*e*x + a*d*e^2*x^3 + (a*e^3*x^5)/5 - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(8*e^2 + 2*c^2*e*
(25*d + 2*e*x^2) + c^4*(225*d^2 + 25*d*e*x^2 + 3*e^2*x^4)))/(75*c^5) + (b*(-5*d^3 + 15*d^2*e*x^2 + 5*d*e^2*x^4
+ e^3*x^6)*ArcCosh[c*x])/(5*x) - b*c*d^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]

________________________________________________________________________________________

Maple [A]
time = 2.11, size = 309, normalized size = 1.17

method	result	size
derivativedivides	\(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) d^{2} e x}{c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d \,e^{2} x^{3}}{c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{3} x^{5}}{5 c}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{3}}{c x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} e}{c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,e^{2} x^{2}}{3 c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{4}}{25 c^{2}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,e^{2}}{3 c^{4}}-\frac {4 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{2}}{75 c^{4}}-\frac {8 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3}}{75 c^{6}}\right )\)	\(309\)
default	\(c \left (\frac {a \left (3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-\frac {c^{5} d^{3}}{x}\right )}{c^{6}}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) d^{2} e x}{c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d \,e^{2} x^{3}}{c}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e^{3} x^{5}}{5 c}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{3}}{c x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} e}{c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,e^{2} x^{2}}{3 c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{4}}{25 c^{2}}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}\, d \,e^{2}}{3 c^{4}}-\frac {4 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{2}}{75 c^{4}}-\frac {8 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3}}{75 c^{6}}\right )\)	\(309\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(a/c^6*(3*c^5*d^2*e*x+c^5*d*e^2*x^3+1/5*e^3*c^5*x^5-c^5*d^3/x)+3*b/c*arccosh(c*x)*d^2*e*x+b/c*arccosh(c*x)*d
*e^2*x^3+1/5*b/c*arccosh(c*x)*e^3*x^5-b*arccosh(c*x)*d^3/c/x-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d
^3*arctan(1/(c^2*x^2-1)^(1/2))-3*b/c^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^2*e-1/3*b/c^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)
*d*e^2*x^2-1/25*b/c^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^4-2/3*b/c^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d*e^2-4/75*b/c
^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^2-8/75*b/c^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3)

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 220, normalized size = 0.83 \begin {gather*} \frac {1}{5} \, a x^{5} e^{3} + a d x^{3} e^{2} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{3} + 3 \, a d^{2} x e + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2} e}{c} - \frac {a d^{3}}{x} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5*e^3 + a*d*x^3*e^2 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b*d^3 + 3*a*d^2*x*e + 1/3*(3*x^3*arcco
sh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d*e^2 + 3*(c*x*arccosh(c*x) - sqrt(c^2*x^
2 - 1))*b*d^2*e/c - a*d^3/x + 1/75*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x
^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*e^3

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 771 vs. \(2 (233) = 466\).
time = 0.41, size = 771, normalized size = 2.91 \begin {gather*} \frac {15 \, a c^{5} x^{6} \cosh \left (1\right )^{3} + 15 \, a c^{5} x^{6} \sinh \left (1\right )^{3} + 75 \, a c^{5} d x^{4} \cosh \left (1\right )^{2} + 150 \, b c^{6} d^{3} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 225 \, a c^{5} d^{2} x^{2} \cosh \left (1\right ) - 75 \, a c^{5} d^{3} + 15 \, {\left (3 \, a c^{5} x^{6} \cosh \left (1\right ) + 5 \, a c^{5} d x^{4}\right )} \sinh \left (1\right )^{2} + 15 \, {\left (5 \, b c^{5} d^{3} x - 5 \, b c^{5} d^{3} + {\left (b c^{5} x^{6} - b c^{5} x\right )} \cosh \left (1\right )^{3} + {\left (b c^{5} x^{6} - b c^{5} x\right )} \sinh \left (1\right )^{3} + 5 \, {\left (b c^{5} d x^{4} - b c^{5} d x\right )} \cosh \left (1\right )^{2} + {\left (5 \, b c^{5} d x^{4} - 5 \, b c^{5} d x + 3 \, {\left (b c^{5} x^{6} - b c^{5} x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 15 \, {\left (b c^{5} d^{2} x^{2} - b c^{5} d^{2} x\right )} \cosh \left (1\right ) + {\left (15 \, b c^{5} d^{2} x^{2} - 15 \, b c^{5} d^{2} x + 3 \, {\left (b c^{5} x^{6} - b c^{5} x\right )} \cosh \left (1\right )^{2} + 10 \, {\left (b c^{5} d x^{4} - b c^{5} d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 15 \, {\left (5 \, b c^{5} d^{3} x - 15 \, b c^{5} d^{2} x \cosh \left (1\right ) - 5 \, b c^{5} d x \cosh \left (1\right )^{2} - b c^{5} x \cosh \left (1\right )^{3} - b c^{5} x \sinh \left (1\right )^{3} - {\left (5 \, b c^{5} d x + 3 \, b c^{5} x \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} - {\left (15 \, b c^{5} d^{2} x + 10 \, b c^{5} d x \cosh \left (1\right ) + 3 \, b c^{5} x \cosh \left (1\right )^{2}\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 15 \, {\left (3 \, a c^{5} x^{6} \cosh \left (1\right )^{2} + 10 \, a c^{5} d x^{4} \cosh \left (1\right ) + 15 \, a c^{5} d^{2} x^{2}\right )} \sinh \left (1\right ) - {\left (225 \, b c^{4} d^{2} x \cosh \left (1\right ) + {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \cosh \left (1\right )^{3} + {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sinh \left (1\right )^{3} + 25 \, {\left (b c^{4} d x^{3} + 2 \, b c^{2} d x\right )} \cosh \left (1\right )^{2} + {\left (25 \, b c^{4} d x^{3} + 50 \, b c^{2} d x + 3 \, {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (225 \, b c^{4} d^{2} x + 3 \, {\left (3 \, b c^{4} x^{5} + 4 \, b c^{2} x^{3} + 8 \, b x\right )} \cosh \left (1\right )^{2} + 50 \, {\left (b c^{4} d x^{3} + 2 \, b c^{2} d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{75 \, c^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/75*(15*a*c^5*x^6*cosh(1)^3 + 15*a*c^5*x^6*sinh(1)^3 + 75*a*c^5*d*x^4*cosh(1)^2 + 150*b*c^6*d^3*x*arctan(-c*x
+ sqrt(c^2*x^2 - 1)) + 225*a*c^5*d^2*x^2*cosh(1) - 75*a*c^5*d^3 + 15*(3*a*c^5*x^6*cosh(1) + 5*a*c^5*d*x^4)*si
nh(1)^2 + 15*(5*b*c^5*d^3*x - 5*b*c^5*d^3 + (b*c^5*x^6 - b*c^5*x)*cosh(1)^3 + (b*c^5*x^6 - b*c^5*x)*sinh(1)^3
+ 5*(b*c^5*d*x^4 - b*c^5*d*x)*cosh(1)^2 + (5*b*c^5*d*x^4 - 5*b*c^5*d*x + 3*(b*c^5*x^6 - b*c^5*x)*cosh(1))*sinh
(1)^2 + 15*(b*c^5*d^2*x^2 - b*c^5*d^2*x)*cosh(1) + (15*b*c^5*d^2*x^2 - 15*b*c^5*d^2*x + 3*(b*c^5*x^6 - b*c^5*x
)*cosh(1)^2 + 10*(b*c^5*d*x^4 - b*c^5*d*x)*cosh(1))*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + 15*(5*b*c^5*d^3*x
- 15*b*c^5*d^2*x*cosh(1) - 5*b*c^5*d*x*cosh(1)^2 - b*c^5*x*cosh(1)^3 - b*c^5*x*sinh(1)^3 - (5*b*c^5*d*x + 3*b*
c^5*x*cosh(1))*sinh(1)^2 - (15*b*c^5*d^2*x + 10*b*c^5*d*x*cosh(1) + 3*b*c^5*x*cosh(1)^2)*sinh(1))*log(-c*x + s
qrt(c^2*x^2 - 1)) + 15*(3*a*c^5*x^6*cosh(1)^2 + 10*a*c^5*d*x^4*cosh(1) + 15*a*c^5*d^2*x^2)*sinh(1) - (225*b*c^
4*d^2*x*cosh(1) + (3*b*c^4*x^5 + 4*b*c^2*x^3 + 8*b*x)*cosh(1)^3 + (3*b*c^4*x^5 + 4*b*c^2*x^3 + 8*b*x)*sinh(1)^
3 + 25*(b*c^4*d*x^3 + 2*b*c^2*d*x)*cosh(1)^2 + (25*b*c^4*d*x^3 + 50*b*c^2*d*x + 3*(3*b*c^4*x^5 + 4*b*c^2*x^3 +
8*b*x)*cosh(1))*sinh(1)^2 + (225*b*c^4*d^2*x + 3*(3*b*c^4*x^5 + 4*b*c^2*x^3 + 8*b*x)*cosh(1)^2 + 50*(b*c^4*d*
x^3 + 2*b*c^2*d*x)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/(c^5*x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________