\(\int \frac {(d-c^2 d x^2)^3 (a+b \text {arccosh}(c x))}{x^4} \, dx\) [27]

Optimal result

Integrand size = 25, antiderivative size = 195 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {8}{3} b c^3 d^3 \sqrt {-1+c x} \sqrt {1+c x}+\frac {b c d^3 \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}+\frac {1}{9} b c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {17}{6} b c^3 d^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]

[Out]

1/9*b*c^3*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)-1/3*d^3*(a+b*arccosh(c*x))/x^3+3*c^2*d^3*(a+b*arccosh(c*x))/x+3*c^4*
d^3*x*(a+b*arccosh(c*x))-1/3*c^6*d^3*x^3*(a+b*arccosh(c*x))-17/6*b*c^3*d^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))
-8/3*b*c^3*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/6*b*c*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.29, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {276, 5921, 12, 1624, 1813, 1635, 911, 1167, 211} \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {17 b c^3 d^3 \sqrt {c^2 x^2-1} \arctan \left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]

[Out]

(8*b*c^3*d^3*(1 - c^2*x^2))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) + (b*c^3*d^3*(1 - c^2*x^2)^2)/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/(3*x
^3) + (3*c^2*d^3*(a + b*ArcCosh[c*x]))/x + 3*c^4*d^3*x*(a + b*ArcCosh[c*x]) - (c^6*d^3*x^3*(a + b*ArcCosh[c*x]
))/3 - (17*b*c^3*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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 1635

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(
b*c - a*d))), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expo
n[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 5921

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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-(b c) \int \frac {d^3 \left (-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6\right )}{3 x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} \left (b c d^3\right ) \int \frac {-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6}{x^3 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-1+9 c^2 x+9 c^4 x^2-c^6 x^3}{x^2 \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\frac {17 c^2}{2}+9 c^4 x-c^6 x^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\frac {33 c^2}{2}+7 c^2 x^2-c^2 x^4}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (8 c^4-c^4 x^2+\frac {17 c^2}{2 \left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {\left (17 b c 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 )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {17 b c^3 d^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.73 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {d^3 \left (-6 a+54 a c^2 x^2+54 a c^4 x^4-6 a c^6 x^6+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (3-50 c^2 x^2+2 c^4 x^4\right )-6 b \left (1-9 c^2 x^2-9 c^4 x^4+c^6 x^6\right ) \text {arccosh}(c x)+51 b c^3 x^3 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right )}{18 x^3} \]

[In]

Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]

[Out]

(d^3*(-6*a + 54*a*c^2*x^2 + 54*a*c^4*x^4 - 6*a*c^6*x^6 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - 50*c^2*x^2 +
2*c^4*x^4) - 6*b*(1 - 9*c^2*x^2 - 9*c^4*x^4 + c^6*x^6)*ArcCosh[c*x] + 51*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sq
rt[1 + c*x])]))/(18*x^3)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00

method	result	size
parts	\(-d^{3} a \left (\frac {x^{3} c^{6}}{3}-3 c^{4} x -\frac {3 c^{2}}{x}+\frac {1}{3 x^{3}}\right )-d^{3} b \,c^{3} \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+51 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-50 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\)	\(195\)
derivativedivides	\(c^{3} \left (-d^{3} a \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-d^{3} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+51 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-50 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(197\)
default	\(c^{3} \left (-d^{3} a \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-d^{3} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-3 c x \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+51 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-50 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(197\)

[In]

int((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x,method=_RETURNVERBOSE)

[Out]

-d^3*a*(1/3*x^3*c^6-3*c^4*x-3*c^2/x+1/3/x^3)-d^3*b*c^3*(1/3*c^3*x^3*arccosh(c*x)-3*c*x*arccosh(c*x)+1/3/c^3/x^
3*arccosh(c*x)-3*arccosh(c*x)/c/x-1/18*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(2*c^4*x^4*(c^2*x^2-1)^(1/2)+51*arctan(1/(c
^2*x^2-1)^(1/2))*c^2*x^2-50*c^2*x^2*(c^2*x^2-1)^(1/2)+3*(c^2*x^2-1)^(1/2))/c^2/x^2/(c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {6 \, a c^{6} d^{3} x^{6} - 54 \, a c^{4} d^{3} x^{4} + 102 \, b c^{3} d^{3} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 54 \, a c^{2} d^{3} x^{2} - 6 \, {\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, a d^{3} + 6 \, {\left (b c^{6} d^{3} x^{6} - 9 \, b c^{4} d^{3} x^{4} - 9 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} + b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d^{3} x^{5} - 50 \, b c^{3} d^{3} x^{3} + 3 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, x^{3}} \]

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="fricas")

[Out]

-1/18*(6*a*c^6*d^3*x^6 - 54*a*c^4*d^3*x^4 + 102*b*c^3*d^3*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 54*a*c^2*d^3*
x^2 - 6*(b*c^6 - 9*b*c^4 - 9*b*c^2 + b)*d^3*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) + 6*a*d^3 + 6*(b*c^6*d^3*x^6 - 9
*b*c^4*d^3*x^4 - 9*b*c^2*d^3*x^2 - (b*c^6 - 9*b*c^4 - 9*b*c^2 + b)*d^3*x^3 + b*d^3)*log(c*x + sqrt(c^2*x^2 - 1
)) - (2*b*c^5*d^3*x^5 - 50*b*c^3*d^3*x^3 + 3*b*c*d^3*x)*sqrt(c^2*x^2 - 1))/x^3

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=- d^{3} \left (\int \left (- 3 a c^{4}\right )\, dx + \int \left (- \frac {a}{x^{4}}\right )\, dx + \int \frac {3 a c^{2}}{x^{2}}\, dx + \int a c^{6} x^{2}\, dx + \int \left (- 3 b c^{4} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \frac {3 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{6} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]

[In]

integrate((-c**2*d*x**2+d)**3*(a+b*acosh(c*x))/x**4,x)

[Out]

-d**3*(Integral(-3*a*c**4, x) + Integral(-a/x**4, x) + Integral(3*a*c**2/x**2, x) + Integral(a*c**6*x**2, x) +
Integral(-3*b*c**4*acosh(c*x), x) + Integral(-b*acosh(c*x)/x**4, x) + Integral(3*b*c**2*acosh(c*x)/x**2, x) +
Integral(b*c**6*x**2*acosh(c*x), x))

Maxima [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {1}{3} \, a c^{6} d^{3} x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{6} d^{3} + 3 \, a c^{4} d^{3} x + 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c^{3} d^{3} + 3 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{3} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{3} + \frac {3 \, a c^{2} d^{3}}{x} - \frac {a d^{3}}{3 \, x^{3}} \]

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="maxima")

[Out]

-1/3*a*c^6*d^3*x^3 - 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^6*
d^3 + 3*a*c^4*d^3*x + 3*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c^3*d^3 + 3*(c*arcsin(1/(c*abs(x))) + arccosh
(c*x)/x)*b*c^2*d^3 - 1/6*((c^2*arcsin(1/(c*abs(x))) - sqrt(c^2*x^2 - 1)/x^2)*c + 2*arccosh(c*x)/x^3)*b*d^3 + 3
*a*c^2*d^3/x - 1/3*a*d^3/x^3

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^4} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^3)/x^4,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^3)/x^4, x)