\(\int \frac {(d-c^2 d x^2)^2 (a+b \text {arccosh}(c x))}{x^4} \, dx\) [18]

Optimal result

Integrand size = 25, antiderivative size = 142 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=-b c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}+\frac {b c d^2 \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {11}{6} b c^3 d^2 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]

[Out]

-1/3*d^2*(a+b*arccosh(c*x))/x^3+2*c^2*d^2*(a+b*arccosh(c*x))/x+c^4*d^2*x*(a+b*arccosh(c*x))-11/6*b*c^3*d^2*arc
tan((c*x-1)^(1/2)*(c*x+1)^(1/2))-b*c^3*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/6*b*c*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/x^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.31, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {276, 5921, 12, 534, 1265, 911, 1171, 396, 211} \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}-\frac {11 b c^3 d^2 \sqrt {c^2 x^2-1} \arctan \left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(b*c^3*d^2*(1 - c^2*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/(3*x^3) + (2*c^2*d^2*(a + b*ArcCosh[c*x]))/x + c^4*d^2*x*(a + b*ArcCosh
[c*x]) - (11*b*c^3*d^2*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 396

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

Rule 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(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)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 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 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
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^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-(b c) \int \frac {d^2 \left (-1+6 c^2 x^2+3 c^4 x^4\right )}{3 x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {1}{3} \left (b c d^2\right ) \int \frac {-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-1+6 c^2 x+3 c^4 x^2}{x^2 \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {8+12 x^2+3 x^4}{\left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))+\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {-17-6 x^2}{\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 {b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {\left (11 b c d^2 \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 {b c^3 d^2 \left (1-c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {arccosh}(c x))}{x}+c^4 d^2 x (a+b \text {arccosh}(c x))-\frac {11 b c^3 d^2 \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 = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {d^2 \left (-2 a+12 a c^2 x^2+6 a c^4 x^4+b c x \sqrt {-1+c x} \sqrt {1+c x}-6 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (-1+6 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)+11 b c^3 x^3 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right )}{6 x^3} \]

[In]

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

[Out]

(d^2*(-2*a + 12*a*c^2*x^2 + 6*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 6*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt
[1 + c*x] + 2*b*(-1 + 6*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] + 11*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x
])]))/(6*x^3)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06

method	result	size
parts	\(d^{2} a \left (c^{4} x +\frac {2 c^{2}}{x}-\frac {1}{3 x^{3}}\right )+d^{2} b \,c^{3} \left (c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (11 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\right )}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\)	\(150\)
derivativedivides	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (11 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\right )}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(152\)
default	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (11 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\right )}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(152\)

[In]

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

[Out]

d^2*a*(c^4*x+2*c^2/x-1/3/x^3)+d^2*b*c^3*(c*x*arccosh(c*x)-1/3/c^3/x^3*arccosh(c*x)+2*arccosh(c*x)/c/x+1/6*(c*x
-1)^(1/2)*(c*x+1)^(1/2)*(11*arctan(1/(c^2*x^2-1)^(1/2))*c^2*x^2-6*c^2*x^2*(c^2*x^2-1)^(1/2)+(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 = 0.29 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {6 \, a c^{4} d^{2} x^{4} - 22 \, b c^{3} d^{2} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, a c^{2} d^{2} x^{2} - 2 \, {\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, a d^{2} + 2 \, {\left (3 \, b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - {\left (3 \, b c^{4} + 6 \, b c^{2} - b\right )} d^{2} x^{3} - b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{3} d^{2} x^{3} - b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, x^{3}} \]

[In]

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

[Out]

1/6*(6*a*c^4*d^2*x^4 - 22*b*c^3*d^2*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 12*a*c^2*d^2*x^2 - 2*(3*b*c^4 + 6*b
*c^2 - b)*d^2*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) - 2*a*d^2 + 2*(3*b*c^4*d^2*x^4 + 6*b*c^2*d^2*x^2 - (3*b*c^4 +
6*b*c^2 - b)*d^2*x^3 - b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (6*b*c^3*d^2*x^3 - b*c*d^2*x)*sqrt(c^2*x^2 - 1))/
x^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^4} \, dx=a c^{4} d^{2} x + {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c^{3} d^{2} + 2 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{2} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{2} + \frac {2 \, a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 \, x^{3}} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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