\(\int \frac {(d-c^2 d x^2)^2 (a+b \text {arccosh}(c x))}{x} \, dx\) [15]

Optimal result

Integrand size = 25, antiderivative size = 184 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \text {arccosh}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}+d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]

[Out]

-1/16*b*c*d^2*x*(c*x-1)^(3/2)*(c*x+1)^(3/2)-11/32*b*d^2*arccosh(c*x)+1/2*d^2*(-c^2*x^2+1)*(a+b*arccosh(c*x))+1
/4*d^2*(-c^2*x^2+1)^2*(a+b*arccosh(c*x))+1/2*d^2*(a+b*arccosh(c*x))^2/b+d^2*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))^2)-1/2*b*d^2*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+11/32*b*c*d^2*x*(c*x-
1)^(1/2)*(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5919, 5882, 3799, 2221, 2317, 2438, 38, 54} \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}+d^2 \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-\frac {11}{32} b d^2 \text {arccosh}(c x)-\frac {1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac {11}{32} b c d^2 x \sqrt {c x-1} \sqrt {c x+1} \]

[In]

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

[Out]

(11*b*c*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/32 - (b*c*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/16 - (11*b*d^2*A
rcCosh[c*x])/32 + (d^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/2 + (d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/4 +
(d^2*(a + b*ArcCosh[c*x])^2)/(2*b) + d^2*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (b*d^2*PolyLog[2,
-E^(-2*ArcCosh[c*x])])/2

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5919

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+d \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx-\frac {1}{4} \left (b c d^2\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx \\ & = -\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+d^2 \int \frac {a+b \text {arccosh}(c x)}{x} \, dx+\frac {1}{16} \left (3 b c d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx+\frac {1}{2} \left (b c d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx \\ & = \frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))-\frac {d^2 \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b}-\frac {1}{32} \left (3 b c d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{4} \left (b c d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \text {arccosh}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b} \\ & = \frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \text {arccosh}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}+d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-d^2 \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right ) \\ & = \frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \text {arccosh}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}+d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\frac {1}{2} \left (b d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ & = \frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \text {arccosh}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))+\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 b}+d^2 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right ) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{4} d^2 \left (-4 a c^2 x^2+a c^4 x^4-4 b c^2 x^2 \text {arccosh}(c x)+b c^4 x^4 \text {arccosh}(c x)+2 b \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )-\frac {1}{8} b \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+2 b \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+4 a \log (x)-2 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]

[In]

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

[Out]

(d^2*(-4*a*c^2*x^2 + a*c^4*x^4 - 4*b*c^2*x^2*ArcCosh[c*x] + b*c^4*x^4*ArcCosh[c*x] + 2*b*(c*x*Sqrt[-1 + c*x]*S
qrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]) - (b*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(3 + 3*c*x + 2*c^2*
x^2 + 2*c^3*x^3) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/8 + 2*b*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-
2*ArcCosh[c*x])]) + 4*a*Log[x] - 2*b*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/4

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.04

method	result	size
parts	\(d^{2} a \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (x \right )\right )+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}+\frac {d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {13 b \,d^{2} \operatorname {arccosh}\left (c x \right )}{32}-\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\)	\(192\)
derivativedivides	\(d^{2} a \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 b \,d^{2} \operatorname {arccosh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}\)	\(194\)
default	\(d^{2} a \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+\frac {13 b \,d^{2} \operatorname {arccosh}\left (c x \right )}{32}+d^{2} b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {d^{2} b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}\)	\(194\)

[In]

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

[Out]

d^2*a*(1/4*c^4*x^4-c^2*x^2+ln(x))+1/4*d^2*b*arccosh(c*x)*c^4*x^4-d^2*b*arccosh(c*x)*c^2*x^2+1/2*d^2*b*polylog(
2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)-1/2*d^2*b*arccosh(c*x)^2+13/32*b*d^2*arccosh(c*x)-1/16*d^2*b*(c*x+1)^(
1/2)*(c*x-1)^(1/2)*c^3*x^3+13/32*b*c*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)+d^2*b*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))^2)

Fricas [F]

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

[In]

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

[Out]

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c^2*d^2*x^2 + b*d^2)*arccosh(c*x))/x,
x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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