\(\int \frac {a+b \text {arccosh}(c x)}{x^4 (d-c^2 d x^2)} \, dx\) [36]

Optimal result

Integrand size = 25, antiderivative size = 157 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}+\frac {7 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}+\frac {2 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d}+\frac {b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d} \]

[Out]

1/3*(-a-b*arccosh(c*x))/d/x^3-c^2*(a+b*arccosh(c*x))/d/x+7/6*b*c^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d+2*c^3
*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d+b*c^3*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2
))/d-b*c^3*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d+1/6*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5932, 5903, 4267, 2317, 2438, 94, 211, 105, 12} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )} \, dx=\frac {2 c^3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d}+\frac {7 b c^3 \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x^3) - (c^2*(a + b*ArcCosh[c*x]))/(d*
x) + (7*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d) + (2*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]
])/d + (b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/d - (b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d

Rule 12

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

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

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 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 4267

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

Rule 5903

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

Rule 5932

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{3 d x^3}+c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {1}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}+c^4 \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx+\frac {(b c) \int \frac {c^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}-\frac {c^3 \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{d}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{d}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}+\frac {7 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}+\frac {2 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{d} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {c^2 (a+b \text {arccosh}(c x))}{d x}+\frac {7 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}+\frac {2 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d}+\frac {b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )} \, dx=\frac {-\frac {2 a}{x^3}-\frac {6 a c^2}{x}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{x^2}-\frac {2 b \text {arccosh}(c x)}{x^3}-\frac {6 b c^2 \text {arccosh}(c x)}{x}+\frac {7 b c^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-6 a c^3 \log \left (1-e^{\text {arccosh}(c x)}\right )-6 b c^3 \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+6 a c^3 \log \left (1+e^{\text {arccosh}(c x)}\right )+6 b c^3 \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+6 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-6 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{6 d} \]

[In]

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

[Out]

((-2*a)/x^3 - (6*a*c^2)/x + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x^2 - (2*b*ArcCosh[c*x])/x^3 - (6*b*c^2*ArcCosh
[c*x])/x + (7*b*c^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 6*a*c^3*Lo
g[1 - E^ArcCosh[c*x]] - 6*b*c^3*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 6*a*c^3*Log[1 + E^ArcCosh[c*x]] + 6*b*c
^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 6*b*c^3*PolyLog[2, -E^ArcCosh[c*x]] - 6*b*c^3*PolyLog[2, E^ArcCosh[c
*x]])/(6*d)

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a \left (\frac {1}{3 c^{3} x^{3}}+\frac {1}{c x}-\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (\frac {6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {7 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(190\)
default	\(c^{3} \left (-\frac {a \left (\frac {1}{3 c^{3} x^{3}}+\frac {1}{c x}-\frac {\ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \left (\frac {6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {7 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(190\)
parts	\(-\frac {a \left (-\frac {c^{3} \ln \left (c x +1\right )}{2}+\frac {1}{3 x^{3}}+\frac {c^{2}}{x}+\frac {c^{3} \ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \,c^{3} \left (\frac {6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )}{6 c^{3} x^{3}}-\frac {7 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(192\)

[In]

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

[Out]

c^3*(-a/d*(1/3/c^3/x^3+1/c/x-1/2*ln(c*x+1)+1/2*ln(c*x-1))-b/d*(1/6*(6*c^2*x^2*arccosh(c*x)-(c*x-1)^(1/2)*(c*x+
1)^(1/2)*c*x+2*arccosh(c*x))/c^3/x^3-7/3*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-dilog(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))-dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))))

Fricas [F]

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

[In]

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

[Out]

integral(-(b*arccosh(c*x) + a)/(c^2*d*x^6 - d*x^4), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/6*(3*c^3*log(c*x + 1)/d - 3*c^3*log(c*x - 1)/d - 2*(3*c^2*x^2 + 1)/(d*x^3))*a + 1/24*(216*c^5*integrate(1/12
*x^3*log(c*x - 1)/(c^2*d*x^4 - d*x^2), x) - 12*c^4*(log(c*x + 1)/(c*d) - log(c*x - 1)/(c*d)) - 72*c^4*integrat
e(1/12*x^2*log(c*x - 1)/(c^2*d*x^4 - d*x^2), x) - 4*c^2*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x)) - (3*c
^3*x^3*log(c*x + 1)^2 + 6*c^3*x^3*log(c*x + 1)*log(c*x - 1) - 4*(3*c^3*x^3*log(c*x + 1) - 3*c^3*x^3*log(c*x -
1) - 6*c^2*x^2 - 2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(d*x^3) + 24*integrate(1/6*(3*c^4*x^3*log(c*x + 1)
- 3*c^4*x^3*log(c*x - 1) - 6*c^3*x^2 - 2*c)/(c^3*d*x^6 - c*d*x^4 + (c^2*d*x^5 - d*x^3)*sqrt(c*x + 1)*sqrt(c*x
- 1)), x))*b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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