\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx\) [28]

Optimal result

Integrand size = 25, antiderivative size = 158 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^5 d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^5 d} \]

[Out]

-x*(a+b*arccosh(c*x))/c^4/d-1/3*x^3*(a+b*arccosh(c*x))/c^2/d+2*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c
*x+1)^(1/2))/c^5/d+b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d-b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(
1/2))/c^5/d+11/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d+1/9*b*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 158, 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 = {5938, 5903, 4267, 2317, 2438, 75, 102, 12} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {2 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^5 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^5 d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^5 d}+\frac {11 b \sqrt {c x-1} \sqrt {c x+1}}{9 c^5 d}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c^3 d} \]

[In]

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

[Out]

(11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^5*d) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3*d) - (x*(a + b*Arc
Cosh[c*x]))/(c^4*d) - (x^3*(a + b*ArcCosh[c*x]))/(3*c^2*d) + (2*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/
(c^5*d) + (b*PolyLog[2, -E^ArcCosh[c*x]])/(c^5*d) - (b*PolyLog[2, E^ArcCosh[c*x]])/(c^5*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 75

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

Rule 102

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)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

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 5938

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {\int \frac {x^2 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{c^2}+\frac {b \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c d} \\ & = \frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {\int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{c^4}+\frac {b \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^3 d}+\frac {b \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^3 d} \\ & = \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}-\frac {\text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{c^5 d}+\frac {(2 b) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^3 d} \\ & = \frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^5 d}-\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^5 d} \\ & = \frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{c^5 d}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{c^5 d} \\ & = \frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^5 d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=-\frac {18 a c x+6 a c^3 x^3-18 b \sqrt {\frac {-1+c x}{1+c x}}-18 b c x \sqrt {\frac {-1+c x}{1+c x}}-4 b \sqrt {-1+c x} \sqrt {1+c x}-2 b c^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}+18 b c x \text {arccosh}(c x)+6 b c^3 x^3 \text {arccosh}(c x)-9 b \text {arccosh}(c x)^2-18 b \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+18 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+9 a \log (1-c x)-9 a \log (1+c x)+18 b \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )+18 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{18 c^5 d} \]

[In]

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

[Out]

-1/18*(18*a*c*x + 6*a*c^3*x^3 - 18*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 18*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*b*Sq
rt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 18*b*c*x*ArcCosh[c*x] + 6*b*c^3*x^3*Ar
cCosh[c*x] - 9*b*ArcCosh[c*x]^2 - 18*b*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 18*b*ArcCosh[c*x]*Log[1 - E^A
rcCosh[c*x]] + 9*a*Log[1 - c*x] - 9*a*Log[1 + c*x] + 18*b*PolyLog[2, -E^(-ArcCosh[c*x])] + 18*b*PolyLog[2, E^A
rcCosh[c*x]])/(c^5*d)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{3} x^{3}}{3}+c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{3} x^{3}}{3 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}}{9 d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 d}}{c^{5}}\)	\(232\)
default	\(\frac {-\frac {a \left (\frac {c^{3} x^{3}}{3}+c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{3} x^{3}}{3 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}}{9 d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 d}}{c^{5}}\)	\(232\)
parts	\(-\frac {a \left (\frac {\frac {1}{3} x^{3} c^{2}+x}{c^{4}}-\frac {\ln \left (c x +1\right )}{2 c^{5}}+\frac {\ln \left (c x -1\right )}{2 c^{5}}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{5}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{5}}+\frac {b \,x^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3} d}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{5} d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{3}}{3 d \,c^{2}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{d \,c^{4}}\)	\(254\)

[In]

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

[Out]

1/c^5*(-a/d*(1/3*c^3*x^3+c*x+1/2*ln(c*x-1)-1/2*ln(c*x+1))-1/3*b/d*arccosh(c*x)*c^3*x^3-b/d*arccosh(c*x)*c*x-b/
d*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+b/d*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+1/
9*b/d*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2-b/d*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+b/d*polylog(2,-c*x-(c
*x-1)^(1/2)*(c*x+1)^(1/2))+11/9*b/d*(c*x-1)^(1/2)*(c*x+1)^(1/2))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a x^{4}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]

[In]

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

[Out]

-(Integral(a*x**4/(c**2*x**2 - 1), x) + Integral(b*x**4*acosh(c*x)/(c**2*x**2 - 1), x))/d

Maxima [F]

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

[In]

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

[Out]

1/72*(4*c^4*(2*(c^2*x^3 + 3*x)/(c^8*d) - 3*log(c*x + 1)/(c^9*d) + 3*log(c*x - 1)/(c^9*d)) + 36*c^2*(2*x/(c^6*d
) - log(c*x + 1)/(c^7*d) + log(c*x - 1)/(c^7*d)) + 648*c*integrate(1/12*x*log(c*x - 1)/(c^6*d*x^2 - c^4*d), x)
- 3*(4*(2*c^3*x^3 + 6*c*x - 3*log(c*x + 1) + 3*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + 3*log(c
*x + 1)^2 + 6*log(c*x + 1)*log(c*x - 1))/(c^5*d) + 72*integrate(-1/6*(2*c^3*x^3 + 6*c*x - 3*log(c*x + 1) + 3*l
og(c*x - 1))/(c^7*d*x^3 - c^5*d*x + (c^6*d*x^2 - c^4*d)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) - 216*integrate(1/12*
log(c*x - 1)/(c^6*d*x^2 - c^4*d), x))*b - 1/6*a*(2*(c^2*x^3 + 3*x)/(c^4*d) - 3*log(c*x + 1)/(c^5*d) + 3*log(c*
x - 1)/(c^5*d))

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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