3.1.8 \(\int \frac {(d-c^2 d x^2) (a+b \cosh ^{-1}(c x))}{x^3} \, dx\) [8]

Optimal. Leaf size=135 \[ \frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {1}{2} b c^2 d \cosh ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} b c^2 d \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5920, 99, 12, 54, 5882, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} b c^2 d \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} b c^2 d \cosh ^{-1}(c x)+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (b*c^2*d*ArcCosh[c*x])/2 - (d*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]))
/(2*x^2) - (c^2*d*(a + b*ArcCosh[c*x])^2)/(2*b) - c^2*d*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] + (b
*c^2*d*PolyLog[2, -E^(-2*ArcCosh[c*x])])/2

Rule 12

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

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 99

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

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 5920

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

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {1}{2} (b c d) \int \frac {\sqrt {-1+c x} \sqrt {1+c x}}{x^2} \, dx-\left (c^2 d\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {1}{2} (b c d) \int \frac {c^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\left (c^2 d\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-\left (2 c^2 d\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{2} \left (b c^3 d\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {1}{2} b c^2 d \cosh ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (b c^2 d\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {1}{2} b c^2 d \cosh ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} \left (b c^2 d\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {1}{2} b c^2 d \cosh ^{-1}(c x)-\frac {d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-c^2 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} b c^2 d \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 106, normalized size = 0.79 \begin {gather*} -\frac {d \left (a-b c x \sqrt {-1+c x} \sqrt {1+c x}+b c^2 x^2 \cosh ^{-1}(c x)^2+b \cosh ^{-1}(c x) \left (1+2 c^2 x^2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )+2 a c^2 x^2 \log (x)-b c^2 x^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(d*(a - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*c^2*x^2*ArcCosh[c*x]^2 + b*ArcCosh[c*x]*(1 + 2*c^2*x^2*Log
[1 + E^(-2*ArcCosh[c*x])]) + 2*a*c^2*x^2*Log[x] - b*c^2*x^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/x^2

________________________________________________________________________________________

Maple [A]
time = 7.59, size = 137, normalized size = 1.01

method	result	size
derivativedivides	\(c^{2} \left (-a d \ln \left (c x \right )-\frac {a d}{2 c^{2} x^{2}}+\frac {b d \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{2 c x}-\frac {b d}{2}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-b d \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b d \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\)	\(137\)
default	\(c^{2} \left (-a d \ln \left (c x \right )-\frac {a d}{2 c^{2} x^{2}}+\frac {b d \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{2 c x}-\frac {b d}{2}-\frac {b d \,\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-b d \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b d \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))/x^3, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {a c^{2}}{x}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)*(a+b*arccosh(c*x))/x^3,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]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________