3.1.6 \(\int \frac {(d-c^2 d x^2) (a+b \cosh ^{-1}(c x))}{x} \, dx\) [6]

Optimal. Leaf size=117 \[ \frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} b d \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) \]

[Out]

-1/4*b*d*arccosh(c*x)+1/2*d*(-c^2*x^2+1)*(a+b*arccosh(c*x))+1/2*d*(a+b*arccosh(c*x))^2/b+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*d*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+1/4*b*c*
d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5919, 5882, 3799, 2221, 2317, 2438, 38, 54} \begin {gather*} \frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{2} b d \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+\frac {1}{4} b c d x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{4} b d \cosh ^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/4 - (b*d*ArcCosh[c*x])/4 + (d*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/2 + (
d*(a + b*ArcCosh[c*x])^2)/(2*b) + d*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (b*d*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*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+d \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx+\frac {1}{2} (b c d) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+d \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{4} (b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+(2 d) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-(b d) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} (b d) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \cosh ^{-1}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} b d \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 116, normalized size = 0.99 \begin {gather*} -\frac {1}{4} d \left (2 a c^2 x^2-b c x \sqrt {-1+c x} \sqrt {1+c x}-2 b \cosh ^{-1}(c x)^2-2 b \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )+2 b \cosh ^{-1}(c x) \left (c^2 x^2-2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )-4 a \log (x)+2 b \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 5.02, size = 131, normalized size = 1.12

method	result	size
derivativedivides	\(-\frac {a \,c^{2} d \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {b d \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\mathrm {arccosh}\left (c x \right )}{4}+b d \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b d \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(131\)
default	\(-\frac {a \,c^{2} d \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {b d \mathrm {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b d \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\mathrm {arccosh}\left (c x \right )}{4}+b d \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {b d \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*c^2*d*x^2+a*d*ln(c*x)-1/2*b*d*arccosh(c*x)^2+1/4*b*c*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/2*b*d*arccosh(c*
x)*c^2*x^2+1/4*b*d*arccosh(c*x)+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)

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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,x, algorithm="maxima")

[Out]

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

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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,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, x)

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

[Out]

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

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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,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

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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} \,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,x)

[Out]

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

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