3.3.1 \(\int \frac {(a+b \cosh ^{-1}(c x))^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx\) [201]

Optimal. Leaf size=186 \[ -\frac {c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{d x}-\frac {2 b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5917, 5882, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{d x}-\frac {c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}}-\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 c \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5917

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[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*A
rcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m
+ 2*p + 3, 0] && NeQ[m, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.52, size = 237, normalized size = 1.27 \begin {gather*} -\frac {a^2 \sqrt {-d \left (-1+c^2 x^2\right )}}{d x}-2 a b c \left (\frac {\sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{c d x}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\log \left (-1+\sqrt {1+c x}\right )+\log \left (1+\sqrt {1+c x}\right )\right )}{\sqrt {d-c^2 d x^2}}\right )+\frac {b^2 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{c x}-2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )}{\sqrt {-d (-1+c x) (1+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(512\) vs. \(2(192)=384\).
time = 2.04, size = 513, normalized size = 2.76

method	result	size
default	\(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{d x}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2} c}{d \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x \,c^{2}}{\left (c^{2} x^{2}-1\right ) d}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right ) d}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}-1\right ) d}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{x \left (c^{2} x^{2}-1\right ) d}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c}{d \left (c^{2} x^{2}-1\right )}\)	\(513\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/x^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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((a+b*arccosh(c*x))^2/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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((a+b*arccosh(c*x))^2/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [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((a+b*arccosh(c*x))^2/x^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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 )}^2}{x^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))^2/(x^2*(d - c^2*d*x^2)^(1/2)),x)

[Out]

int((a + b*acosh(c*x))^2/(x^2*(d - c^2*d*x^2)^(1/2)), x)

________________________________________________________________________________________