3.5.67 \(\int \frac {(d+e x^2) (a+b \cosh ^{-1}(c x))}{x^2} \, dx\) [467]

Optimal. Leaf size=75 \[ -\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+b c d \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]

[Out]

-d*(a+b*arccosh(c*x))/x+e*x*(a+b*arccosh(c*x))+b*c*d*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))-b*e*(c*x-1)^(1/2)*(c*
x+1)^(1/2)/c

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Rubi [A]
time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5956, 471, 94, 211} \begin {gather*} -\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{x}+e x \left (a+b \cosh ^{-1}(c x)\right )+b c d \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {b e \sqrt {c x-1} \sqrt {c x+1}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c) - (d*(a + b*ArcCosh[c*x]))/x + e*x*(a + b*ArcCosh[c*x]) + b*c*d*ArcTan
[Sqrt[-1 + c*x]*Sqrt[1 + c*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 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 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 5956

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[d*(f*x)^(m
+ 1)*((a + b*ArcCosh[c*x])/(f*(m + 1))), x] + (-Dist[b*(c/(f*(m + 1)*(m + 3))), Int[(f*x)^(m + 1)*((d*(m + 3)
+ e*(m + 1)*x^2)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[e*(f*x)^(m + 3)*((a + b*ArcCosh[c*x])/(f^3*(m
+ 3))), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 105, normalized size = 1.40 \begin {gather*} -\frac {a d}{x}+a e x-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {b d \cosh ^{-1}(c x)}{x}+b e x \cosh ^{-1}(c x)+\frac {b c d \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d)/x) + a*e*x - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (b*d*ArcCosh[c*x])/x + b*e*x*ArcCosh[c*x] + (b*c*d
*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [A]
time = 1.95, size = 108, normalized size = 1.44

method	result	size
derivativedivides	\(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e x}{c}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{c x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e}{c^{2}}\right )\)	\(108\)
default	\(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) e x}{c}-\frac {b \,\mathrm {arccosh}\left (c x \right ) d}{c x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{\sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e}{c^{2}}\right )\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(a/c^2*(e*c*x-d*c/x)+b/c*arccosh(c*x)*e*x-b*arccosh(c*x)*d/c/x-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/
2)*d*arctan(1/(c^2*x^2-1)^(1/2))-b/c^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e)

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Maxima [A]
time = 0.48, size = 65, normalized size = 0.87 \begin {gather*} -{\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d + a x e + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b e}{c} - \frac {a d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x)*b*d + a*x*e + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*e/c - a*d/x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (69) = 138\).
time = 0.41, size = 174, normalized size = 2.32 \begin {gather*} \frac {2 \, b c^{2} d x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + a c x^{2} \cosh \left (1\right ) + a c x^{2} \sinh \left (1\right ) - a c d + {\left (b c d x - b c d + {\left (b c x^{2} - b c x\right )} \cosh \left (1\right ) + {\left (b c x^{2} - b c x\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c d x - b c x \cosh \left (1\right ) - b c x \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} {\left (b x \cosh \left (1\right ) + b x \sinh \left (1\right )\right )}}{c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b*c^2*d*x*arctan(-c*x + sqrt(c^2*x^2 - 1)) + a*c*x^2*cosh(1) + a*c*x^2*sinh(1) - a*c*d + (b*c*d*x - b*c*d +
(b*c*x^2 - b*c*x)*cosh(1) + (b*c*x^2 - b*c*x)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + (b*c*d*x - b*c*x*cosh(1
) - b*c*x*sinh(1))*log(-c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2*x^2 - 1)*(b*x*cosh(1) + b*x*sinh(1)))/(c*x)

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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 ) \left (d + e x^{2}\right )}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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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 (e\,x^2+d\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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