3.4.17 \(\int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx\) [317]

Optimal. Leaf size=58 \[ \frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]

[Out]

c*(a*x-1)^(3/2)*(a*x+1)^(3/2)/a/arccosh(a*x)+3/4*c*Chi(arccosh(a*x))/a-3/4*c*Chi(3*arccosh(a*x))/a

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Rubi [A]
time = 0.15, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5904, 5953, 5556, 3382} \begin {gather*} \frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}+\frac {c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)/ArcCosh[a*x]^2,x]

[Out]

(c*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2))/(a*ArcCosh[a*x]) + (3*c*CoshIntegral[ArcCosh[a*x]])/(4*a) - (3*c*CoshInte
gral[3*ArcCosh[a*x]])/(4*a)

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5904

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

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-(3 a c) \int \frac {x \sqrt {-1+a x} \sqrt {1+a x}}{\cosh ^{-1}(a x)} \, dx\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}-\frac {(3 c) \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}-\frac {(3 c) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2}}{a \cosh ^{-1}(a x)}+\frac {3 c \text {Chi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {3 c \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(58)=116\).
time = 1.29, size = 217, normalized size = 3.74 \begin {gather*} \frac {c \sqrt {-1+a x} \left (4 \left (\frac {-1+a x}{1+a x}\right )^{5/2} (1+a x)^5-3 \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )-(-1+a x) \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right ) \left (1+a x-4 \sqrt {-1+a x} \sqrt {1+a x} \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )+4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)+(1-a x) \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right ) \log \left (\cosh ^{-1}(a x)\right )\right )}{4 a \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{5/2} \cosh ^{-1}(a x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[-1 + a*x]*(4*((-1 + a*x)/(1 + a*x))^(5/2)*(1 + a*x)^5 - 3*(-1 + a^2*x^2)*ArcCosh[a*x]*CoshIntegral[3*A
rcCosh[a*x]] - (-1 + a*x)*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]]*(1 + a*x - 4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Co
th[ArcCosh[a*x]/2]) + 4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + (1 -
a*x)*Coth[ArcCosh[a*x]/2])*Log[ArcCosh[a*x]]))/(4*a*((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^(5/2)*ArcCosh[a*x]
)

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Maple [A]
time = 2.12, size = 61, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {c \left (3 \hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \sqrt {a x -1}\, \sqrt {a x +1}+\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a \,\mathrm {arccosh}\left (a x \right )}\)	\(61\)
default	\(\frac {c \left (3 \hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-3 \sqrt {a x -1}\, \sqrt {a x +1}+\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a \,\mathrm {arccosh}\left (a x \right )}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*c*(3*Chi(arccosh(a*x))*arccosh(a*x)-3*Chi(3*arccosh(a*x))*arccosh(a*x)-3*(a*x-1)^(1/2)*(a*x+1)^(1/2)+sin
h(3*arccosh(a*x)))/arccosh(a*x)

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

[Out]

(a^5*c*x^5 - 2*a^3*c*x^3 + a*c*x + (a^4*c*x^4 - 2*a^2*c*x^2 + c)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt
(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) - integrate((3*a^6*c*x^6 - 7*a^4*c*
x^4 + 5*a^2*c*x^2 + (3*a^4*c*x^4 - 2*a^2*c*x^2 - c)*(a*x + 1)*(a*x - 1) + 3*(2*a^5*c*x^5 - 3*a^3*c*x^3 + a*c*x
)*sqrt(a*x + 1)*sqrt(a*x - 1) - c)/((a^4*x^4 + (a*x + 1)*(a*x - 1)*a^2*x^2 - 2*a^2*x^2 + 2*(a^3*x^3 - a*x)*sqr
t(a*x + 1)*sqrt(a*x - 1) + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), 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((-a^2*c*x^2+c)/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(-(a^2*c*x^2 - c)/arccosh(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(Integral(a**2*x**2/acosh(a*x)**2, x) + Integral(-1/acosh(a*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((-a^2*c*x^2+c)/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(-(a^2*c*x^2 - c)/arccosh(a*x)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {c-a^2\,c\,x^2}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a^2*c*x^2)/acosh(a*x)^2,x)

[Out]

int((c - a^2*c*x^2)/acosh(a*x)^2, x)

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