3.4.15 \(\int \frac {(c-a^2 c x^2)^3}{\cosh ^{-1}(a x)^2} \, dx\) [315]

Optimal. Leaf size=98 \[ \frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}+\frac {35 c^3 \text {Chi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac {35 c^3 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Chi}\left (7 \cosh ^{-1}(a x)\right )}{64 a} \]

[Out]

c^3*(a*x-1)^(7/2)*(a*x+1)^(7/2)/a/arccosh(a*x)+35/64*c^3*Chi(arccosh(a*x))/a-63/64*c^3*Chi(3*arccosh(a*x))/a+3
5/64*c^3*Chi(5*arccosh(a*x))/a-7/64*c^3*Chi(7*arccosh(a*x))/a

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Rubi [A]
time = 0.22, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5904, 5953, 5556, 3382} \begin {gather*} \frac {35 c^3 \text {Chi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac {35 c^3 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Chi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}+\frac {c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2))/(a*ArcCosh[a*x]) + (35*c^3*CoshIntegral[ArcCosh[a*x]])/(64*a) - (63*c^3
*CoshIntegral[3*ArcCosh[a*x]])/(64*a) + (35*c^3*CoshIntegral[5*ArcCosh[a*x]])/(64*a) - (7*c^3*CoshIntegral[7*A
rcCosh[a*x]])/(64*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 {\left (c-a^2 c x^2\right )^3}{\cosh ^{-1}(a x)^2} \, dx &=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\left (7 a c^3\right ) \int \frac {x (-1+a x)^{5/2} (1+a x)^{5/2}}{\cosh ^{-1}(a x)} \, dx\\ &=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^6(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \left (-\frac {5 \cosh (x)}{64 x}+\frac {9 \cosh (3 x)}{64 x}-\frac {5 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}-\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}-\frac {\left (63 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}\\ &=\frac {c^3 (-1+a x)^{7/2} (1+a x)^{7/2}}{a \cosh ^{-1}(a x)}+\frac {35 c^3 \text {Chi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Chi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac {35 c^3 \text {Chi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Chi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(257\) vs. \(2(98)=196\).
time = 0.27, size = 257, normalized size = 2.62 \begin {gather*} \frac {c^3 \left (-64 \sqrt {\frac {-1+a x}{1+a x}}-64 a x \sqrt {\frac {-1+a x}{1+a x}}+192 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}}+192 a^3 x^3 \sqrt {\frac {-1+a x}{1+a x}}-192 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-192 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}+64 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}}+64 a^7 x^7 \sqrt {\frac {-1+a x}{1+a x}}+35 \cosh ^{-1}(a x) \text {Chi}\left (\cosh ^{-1}(a x)\right )-63 \cosh ^{-1}(a x) \text {Chi}\left (3 \cosh ^{-1}(a x)\right )+35 \cosh ^{-1}(a x) \text {Chi}\left (5 \cosh ^{-1}(a x)\right )-7 \cosh ^{-1}(a x) \text {Chi}\left (7 \cosh ^{-1}(a x)\right )\right )}{64 a \cosh ^{-1}(a x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-64*Sqrt[(-1 + a*x)/(1 + a*x)] - 64*a*x*Sqrt[(-1 + a*x)/(1 + a*x)] + 192*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*
x)] + 192*a^3*x^3*Sqrt[(-1 + a*x)/(1 + a*x)] - 192*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 192*a^5*x^5*Sqrt[(-1 +
a*x)/(1 + a*x)] + 64*a^6*x^6*Sqrt[(-1 + a*x)/(1 + a*x)] + 64*a^7*x^7*Sqrt[(-1 + a*x)/(1 + a*x)] + 35*ArcCosh[
a*x]*CoshIntegral[ArcCosh[a*x]] - 63*ArcCosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] + 35*ArcCosh[a*x]*CoshIntegral[
5*ArcCosh[a*x]] - 7*ArcCosh[a*x]*CoshIntegral[7*ArcCosh[a*x]]))/(64*a*ArcCosh[a*x])

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Maple [A]
time = 4.33, size = 109, normalized size = 1.11

method	result	size
derivativedivides	\(-\frac {c^{3} \left (7 \hyperbolicCosineIntegral \left (7 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-35 \hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )+63 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-35 \hyperbolicCosineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )+35 \sqrt {a x -1}\, \sqrt {a x +1}-21 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )+7 \sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )-\sinh \left (7 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{64 a \,\mathrm {arccosh}\left (a x \right )}\)	\(109\)
default	\(-\frac {c^{3} \left (7 \hyperbolicCosineIntegral \left (7 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-35 \hyperbolicCosineIntegral \left (\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )+63 \hyperbolicCosineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )-35 \hyperbolicCosineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right ) \mathrm {arccosh}\left (a x \right )+35 \sqrt {a x -1}\, \sqrt {a x +1}-21 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )+7 \sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )-\sinh \left (7 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{64 a \,\mathrm {arccosh}\left (a x \right )}\)	\(109\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64/a*c^3*(7*Chi(7*arccosh(a*x))*arccosh(a*x)-35*Chi(arccosh(a*x))*arccosh(a*x)+63*Chi(3*arccosh(a*x))*arcco
sh(a*x)-35*Chi(5*arccosh(a*x))*arccosh(a*x)+35*(a*x-1)^(1/2)*(a*x+1)^(1/2)-21*sinh(3*arccosh(a*x))+7*sinh(5*ar
ccosh(a*x))-sinh(7*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)^3/arccosh(a*x)^2,x, algorithm="maxima")

[Out]

(a^9*c^3*x^9 - 4*a^7*c^3*x^7 + 6*a^5*c^3*x^5 - 4*a^3*c^3*x^3 + a*c^3*x + (a^8*c^3*x^8 - 4*a^6*c^3*x^6 + 6*a^4*
c^3*x^4 - 4*a^2*c^3*x^2 + c^3)*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((7*a^10*c^3*x^10 - 29*a^8*c^3*x^8 + 46*a^6*c^3*x^6 - 34*a
^4*c^3*x^4 + 11*a^2*c^3*x^2 + (7*a^8*c^3*x^8 - 20*a^6*c^3*x^6 + 18*a^4*c^3*x^4 - 4*a^2*c^3*x^2 - c^3)*(a*x + 1
)*(a*x - 1) - c^3 + 7*(2*a^9*c^3*x^9 - 7*a^7*c^3*x^7 + 9*a^5*c^3*x^5 - 5*a^3*c^3*x^3 + a*c^3*x)*sqrt(a*x + 1)*
sqrt(a*x - 1))/((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)*sqrt(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)^3/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arccosh(a*x)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\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)**3/acosh(a*x)**2,x)

[Out]

-c**3*(Integral(3*a**2*x**2/acosh(a*x)**2, x) + Integral(-3*a**4*x**4/acosh(a*x)**2, x) + Integral(a**6*x**6/a
cosh(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)^3/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(-(a^2*c*x^2 - c)^3/arccosh(a*x)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{{\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)^3/acosh(a*x)^2,x)

[Out]

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

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