3.131 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^2 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=248 \[ \frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^3 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c \sqrt{d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b*c*Sqrt[d - c^2*d*x^2])/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)) - (a + b*ArcCosh[c*x])/(d*x*(d -
c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcCosh[c*x]
))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c*
Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.44195, antiderivative size = 279, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 103, 12, 40, 39, 5733, 1251, 893} \[ \frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \log (x)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b c \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (8*c^2*x*(a + b*ArcCosh[c*x]))/
(3*d^2*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])/(d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) + (4*c^2*x*
(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[
x])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(6*d^2*Sqrt[d - c^2*d*x^
2])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m +
1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; F
reeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{3-12 c^2 x^2+8 c^4 x^4}{3 x \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{3-12 c^2 x^2+8 c^4 x^4}{x \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{3-12 c^2 x+8 c^4 x^2}{x \left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{x}-\frac{c^2}{\left (-1+c^2 x\right )^2}+\frac{5 c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{8 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \log (x)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b c \sqrt{-1+c x} \sqrt{1+c x} \log \left (1-c^2 x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.336904, size = 147, normalized size = 0.59 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (\frac{4 c^2 x \left (2 c^2 x^2-3\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{x (c x-1)^{3/2} (c x+1)^{3/2}}-\frac{1}{6} b c \left (\frac{1}{c^2 x^2-1}+5 \log \left (1-c^2 x^2\right )+6 \log (x)\right )\right )}{d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/(x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (4*c^2*x*(-3 + 2*c^
2*x^2)*(a + b*ArcCosh[c*x]))/(3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (b*c*((-1 + c^2*x^2)^(-1) + 6*Log[x] + 5*L
og[1 - c^2*x^2]))/6))/(d^2*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.194, size = 1350, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-a/d/x/(-c^2*d*x^2+d)^(3/2)+4/3*a*c^2/d*x/(-c^2*d*x^2+d)^(3/2)+8/3*a*c^2/d^2*x/(-c^2*d*x^2+d)^(1/2)-16/3*b*(-d
*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*c+32/3*b*(-d*(c^2*x^2-1))^(1/2)/d
^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^7*(c*x+1)*(c*x-1)*c^8-32/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25
*c^4*x^4+26*c^2*x^2-9)*x^9*c^10-80/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^5*(c*x
+1)*(c*x-1)*c^6+112/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^7*c^8+64/3*b*(-d*(c^2
*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^4*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-64/3*b
*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^5*arccosh(c*x)*c^6+20*b*(-d*(c^2*x^2-1))^(1/
2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^3*(c*x+1)*(c*x-1)*c^4-140/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x
^6-25*c^4*x^4+26*c^2*x^2-9)*x^5*c^6-136/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^2
*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+56*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2
-9)*x^3*arccosh(c*x)*c^4-4*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x*(c*x+1)*(c*x-1)*
c^2+4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+2
4*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x^3*c^4+24*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*
c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c-44*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8
*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x*arccosh(c*x)*c^2-3/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26
*c^2*x^2-9)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c-4*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)*x
*c^2+9*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(8*c^6*x^6-25*c^4*x^4+26*c^2*x^2-9)/x*arccosh(c*x)+5/3*b*(-d*(c^2*x^2-1))^
(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*c+b*(-d*(c^2*x^2-1
))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{6} d^{3} x^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x^2), x)