3.123 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=250 \[ \frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^2 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

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

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Rubi [A]  time = 0.458335, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {5798, 103, 12, 39, 5733, 1251, 893} \[ \frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2 \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log (x)}{3 d \sqrt{d-c^2 d x^2}}-\frac{b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{2 d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])/(3*d*x^3*Sqrt[d - c^2*
d*x^2]) - (4*c^2*(a + b*ArcCosh[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3*d*Sqrt[
d - c^2*d*x^2]) - (5*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x])/(3*d*Sqrt[d - c^2*d*x^2]) - (b*c^3*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*d*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 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^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^4 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{-1-4 c^2 x^2+8 c^4 x^4}{3 x^3 \left (1-c^2 x^2\right )} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{-1-4 c^2 x^2+8 c^4 x^4}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \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{-1-4 c^2 x+8 c^4 x^2}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{6 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \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{1}{x^2}-\frac{5 c^2}{x}-\frac{3 c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2 \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \log (x)}{3 d \sqrt{d-c^2 d x^2}}-\frac{b c^3 \sqrt{-1+c x} \sqrt{1+c x} \log \left (1-c^2 x^2\right )}{2 d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.116255, size = 161, normalized size = 0.64 \[ \frac{16 a c^4 x^4-8 a c^2 x^2-2 a-10 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \log (x)-3 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )+2 b \left (8 c^4 x^4-4 c^2 x^2-1\right ) \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1}}{6 d x^3 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

(-2*a - 8*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(-1 - 4*c^2*x^2 + 8*c^4*x^4)*Arc
Cosh[c*x] - 10*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] - 3*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1
- c^2*x^2])/(6*d*x^3*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.211, size = 1050, normalized size = 4.2 \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^4/(-c^2*d*x^2+d)^(3/2),x)

[Out]

-1/3*a/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*a*c^2/d/x/(-c^2*d*x^2+d)^(1/2)+8/3*a*c^4/d*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^2/(c^2*x^2-1)*arccosh(c*x)*c^3+32/3*b*(-d*(c^2*x^2-1))^(
1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^5*(c*x+1)*(c*x-1)*c^8-32/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^
2-1)*x^7*c^10-16/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^3*(c*x+1)*(c*x-1)*c^6+16*b*(-d*(c^2*
x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^5*c^8+64/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^
2*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^2/(8*c^4*x^4-7*c^2*x^2-1)*x^3*a
rccosh(c*x)*c^6-4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*(c*x+1)*(c*x-1)*c^4-4*b*(-d*(c^2*x^
2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x^3*c^6+8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*arcco
sh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+8*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*arccosh(c*x)*
c^4-4/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3-4/3*b*(-d*(c^2*x^
2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)*x*c^4+4*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)/x*arccosh
(c*x)*c^2-1/6*b*(-d*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)/x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c+1/3*b*(-d
*(c^2*x^2-1))^(1/2)/d^2/(8*c^4*x^4-7*c^2*x^2-1)/x^3*arccosh(c*x)+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/d^2/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*c^3+5/3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/
2)*(c*x+1)^(1/2)/d^2/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c^3

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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^4/(-c^2*d*x^2+d)^(3/2),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^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(3/2)*x^4), x)