3.133 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=338 \[ \frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{b c^3 \sqrt{d-c^2 d x^2}}{6 d^3 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^3 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b c^3 \sqrt{d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

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

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Rubi [A]  time = 0.541376, antiderivative size = 383, normalized size of antiderivative = 1.13, 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, 1799, 1620} \[ \frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b c^3 \sqrt{c x-1} \sqrt{c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d^2 x^2 \sqrt{d-c^2 d x^2}}-\frac{8 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log (x)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-c^2 x^2\right )}{3 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d^2*x^2*Sqrt[d - c^2*d*x^2]) + (b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d
^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (16*c^4*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) - (a + b*A
rcCosh[c*x])/(3*d^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (2*c^2*(a + b*ArcCosh[c*x]))/(d^2*x*(1 - c*
x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x
^2]) - (8*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (4*b*c^3*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]*Log[1 - c^2*x^2])/(3*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 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^4 \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^4 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 c^4 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)}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 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{1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6}{3 x^3 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 c^4 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)}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 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{1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6}{x^3 \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 c^4 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)}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 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{1+6 c^2 x-24 c^4 x^2+16 c^6 x^3}{x^2 \left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 c^4 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)}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 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{1}{x^2}+\frac{8 c^2}{x}-\frac{c^4}{\left (-1+c^2 x\right )^2}+\frac{8 c^4}{-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 x^2 \sqrt{d-c^2 d x^2}}+\frac{b c^3 \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{16 c^4 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)}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 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{8 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \log (x)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \log \left (1-c^2 x^2\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.397771, size = 169, normalized size = 0.5 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (\frac{2 c^2 \left (8 c^4 x^4-12 c^2 x^2+3\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{x^3 (c x-1)^{3/2} (c x+1)^{3/2}}-b c \left (\frac{1}{2 x^2 \left (c^2 x^2-1\right )}+4 c^2 \log \left (1-c^2 x^2\right )+8 c^2 \log (x)\right )\right )}{3 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/(x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (2*c^2*(3 - 12*c^
2*x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]))/(x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - b*c*(1/(2*x^2*(-1 + c^2*x^2))
+ 8*c^2*Log[x] + 4*c^2*Log[1 - c^2*x^2])))/(3*d^2*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.209, size = 1878, normalized size = 5.6 \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)^(5/2),x)

[Out]

-8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x*c^4+8/3*a*c^4/d*x/(-c^2*d*
x^2+d)^(3/2)+16/3*a*c^4/d^2*x/(-c^2*d*x^2+d)^(1/2)+16/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35
*c^4*x^4-10*c^2*x^2-1)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+128/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8
*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^9*(c*x+1)*(c*x-1)*c^12-320/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x
^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*(c*x+1)*(c*x-1)*c^10+80*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36
*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*(c*x+1)*(c*x-1)*c^8-40/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6
*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*(c*x+1)*(c*x-1)*c^6-8/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+
35*c^4*x^4-10*c^2*x^2-1)*x*(c*x+1)*(c*x-1)*c^4+2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^
4-10*c^2*x^2-1)*x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-1/6*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35
*c^4*x^4-10*c^2*x^2-1)/x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c-32/3*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^3+8/3*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))^4-1)*c^3-2*a*c^2/d/x/(-c^2*d*x^2+d)^(3/2)-1/3*a/d/x^3/(-c^2*d*x^2+d)^(3/
2)+64*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^6*arccosh(c*x)*(c*x+1)^(1
/2)*(c*x-1)^(1/2)*c^9-128*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^4*arc
cosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^7+176/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4
-10*c^2*x^2-1)*x^2*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*
c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3-64*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-3
6*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*arccosh(c*x)*c^10+160*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x
^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*arccosh(c*x)*c^8-344/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*
c^4*x^4-10*c^2*x^2-1)*x^3*arccosh(c*x)*c^6+12*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-1
0*c^2*x^2-1)*x*arccosh(c*x)*c^4+6*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)
/x*arccosh(c*x)*c^2+1/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/x^3*arcco
sh(c*x)-128/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^11*c^14+448/3*b*(
-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^9*c^12-560/3*b*(-d*(c^2*x^2-1))^(1
/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*c^10+280/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^
8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*c^8-32/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*
x^4-10*c^2*x^2-1)*x^3*c^6

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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)^(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^{10} - 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} - d^{3} 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)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 - d^3*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)**(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^{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)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x^4), x)