3.2.33 \(\int \frac {a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\) [133]

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

[Out]

1/3*(-a-b*arccosh(c*x))/d/x^3/(-c^2*d*x^2+d)^(3/2)-2*c^2*(a+b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(3/2)+8/3*c^4*x
*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(3/2)+16/3*c^4*x*(a+b*arccosh(c*x))/d^2/(-c^2*d*x^2+d)^(1/2)-1/6*b*c*(-c^
2*d*x^2+d)^(1/2)/d^3/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/6*b*c^3*(-c^2*d*x^2+d)^(1/2)/d^3/(-c^2*x^2+1)/(c*x-1)^(
1/2)/(c*x+1)^(1/2)+8/3*b*c^3*ln(x)*(-c^2*d*x^2+d)^(1/2)/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+4/3*b*c^3*ln(-c^2*x^2+
1)*(-c^2*d*x^2+d)^(1/2)/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {277, 198, 197, 5922, 12, 1813, 1634} \begin {gather*} -\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 {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 {b c \sqrt {d-c^2 d x^2}}{6 d^3 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\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 {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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*c*Sqrt[d - c^2*d*x^2])/(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*Sqr
t[-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)) + (1
6*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])

Rule 12

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 1634

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]

Rule 1813

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 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.45, size = 218, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (b c x \sqrt {-1+c x} \sqrt {1+c x}-2 a \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )-2 b \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \cosh ^{-1}(c x)\right )}{6 d^3 x^3 \left (-1+c^2 x^2\right )^2}+\frac {8 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \left (\log (-1+c x)+\log (1+c x)+2 \left (\log \left (-1+\sqrt {1+c x}\right )+\log \left (1+\sqrt {1+c x}\right )\right )\right )}{3 d^3 \left (-2+2 c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*a*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6) - 2*b
*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*ArcCosh[c*x]))/(6*d^3*x^3*(-1 + c^2*x^2)^2) + (8*b*c^3*Sqrt[-1 + c*
x]*Sqrt[1 + c*x]*Sqrt[d - c^2*d*x^2]*(Log[-1 + c*x] + Log[1 + c*x] + 2*(Log[-1 + Sqrt[1 + c*x]] + Log[1 + Sqrt
[1 + c*x]])))/(3*d^3*(-2 + 2*c^2*x^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1879\) vs. \(2(295)=590\).
time = 3.47, size = 1880, normalized size = 5.56

method	result	size
default	\(\text {Expression too large to display}\)	\(1880\)

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,method=_RETURNVERBOSE)

[Out]

a*(-1/3/d/x^3/(-c^2*d*x^2+d)^(3/2)+2*c^2*(-1/d/x/(-c^2*d*x^2+d)^(3/2)+4*c^2*(1/3*x/d/(-c^2*d*x^2+d)^(3/2)+2/3/
d^2*x/(-c^2*d*x^2+d)^(1/2))))-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^10-128*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*(c*x+1)^(1/2)*(c*x-1)^(1/2)*arccosh(c*x)*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*(c*x+1)^(1/2)*(c*x-1)^(1/2)*arccosh(c*x)*c^5+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*(c*x+1)^(1/2)*(c*x-1)^(1/2)*arccosh(c*x)*c^9+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^4-10*c^2*x^2-1)*x*arccosh(c*x)*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)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+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^2-64*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^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+3
5*c^4*x^4-10*c^2*x^2-1)*x^3*arccosh(c*x)*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+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)
*(c*x+1)^(1/2)*(c*x-1)^(1/2)*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+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+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^6-128/3*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^6-8/3*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+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*arccosh(c*x)

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Maxima [A]
time = 0.28, size = 276, normalized size = 0.82 \begin {gather*} \frac {1}{6} \, b c {\left (\frac {8 \, c^{2} \sqrt {-d} \log \left (c x + 1\right )}{d^{3}} + \frac {8 \, c^{2} \sqrt {-d} \log \left (c x - 1\right )}{d^{3}} + \frac {16 \, c^{2} \sqrt {-d} \log \left (x\right )}{d^{3}} + \frac {\sqrt {-d}}{c^{2} d^{3} x^{4} - d^{3} x^{2}}\right )} + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} b \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a \end {gather*}

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]

1/6*b*c*(8*c^2*sqrt(-d)*log(c*x + 1)/d^3 + 8*c^2*sqrt(-d)*log(c*x - 1)/d^3 + 16*c^2*sqrt(-d)*log(x)/d^3 + sqrt
(-d)/(c^2*d^3*x^4 - d^3*x^2)) + 1/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d)
- 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*b*arccosh(c*x) + 1/3*(16*c^4*x/(sqrt(
-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^
2 + d)^(3/2)*d*x^3))*a

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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+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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

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]

Integral((a + b*acosh(c*x))/(x**4*(-d*(c*x - 1)*(c*x + 1))**(5/2)), 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+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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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