3.2.31 \(\int \frac {a+b \cosh ^{-1}(c x)}{x^2 (d-c^2 d x^2)^{5/2}} \, dx\) [131]

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

[Out]

(-a-b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(3/2)+4/3*c^2*x*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(3/2)+8/3*c^2*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/(-c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(
1/2)+b*c*ln(x)*(-c^2*d*x^2+d)^(1/2)/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/6*b*c*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 = 0.14, antiderivative size = 248, 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, 1265, 907} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(b*c*Sqrt[d - c^2*d*x^2])/(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])

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 907

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]))

Rule 1265

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 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^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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.27, size = 147, normalized size = 0.59 \begin {gather*} \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\frac {a+b \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {4 c^2 x \left (-3+2 c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {1}{6} b c \left (\frac {1}{-1+c^2 x^2}+6 \log (x)+5 \log \left (1-c^2 x^2\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(1349\) vs. \(2(218)=436\).
time = 2.95, size = 1350, normalized size = 5.44

method	result	size
default	\(\text {Expression too large to display}\)	\(1350\)

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,method=_RETURNVERBOSE)

[Out]

a*(-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)))-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(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c

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

[Out]

1/3*a*(8*c^2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 4*c^2*x/((-c^2*d*x^2 + d)^(3/2)*d) - 3/((-c^2*d*x^2 + d)^(3/2)*d*x
)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x^2), 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+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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2} \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**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral((a + b*acosh(c*x))/(x**2*(-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^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)

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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^2\,{\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^2*(d - c^2*d*x^2)^(5/2)),x)

[Out]

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

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