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

Optimal result

Integrand size = 27, antiderivative size = 248 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \]

[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)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {277, 198, 197, 5922, 12, 1265, 907} \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{6 d^3 \sqrt {c x-1} \sqrt {c x+1}} \]

[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*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-3+12 c^2 x^2-8 c^4 x^4}{3 d^3 x \left (1-c^2 x^2\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\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^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\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^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\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^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\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 \text {arccosh}(c x)}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\frac {a+b \text {arccosh}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {4 c^2 x (a+b \text {arccosh}(c x))}{3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {8 c^2 x (a+b \text {arccosh}(c x))}{3 \sqrt {-1+c x} \sqrt {1+c x}}-b c \left (\frac {1}{6 \left (-1+c^2 x^2\right )}+\log (x)+\frac {5}{6} \log \left (1-c^2 x^2\right )\right )\right )}{d^2 \sqrt {d-c^2 d x^2}} \]

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.75

method	result	size
default	\(a \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+12 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+20 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}-c^{3} x^{3}+6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+16 c x \,\operatorname {arccosh}\left (c x \right )-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c +c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x}\)	\(434\)
parts	\(a \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+12 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+20 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}-c^{3} x^{3}+6 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+16 c x \,\operatorname {arccosh}\left (c x \right )-6 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -10 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c +c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x}\)	\(434\)

[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/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(-c^2*d*x^2+d)^(1/2)))-1/6*b*(-d*
(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(16*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4+16*arccosh
(c*x)*c^5*x^5-6*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^5*c^5-10*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)
*x^5*c^5-24*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-32*c^3*x^3*arccosh(c*x)+12*ln(1+(c*x+(c*x-1)^(1/2
)*(c*x+1)^(1/2))^2)*x^3*c^3+20*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c^3-c^3*x^3+6*arccosh(c*x)*(c*x-1
)^(1/2)*(c*x+1)^(1/2)+16*c*x*arccosh(c*x)-6*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x*c-10*ln((c*x+(c*x-1)^(
1/2)*(c*x+1)^(1/2))^2-1)*x*c+c*x)/d^3/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/x

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[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
)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[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)