\(\int \frac {a+b \text {arccosh}(c x)}{x^2 (d-c^2 d x^2)^{3/2}} \, dx\) [121]

Optimal result

Integrand size = 27, antiderivative size = 158 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

(-a-b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)+b*c*ln(x)*(-c^2
*d*x^2+d)^(1/2)/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*b*c*ln(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/d^2/(c*x-1)^(1/2)/
(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {277, 197, 5922, 12, 457, 78} \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

-((a + b*ArcCosh[c*x])/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d - c^2*d*x^2]) + (
b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2]
)/(2*d^2*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 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 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 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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 \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+2 c^2 x^2}{d^2 x \left (1-c^2 x^2\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+2 c^2 x^2}{x \left (1-c^2 x^2\right )} \, dx}{d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-1+2 c^2 x}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \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 {1}{x}-\frac {c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 a+4 a c^2 x^2+2 b \left (-1+2 c^2 x^2\right ) \text {arccosh}(c x)-2 b c x \sqrt {-1+c x} \sqrt {1+c x} \log (x)-b c x \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 d x \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.68

method	result	size
default	\(a \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{4} c^{4}+\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x c -2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{2} c^{2}+\operatorname {arccosh}\left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{d^{2} \left (c^{2} x^{2}-1\right ) x}\)	\(266\)
parts	\(a \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{4} c^{4}+\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x c -2 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{2} c^{2}+\operatorname {arccosh}\left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{d^{2} \left (c^{2} x^{2}-1\right ) x}\)	\(266\)

[In]

int((a+b*arccosh(c*x))/x^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2/d*x/(-c^2*d*x^2+d)^(1/2))-b*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c*x-
1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^3*c^3+2*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^4*c^4+(c*x-1)^(1/2)*(c*x+1)
^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x*c-2*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^2*c^2+arcco
sh(c*x))*(2*c^2*x^2-1+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)/x

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^2/(-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^6 - 2*c^2*d^2*x^4 + d^2*x^2), x)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*acosh(c*x))/x**2/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))/(x**2*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {1}{2} \, b c {\left (\frac {\sqrt {-d} \log \left (c x + 1\right )}{d^{2}} + \frac {\sqrt {-d} \log \left (c x - 1\right )}{d^{2}} + \frac {2 \, \sqrt {-d} \log \left (x\right )}{d^{2}}\right )} + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} b \operatorname {arcosh}\left (c x\right ) + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} a \]

[In]

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

[Out]

1/2*b*c*(sqrt(-d)*log(c*x + 1)/d^2 + sqrt(-d)*log(c*x - 1)/d^2 + 2*sqrt(-d)*log(x)/d^2) + (2*c^2*x/(sqrt(-c^2*
d*x^2 + d)*d) - 1/(sqrt(-c^2*d*x^2 + d)*d*x))*b*arccosh(c*x) + (2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^
2*d*x^2 + d)*d*x))*a

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

int((a + b*acosh(c*x))/(x^2*(d - c^2*d*x^2)^(3/2)),x)

[Out]

int((a + b*acosh(c*x))/(x^2*(d - c^2*d*x^2)^(3/2)), x)