\(\int \frac {a+b \text {arccosh}(c x)}{x^4 (d-c^2 d x^2)^{3/2}} \, dx\) [123]

Optimal result

Integrand size = 27, antiderivative size = 250 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/3*(-a-b*arccosh(c*x))/d/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*c^2*(a+b*arccosh(c*x))/d/x/(-c^2*d*x^2+d)^(1/2)+8/3*c^4
*x*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)-1/6*b*c*(-c^2*d*x^2+d)^(1/2)/d^2/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+
5/3*b*c^3*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^3*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 = 0.15 (sec) , antiderivative size = 250, 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, 1265, 907} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

-1/6*(b*c*Sqrt[d - c^2*d*x^2])/(d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*d*x^3*Sqrt[d -
c^2*d*x^2]) - (4*c^2*(a + b*ArcCosh[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3*d*
Sqrt[d - c^2*d*x^2]) + (5*b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*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 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 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)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-1-4 c^2 x^2+8 c^4 x^4}{3 d^2 x^3 \left (1-c^2 x^2\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-1-4 c^2 x^2+8 c^4 x^4}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 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 {-1-4 c^2 x+8 c^4 x^2}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{6 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 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^2}-\frac {5 c^2}{x}-\frac {3 c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{6 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {arccosh}(c x))}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \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 = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.64 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 a-8 a c^2 x^2+16 a c^4 x^4+b c x \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (-1-4 c^2 x^2+8 c^4 x^4\right ) \text {arccosh}(c x)-10 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)-3 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{6 d x^3 \sqrt {d-c^2 d x^2}} \]

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]

[Out]

(-2*a - 8*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(-1 - 4*c^2*x^2 + 8*c^4*x^4)*Arc
Cosh[c*x] - 10*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] - 3*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1
- c^2*x^2])/(6*d*x^3*Sqrt[d - c^2*d*x^2])

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.47

method	result	size
default	\(a \left (-\frac {1}{3 d \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 c^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )}{3}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-16 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+c^{3} x^{3}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-c x \right )}{6 d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{3}}\)	\(367\)
parts	\(a \left (-\frac {1}{3 d \,x^{3} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 c^{2} \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )}{3}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (16 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}+16 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}-6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{5} c^{5}-8 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-16 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+6 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+10 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x^{3} c^{3}+c^{3} x^{3}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-c x \right )}{6 d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{3}}\)	\(367\)

[In]

int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a*(-1/3/d/x^3/(-c^2*d*x^2+d)^(1/2)+4/3*c^2*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2/d*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*ar
ccosh(c*x)*c^5*x^5-6*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-10*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2
))^2)*x^5*c^5-8*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-16*c^3*x^3*arccosh(c*x)+6*ln((c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))^2-1)*x^3*c^3+10*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*x^3*c^3+c^3*x^3-2*arccosh(c*x)*(c*
x-1)^(1/2)*(c*x+1)^(1/2)-c*x)/d^2/(c^4*x^4-2*c^2*x^2+1)/x^3

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(-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^8 - 2*c^2*d^2*x^6 + d^2*x^4), x)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))/(x**4*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

1/3*(8*c^4*x/(sqrt(-c^2*d*x^2 + d)*d) - 4*c^2/(sqrt(-c^2*d*x^2 + d)*d*x) - 1/(sqrt(-c^2*d*x^2 + d)*d*x^3))*a +
b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(3/2)*x^4), x)

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]

[In]

int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^(3/2)),x)

[Out]

int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^(3/2)), x)