\(\int \frac {a+b \text {arccosh}(c x)}{(d-c^2 d x^2)^{5/2}} \, dx\) [129]

Optimal result

Integrand size = 24, antiderivative size = 162 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{6 c d \left (d-c^2 d x^2\right )^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5901, 5899, 266, 74, 267} \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Rule 74

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] ||  !IntegerQ[p])))

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 267

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

Rule 5899

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Dist[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Int[x
*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ
[n, 0]

Rule 5901

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(
1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{(-1+c x)^2 (1+c x)^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{6 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {-6 a c x+4 a c^3 x^3-b \sqrt {-1+c x} \sqrt {1+c x}+2 b c x \left (-3+2 c^2 x^2\right ) \text {arccosh}(c x)-2 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )}{6 c d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(138)=276\).

Time = 1.01 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.94

method	result	size
default	\(a \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-3 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-2 \sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{6} c^{6}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-2 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+12 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c -24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +4 c^{2} x^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-2\right )}{6 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3} c}\)	\(477\)
parts	\(a \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-3 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-2 \sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{5} c^{5}-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{6} c^{6}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{3} c^{3}+24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{4} c^{4}+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-2 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+12 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x c -24 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +4 c^{2} x^{2}-8 \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-2\right )}{6 \left (3 c^{6} x^{6}-10 c^{4} x^{4}+11 c^{2} x^{2}-4\right ) d^{3} c}\)	\(477\)

[In]

int((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

a*(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)*(2*c^3*x^3-3*c*x+
2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-2*(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln((c*x+(c
*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^5*c^5-8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^6*c^6-20*(c*x-1)^(1/2)*(
c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^3*c^3+24*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^
4*c^4+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-2*c^4*x^4+6*c^2*x^2*arccosh(c*x)+12*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln
((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x*c-24*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)*x^2*c^2-3*(c*x-1)^(1/
2)*(c*x+1)^(1/2)*c*x+4*c^2*x^2-8*arccosh(c*x)+8*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)-2)/(3*c^6*x^6-10*c^4
*x^4+11*c^2*x^2-4)/d^3/c

Fricas [F]

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

[In]

integrate((a+b*arccosh(c*x))/(-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^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {\sqrt {-d}}{c^{4} d^{3} x^{2} - c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {2 \, \sqrt {-d} \log \left (c x - 1\right )}{c^{2} d^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d}\right )} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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