\(\int (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [73]

Optimal result

Integrand size = 24, antiderivative size = 200 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {5 b c d x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/4*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+3/8*d*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-5/16*b*c*d*x^2*(
-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*c^3*d*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1
/2)-3/16*d*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5897, 5895, 5893, 30, 74, 14} \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c d x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(-5*b*c*d*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^4*Sqrt[d - c^2*d*x^2])/(16*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (x*(d - c^2*d*x^2)^(3/2)*(
a + b*ArcCosh[c*x]))/4 - (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]
)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(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]

Rule 5897

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{4} (3 d) \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x) (1+c x) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 b c d x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {5 b c d x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.18 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{8} a d x \left (-5+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {3 a d^{3/2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{8 c}-\frac {b d \sqrt {d-c^2 d x^2} \left (2 \text {arccosh}(c x)^2+\cosh (2 \text {arccosh}(c x))-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{8 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b d \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{128 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

-1/8*(a*d*x*(-5 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (3*a*d^(3/2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1
+ c^2*x^2))])/(8*c) - (b*d*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh
[2*ArcCosh[c*x]]))/(8*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + C
osh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(128*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs. \(2(168)=336\).

Time = 0.92 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.73

method	result	size
default	\(\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(546\)
parts	\(\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x -1\right ) \left (c x +1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16 \left (c x -1\right ) \left (c x +1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right ) d}{256 \left (c x -1\right ) \left (c x +1\right ) c}\right )\)	\(546\)

[In]

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

[Out]

1/4*a*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c
^2*d*x^2+d)^(1/2))+b*(-3/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*d-1/256*(-d*(c
^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/
2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+4*arccosh(c*x))*d/(c*x-1)/(c*x+1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(2
*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+2*arccosh(c*x))*d/(c*x-1
)/(c*x+1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1
)^(1/2)-2*c*x)*(1+2*arccosh(c*x))*d/(c*x-1)/(c*x+1)/c-1/256*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(
1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(1+
4*arccosh(c*x))*d/(c*x-1)/(c*x+1)/c)

Fricas [F]

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

[In]

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

[Out]

integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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