\(\int (c+d x^2)^3 \text {arccosh}(a x) \, dx\) [42]

Optimal result

Integrand size = 14, antiderivative size = 267 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \left (1-a^2 x^2\right )}{35 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^2}{105 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^3}{175 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x) \]

[Out]

c^3*x*arccosh(a*x)+c^2*d*x^3*arccosh(a*x)+3/5*c*d^2*x^5*arccosh(a*x)+1/7*d^3*x^7*arccosh(a*x)+1/35*(35*a^6*c^3
+35*a^4*c^2*d+21*a^2*c*d^2+5*d^3)*(-a^2*x^2+1)/a^7/(a*x-1)^(1/2)/(a*x+1)^(1/2)-1/105*d*(35*a^4*c^2+42*a^2*c*d+
15*d^2)*(-a^2*x^2+1)^2/a^7/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/175*d^2*(7*a^2*c+5*d)*(-a^2*x^2+1)^3/a^7/(a*x-1)^(1/2
)/(a*x+1)^(1/2)-1/49*d^3*(-a^2*x^2+1)^4/a^7/(a*x-1)^(1/2)/(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {200, 5908, 12, 1624, 1813, 1864} \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {3 d^2 \left (1-a^2 x^2\right )^3 \left (7 a^2 c+5 d\right )}{175 a^7 \sqrt {a x-1} \sqrt {a x+1}}-\frac {d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt {a x-1} \sqrt {a x+1}}-\frac {d \left (1-a^2 x^2\right )^2 \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\left (1-a^2 x^2\right ) \left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right )}{35 a^7 \sqrt {a x-1} \sqrt {a x+1}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x) \]

[In]

Int[(c + d*x^2)^3*ArcCosh[a*x],x]

[Out]

((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*(1 - a^2*x^2))/(35*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d*
(35*a^4*c^2 + 42*a^2*c*d + 15*d^2)*(1 - a^2*x^2)^2)/(105*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*d^2*(7*a^2*c +
5*d)*(1 - a^2*x^2)^3)/(175*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d^3*(1 - a^2*x^2)^4)/(49*a^7*Sqrt[-1 + a*x]*S
qrt[1 + a*x]) + c^3*x*ArcCosh[a*x] + c^2*d*x^3*ArcCosh[a*x] + (3*c*d^2*x^5*ArcCosh[a*x])/5 + (d^3*x^7*ArcCosh[
a*x])/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 5908

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps \begin{align*} \text {integral}& = c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)-a \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{35 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)-\frac {1}{35} a \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt {-1+a^2 x^2}} \, dx}{35 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{70 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3}{a^6 \sqrt {-1+a^2 x}}+\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \sqrt {-1+a^2 x}}{a^6}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (-1+a^2 x\right )^{3/2}}{a^6}+\frac {5 d^3 \left (-1+a^2 x\right )^{5/2}}{a^6}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \left (1-a^2 x^2\right )}{35 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^2}{105 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^3}{175 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+c^3 x \text {arccosh}(a x)+c^2 d x^3 \text {arccosh}(a x)+\frac {3}{5} c d^2 x^5 \text {arccosh}(a x)+\frac {1}{7} d^3 x^7 \text {arccosh}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.58 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (240 d^3+24 a^2 d^2 \left (49 c+5 d x^2\right )+2 a^4 d \left (1225 c^2+294 c d x^2+45 d^2 x^4\right )+a^6 \left (3675 c^3+1225 c^2 d x^2+441 c d^2 x^4+75 d^3 x^6\right )\right )}{3675 a^7}+\frac {1}{35} x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \text {arccosh}(a x) \]

[In]

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

[Out]

-1/3675*(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(240*d^3 + 24*a^2*d^2*(49*c + 5*d*x^2) + 2*a^4*d*(1225*c^2 + 294*c*d*x^2
+ 45*d^2*x^4) + a^6*(3675*c^3 + 1225*c^2*d*x^2 + 441*c*d^2*x^4 + 75*d^3*x^6)))/a^7 + (x*(35*c^3 + 35*c^2*d*x^
2 + 21*c*d^2*x^4 + 5*d^3*x^6)*ArcCosh[a*x])/35

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.63

method	result	size
parts	\(\frac {d^{3} x^{7} \operatorname {arccosh}\left (a x \right )}{7}+\frac {3 c \,d^{2} x^{5} \operatorname {arccosh}\left (a x \right )}{5}+c^{2} d \,x^{3} \operatorname {arccosh}\left (a x \right )+c^{3} x \,\operatorname {arccosh}\left (a x \right )-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{7}}\)	\(168\)
derivativedivides	\(\frac {\operatorname {arccosh}\left (a x \right ) c^{3} a x +a \,\operatorname {arccosh}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccosh}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{3} x^{7}}{7}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{6}}}{a}\)	\(176\)
default	\(\frac {\operatorname {arccosh}\left (a x \right ) c^{3} a x +a \,\operatorname {arccosh}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccosh}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccosh}\left (a x \right ) d^{3} x^{7}}{7}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 a^{6} d^{3} x^{6}+441 a^{6} c \,d^{2} x^{4}+1225 a^{6} c^{2} d \,x^{2}+90 a^{4} d^{3} x^{4}+3675 a^{6} c^{3}+588 a^{4} c \,d^{2} x^{2}+2450 a^{4} c^{2} d +120 a^{2} d^{3} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{6}}}{a}\)	\(176\)

[In]

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

[Out]

1/7*d^3*x^7*arccosh(a*x)+3/5*c*d^2*x^5*arccosh(a*x)+c^2*d*x^3*arccosh(a*x)+c^3*x*arccosh(a*x)-1/3675/a^7*(a*x-
1)^(1/2)*(a*x+1)^(1/2)*(75*a^6*d^3*x^6+441*a^6*c*d^2*x^4+1225*a^6*c^2*d*x^2+90*a^4*d^3*x^4+3675*a^6*c^3+588*a^
4*c*d^2*x^2+2450*a^4*c^2*d+120*a^2*d^3*x^2+1176*a^2*c*d^2+240*d^3)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {105 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (75 \, a^{6} d^{3} x^{6} + 3675 \, a^{6} c^{3} + 2450 \, a^{4} c^{2} d + 1176 \, a^{2} c d^{2} + 9 \, {\left (49 \, a^{6} c d^{2} + 10 \, a^{4} d^{3}\right )} x^{4} + 240 \, d^{3} + {\left (1225 \, a^{6} c^{2} d + 588 \, a^{4} c d^{2} + 120 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{3675 \, a^{7}} \]

[In]

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

[Out]

1/3675*(105*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35*a^7*c^2*d*x^3 + 35*a^7*c^3*x)*log(a*x + sqrt(a^2*x^2 - 1))
- (75*a^6*d^3*x^6 + 3675*a^6*c^3 + 2450*a^4*c^2*d + 1176*a^2*c*d^2 + 9*(49*a^6*c*d^2 + 10*a^4*d^3)*x^4 + 240*d
^3 + (1225*a^6*c^2*d + 588*a^4*c*d^2 + 120*a^2*d^3)*x^2)*sqrt(a^2*x^2 - 1))/a^7

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x**2)**3*acosh(a*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.96 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=-\frac {1}{3675} \, {\left (\frac {75 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{6}}{a^{2}} + \frac {441 \, \sqrt {a^{2} x^{2} - 1} c d^{2} x^{4}}{a^{2}} + \frac {1225 \, \sqrt {a^{2} x^{2} - 1} c^{2} d x^{2}}{a^{2}} + \frac {90 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{4}}{a^{4}} + \frac {3675 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}} + \frac {588 \, \sqrt {a^{2} x^{2} - 1} c d^{2} x^{2}}{a^{4}} + \frac {2450 \, \sqrt {a^{2} x^{2} - 1} c^{2} d}{a^{4}} + \frac {120 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{2}}{a^{6}} + \frac {1176 \, \sqrt {a^{2} x^{2} - 1} c d^{2}}{a^{6}} + \frac {240 \, \sqrt {a^{2} x^{2} - 1} d^{3}}{a^{8}}\right )} a + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right ) \]

[In]

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

[Out]

-1/3675*(75*sqrt(a^2*x^2 - 1)*d^3*x^6/a^2 + 441*sqrt(a^2*x^2 - 1)*c*d^2*x^4/a^2 + 1225*sqrt(a^2*x^2 - 1)*c^2*d
*x^2/a^2 + 90*sqrt(a^2*x^2 - 1)*d^3*x^4/a^4 + 3675*sqrt(a^2*x^2 - 1)*c^3/a^2 + 588*sqrt(a^2*x^2 - 1)*c*d^2*x^2
/a^4 + 2450*sqrt(a^2*x^2 - 1)*c^2*d/a^4 + 120*sqrt(a^2*x^2 - 1)*d^3*x^2/a^6 + 1176*sqrt(a^2*x^2 - 1)*c*d^2/a^6
+ 240*sqrt(a^2*x^2 - 1)*d^3/a^8)*a + 1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*arccosh(a*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.80 \[ \int \left (c+d x^2\right )^3 \text {arccosh}(a x) \, dx=\frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \sqrt {a^{2} x^{2} - 1}}{35 \, a^{7}} - \frac {1225 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{4} c^{2} d + 441 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} a^{2} c d^{2} + 1470 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d^{2} + 75 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {7}{2}} d^{3} + 315 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{3} + 525 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{3}}{3675 \, a^{7}} \]

[In]

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

[Out]

1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/35*(35*a^6*c^3 + 35
*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*sqrt(a^2*x^2 - 1)/a^7 - 1/3675*(1225*(a^2*x^2 - 1)^(3/2)*a^4*c^2*d + 441*(a
^2*x^2 - 1)^(5/2)*a^2*c*d^2 + 1470*(a^2*x^2 - 1)^(3/2)*a^2*c*d^2 + 75*(a^2*x^2 - 1)^(7/2)*d^3 + 315*(a^2*x^2 -
1)^(5/2)*d^3 + 525*(a^2*x^2 - 1)^(3/2)*d^3)/a^7

Mupad [F(-1)]

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

[In]

int(acosh(a*x)*(c + d*x^2)^3,x)

[Out]

int(acosh(a*x)*(c + d*x^2)^3, x)