\(\int x (d+e x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [473]

Optimal result

Integrand size = 19, antiderivative size = 269 \[ \int x \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^6 e \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/6*(e*x^2+d)^3*(a+b*arccosh(c*x))/e+1/288*b*(44*c^4*d^2+44*c^2*d*e+15*e^2)*x*(-c^2*x^2+1)/c^5/(c*x-1)^(1/2)/(
c*x+1)^(1/2)+5/144*b*(2*c^2*d+e)*x*(-c^2*x^2+1)*(e*x^2+d)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/36*b*x*(-c^2*x^2+1
)*(e*x^2+d)^2/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/96*b*(2*c^2*d+e)*(8*c^4*d^2+8*c^2*d*e+5*e^2)*arctanh(c*x/(c^2*x^
2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^6/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5957, 916, 427, 542, 396, 223, 212} \[ \int x \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {b \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right )}{96 c^6 e \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(b*(44*c^4*d^2 + 44*c^2*d*e + 15*e^2)*x*(1 - c^2*x^2))/(288*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*(2*c^2*d
+ e)*x*(1 - c^2*x^2)*(d + e*x^2))/(144*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^2)/(
36*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/(6*e) - (b*(2*c^2*d + e)*(8*c^4*d^2
+ 8*c^2*d*e + 5*e^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(96*c^6*e*Sqrt[-1 + c*x]*Sqrt[1 + c
*x])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 396

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

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 542

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

Rule 916

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_.) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d + e*x)
^FracPart[m]*((f + g*x)^FracPart[m]/(d*f + e*g*x^2)^FracPart[m]), Int[(d*f + e*g*x^2)^m*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 0]

Rule 5957

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 e} \\ & = \frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^3}{\sqrt {-1+c^2 x^2}} \, dx}{6 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right ) \left (d \left (6 c^2 d+e\right )+5 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{36 c e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {d \left (24 c^4 d^2+14 c^2 d e+5 e^2\right )+e \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{144 c^3 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {\left (b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{96 c^5 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {\left (b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^5 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 (a+b \text {arccosh}(c x))}{6 e}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^6 e \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.68 \[ \int x \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {c x \left (48 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (15 e^2+2 c^2 e \left (27 d+5 e x^2\right )+4 c^4 \left (18 d^2+9 d e x^2+2 e^2 x^4\right )\right )\right )+48 b c^6 x^2 \left (3 d^2+3 d e x^2+e^2 x^4\right ) \text {arccosh}(c x)-6 b \left (24 c^4 d^2+18 c^2 d e+5 e^2\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{288 c^6} \]

[In]

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

[Out]

(c*x*(48*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15*e^2 + 2*c^2*e*(27*d + 5*e*
x^2) + 4*c^4*(18*d^2 + 9*d*e*x^2 + 2*e^2*x^4))) + 48*b*c^6*x^2*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCosh[c*x] - 6*
b*(24*c^4*d^2 + 18*c^2*d*e + 5*e^2)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(288*c^6)

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.26

method	result	size
parts	\(\frac {a \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b \left (\frac {c^{2} e^{2} \operatorname {arccosh}\left (c x \right ) x^{6}}{6}+\frac {c^{2} e \,\operatorname {arccosh}\left (c x \right ) x^{4} d}{2}+\frac {\operatorname {arccosh}\left (c x \right ) c^{2} x^{2} d^{2}}{2}+\frac {c^{2} \operatorname {arccosh}\left (c x \right ) d^{3}}{6 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (48 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+72 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+36 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+8 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+72 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+54 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+10 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+54 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+15 e^{3} c x \sqrt {c^{2} x^{2}-1}+15 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 c^{4} e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(338\)
derivativedivides	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {e \,\operatorname {arccosh}\left (c x \right ) c^{6} d \,x^{4}}{2}+\frac {e^{2} \operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (48 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+72 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+36 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+8 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+72 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+54 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+10 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+54 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+15 e^{3} c x \sqrt {c^{2} x^{2}-1}+15 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}}{c^{2}}\)	\(349\)
default	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {e \,\operatorname {arccosh}\left (c x \right ) c^{6} d \,x^{4}}{2}+\frac {e^{2} \operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{6}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (48 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+72 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+36 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+8 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+72 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+54 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+10 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+54 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+15 e^{3} c x \sqrt {c^{2} x^{2}-1}+15 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{288 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}}{c^{2}}\)	\(349\)

[In]

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

[Out]

1/6*a*(e*x^2+d)^3/e+b/c^2*(1/6*c^2*e^2*arccosh(c*x)*x^6+1/2*c^2*e*arccosh(c*x)*x^4*d+1/2*arccosh(c*x)*c^2*x^2*
d^2+1/6*c^2/e*arccosh(c*x)*d^3-1/288/c^4/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(48*c^6*d^3*ln(c*x+(c^2*x^2-1)^(1/2))+7
2*c^5*d^2*e*x*(c^2*x^2-1)^(1/2)+36*c^5*d*e^2*(c^2*x^2-1)^(1/2)*x^3+8*e^3*(c^2*x^2-1)^(1/2)*c^5*x^5+72*c^4*d^2*
e*ln(c*x+(c^2*x^2-1)^(1/2))+54*c^3*d*e^2*x*(c^2*x^2-1)^(1/2)+10*e^3*c^3*x^3*(c^2*x^2-1)^(1/2)+54*c^2*d*e^2*ln(
c*x+(c^2*x^2-1)^(1/2))+15*e^3*c*x*(c^2*x^2-1)^(1/2)+15*e^3*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.72 \[ \int x \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {48 \, a c^{6} e^{2} x^{6} + 144 \, a c^{6} d e x^{4} + 144 \, a c^{6} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} e^{2} x^{6} + 48 \, b c^{6} d e x^{4} + 48 \, b c^{6} d^{2} x^{2} - 24 \, b c^{4} d^{2} - 18 \, b c^{2} d e - 5 \, b e^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (8 \, b c^{5} e^{2} x^{5} + 2 \, {\left (18 \, b c^{5} d e + 5 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (24 \, b c^{5} d^{2} + 18 \, b c^{3} d e + 5 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{6}} \]

[In]

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

[Out]

1/288*(48*a*c^6*e^2*x^6 + 144*a*c^6*d*e*x^4 + 144*a*c^6*d^2*x^2 + 3*(16*b*c^6*e^2*x^6 + 48*b*c^6*d*e*x^4 + 48*
b*c^6*d^2*x^2 - 24*b*c^4*d^2 - 18*b*c^2*d*e - 5*b*e^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5*e^2*x^5 + 2*(18
*b*c^5*d*e + 5*b*c^3*e^2)*x^3 + 3*(24*b*c^5*d^2 + 18*b*c^3*d*e + 5*b*c*e^2)*x)*sqrt(c^2*x^2 - 1))/c^6

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.01 \[ \int x \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d e + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b e^{2} \]

[In]

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

[Out]

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/4*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c
^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*b*d^2 + 1/16*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c
^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*d*e + 1/288*(48*x^6*arccosh(c*x) - (8*sqr
t(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c
^2*x^2 - 1)*c)/c^7)*c)*b*e^2

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int(x*(a + b*acosh(c*x))*(d + e*x^2)^2,x)

[Out]

int(x*(a + b*acosh(c*x))*(d + e*x^2)^2, x)