\(\int x (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [482]

Optimal result

Integrand size = 19, antiderivative size = 358 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1024 c^8 e \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/8*(e*x^2+d)^4*(a+b*arccosh(c*x))/e+5/3072*b*(2*c^2*d+e)*(40*c^4*d^2+40*c^2*d*e+21*e^2)*x*(-c^2*x^2+1)/c^7/(c
*x-1)^(1/2)/(c*x+1)^(1/2)+1/1536*b*(104*c^4*d^2+104*c^2*d*e+35*e^2)*x*(-c^2*x^2+1)*(e*x^2+d)/c^5/(c*x-1)^(1/2)
/(c*x+1)^(1/2)+7/384*b*(2*c^2*d+e)*x*(-c^2*x^2+1)*(e*x^2+d)^2/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/64*b*x*(-c^2*x
^2+1)*(e*x^2+d)^3/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/1024*b*(128*c^8*d^4+256*c^6*d^3*e+288*c^4*d^2*e^2+160*c^2*d*
e^3+35*e^4)*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^8/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5957, 916, 427, 542, 396, 223, 212} \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right )}{1024 c^8 e \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(5*b*(2*c^2*d + e)*(40*c^4*d^2 + 40*c^2*d*e + 21*e^2)*x*(1 - c^2*x^2))/(3072*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
+ (b*(104*c^4*d^2 + 104*c^2*d*e + 35*e^2)*x*(1 - c^2*x^2)*(d + e*x^2))/(1536*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]
) + (7*b*(2*c^2*d + e)*x*(1 - c^2*x^2)*(d + e*x^2)^2)/(384*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x
^2)*(d + e*x^2)^3)/(64*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e) - (b*(128*
c^8*d^4 + 256*c^6*d^3*e + 288*c^4*d^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 +
c^2*x^2]])/(1024*c^8*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 )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 e} \\ & = \frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^4}{\sqrt {-1+c^2 x^2}} \, dx}{8 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 )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^2 \left (d \left (8 c^2 d+e\right )+7 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{64 c e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 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 )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right ) \left (d \left (48 c^4 d^2+20 c^2 d e+7 e^2\right )+e \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{384 c^3 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 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 )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )+5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{1536 c^5 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 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 )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}--\frac {\left (b \left (-5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )-2 c^2 d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{3072 c^7 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 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 )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}--\frac {\left (b \left (-5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )-2 c^2 d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )\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 )}{3072 c^7 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{384 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 )^3}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1024 c^8 e \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.28 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.72 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {c x \left (384 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (105 e^3+10 c^2 e^2 \left (48 d+7 e x^2\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+16 c^6 \left (48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right )\right )\right )+384 b c^8 x^2 \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right ) \text {arccosh}(c x)-6 b \left (256 c^6 d^3+288 c^4 d^2 e+160 c^2 d e^2+35 e^3\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{3072 c^8} \]

[In]

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

[Out]

(c*x*(384*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(105*e^3 + 10
*c^2*e^2*(48*d + 7*e*x^2) + 8*c^4*e*(108*d^2 + 40*d*e*x^2 + 7*e^2*x^4) + 16*c^6*(48*d^3 + 36*d^2*e*x^2 + 16*d*
e^2*x^4 + 3*e^3*x^6))) + 384*b*c^8*x^2*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6)*ArcCosh[c*x] - 6*b*(256*c
^6*d^3 + 288*c^4*d^2*e + 160*c^2*d*e^2 + 35*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(3072*c^8)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.33

method	result	size
parts	\(\frac {a \left (e \,x^{2}+d \right )^{4}}{8 e}+\frac {b \left (\frac {c^{2} e^{3} \operatorname {arccosh}\left (c x \right ) x^{8}}{8}+\frac {c^{2} e^{2} \operatorname {arccosh}\left (c x \right ) x^{6} d}{2}+\frac {3 c^{2} e \,\operatorname {arccosh}\left (c x \right ) x^{4} d^{2}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{2} x^{2} d^{3}}{2}+\frac {c^{2} \operatorname {arccosh}\left (c x \right ) d^{4}}{8 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (384 c^{8} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+768 c^{7} d^{3} e x \sqrt {c^{2} x^{2}-1}+576 c^{7} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+256 c^{7} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{5}+48 e^{4} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+768 c^{6} d^{3} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+864 c^{5} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+320 \sqrt {c^{2} x^{2}-1}\, c^{5} d \,e^{3} x^{3}+56 e^{4} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 c^{4} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+480 c^{3} d \,e^{3} x \sqrt {c^{2} x^{2}-1}+70 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+480 c^{2} d \,e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+105 e^{4} c x \sqrt {c^{2} x^{2}-1}+105 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 c^{6} e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(476\)
derivativedivides	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \operatorname {arccosh}\left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (384 c^{8} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+768 c^{7} d^{3} e x \sqrt {c^{2} x^{2}-1}+576 c^{7} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+256 c^{7} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{5}+48 e^{4} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+768 c^{6} d^{3} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+864 c^{5} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+320 \sqrt {c^{2} x^{2}-1}\, c^{5} d \,e^{3} x^{3}+56 e^{4} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 c^{4} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+480 c^{3} d \,e^{3} x \sqrt {c^{2} x^{2}-1}+70 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+480 c^{2} d \,e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+105 e^{4} c x \sqrt {c^{2} x^{2}-1}+105 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{2}}\)	\(487\)
default	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \operatorname {arccosh}\left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (384 c^{8} d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+768 c^{7} d^{3} e x \sqrt {c^{2} x^{2}-1}+576 c^{7} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+256 c^{7} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{5}+48 e^{4} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+768 c^{6} d^{3} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+864 c^{5} d^{2} e^{2} x \sqrt {c^{2} x^{2}-1}+320 \sqrt {c^{2} x^{2}-1}\, c^{5} d \,e^{3} x^{3}+56 e^{4} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 c^{4} d^{2} e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+480 c^{3} d \,e^{3} x \sqrt {c^{2} x^{2}-1}+70 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+480 c^{2} d \,e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+105 e^{4} c x \sqrt {c^{2} x^{2}-1}+105 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{2}}\)	\(487\)

[In]

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

[Out]

1/8*a*(e*x^2+d)^4/e+b/c^2*(1/8*c^2*e^3*arccosh(c*x)*x^8+1/2*c^2*e^2*arccosh(c*x)*x^6*d+3/4*c^2*e*arccosh(c*x)*
x^4*d^2+1/2*arccosh(c*x)*c^2*x^2*d^3+1/8*c^2/e*arccosh(c*x)*d^4-1/3072/c^6/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(384*
c^8*d^4*ln(c*x+(c^2*x^2-1)^(1/2))+768*c^7*d^3*e*x*(c^2*x^2-1)^(1/2)+576*c^7*d^2*e^2*(c^2*x^2-1)^(1/2)*x^3+256*
c^7*d*e^3*(c^2*x^2-1)^(1/2)*x^5+48*e^4*(c^2*x^2-1)^(1/2)*c^7*x^7+768*c^6*d^3*e*ln(c*x+(c^2*x^2-1)^(1/2))+864*c
^5*d^2*e^2*x*(c^2*x^2-1)^(1/2)+320*(c^2*x^2-1)^(1/2)*c^5*d*e^3*x^3+56*e^4*c^5*x^5*(c^2*x^2-1)^(1/2)+864*c^4*d^
2*e^2*ln(c*x+(c^2*x^2-1)^(1/2))+480*c^3*d*e^3*x*(c^2*x^2-1)^(1/2)+70*e^4*c^3*x^3*(c^2*x^2-1)^(1/2)+480*c^2*d*e
^3*ln(c*x+(c^2*x^2-1)^(1/2))+105*e^4*c*x*(c^2*x^2-1)^(1/2)+105*e^4*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 = 286, normalized size of antiderivative = 0.80 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {384 \, a c^{8} e^{3} x^{8} + 1536 \, a c^{8} d e^{2} x^{6} + 2304 \, a c^{8} d^{2} e x^{4} + 1536 \, a c^{8} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} e^{3} x^{8} + 512 \, b c^{8} d e^{2} x^{6} + 768 \, b c^{8} d^{2} e x^{4} + 512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} - 288 \, b c^{4} d^{2} e - 160 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} e^{3} x^{7} + 8 \, {\left (32 \, b c^{7} d e^{2} + 7 \, b c^{5} e^{3}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} e + 160 \, b c^{5} d e^{2} + 35 \, b c^{3} e^{3}\right )} x^{3} + 3 \, {\left (256 \, b c^{7} d^{3} + 288 \, b c^{5} d^{2} e + 160 \, b c^{3} d e^{2} + 35 \, b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{3072 \, c^{8}} \]

[In]

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

[Out]

1/3072*(384*a*c^8*e^3*x^8 + 1536*a*c^8*d*e^2*x^6 + 2304*a*c^8*d^2*e*x^4 + 1536*a*c^8*d^3*x^2 + 3*(128*b*c^8*e^
3*x^8 + 512*b*c^8*d*e^2*x^6 + 768*b*c^8*d^2*e*x^4 + 512*b*c^8*d^3*x^2 - 256*b*c^6*d^3 - 288*b*c^4*d^2*e - 160*
b*c^2*d*e^2 - 35*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (48*b*c^7*e^3*x^7 + 8*(32*b*c^7*d*e^2 + 7*b*c^5*e^3)*x^
5 + 2*(288*b*c^7*d^2*e + 160*b*c^5*d*e^2 + 35*b*c^3*e^3)*x^3 + 3*(256*b*c^7*d^3 + 288*b*c^5*d^2*e + 160*b*c^3*
d*e^2 + 35*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^8

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.14 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} e + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d e^{2} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b e^{3} \]

[In]

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

[Out]

1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/2*a*d^3*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^3 + 3/32*(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^2*e + 1/96*(48*x^6*
arccosh(c*x) - (8*sqrt(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*l
og(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*d*e^2 + 1/3072*(384*x^8*arccosh(c*x) - (48*sqrt(c^2*x^2 - 1)*x^7
/c^2 + 56*sqrt(c^2*x^2 - 1)*x^5/c^4 + 70*sqrt(c^2*x^2 - 1)*x^3/c^6 + 105*sqrt(c^2*x^2 - 1)*x/c^8 + 105*log(2*c
^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^9)*c)*b*e^3

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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