3.5.73 \(\int x (d+e x^2)^2 (a+b \cosh ^{-1}(c x)) \, dx\) [473]

Optimal. Leaf size=269 \[ \frac {b \left (44 c^4 d^2+44 c^2 d e+15 e^2\right ) x \left (1-c^2 x^2\right )}{288 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b \left (2 c^2 d+e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{144 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}-\frac {b \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{96 c^6 e \sqrt {-1+c x} \sqrt {1+c x}} \]

[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)

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Rubi [A]
time = 0.17, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5957, 916, 427, 542, 396, 223, 212} \begin {gather*} \frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}+\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 \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{96 c^6 e \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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int x \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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} \tanh ^{-1}\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*}

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Mathematica [A]
time = 0.17, size = 189, normalized size = 0.70 \begin {gather*} \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 ) \cosh ^{-1}(c x)-3 b \left (24 c^4 d^2+18 c^2 d e+5 e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{288 c^6} \end {gather*}

Antiderivative was successfully verified.

[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] - 3*
b*(24*c^4*d^2 + 18*c^2*d*e + 5*e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(288*c^6)

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Maple [A]
time = 2.80, size = 435, normalized size = 1.62

method	result	size
derivativedivides	\(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{3} a}{6 c^{4} e}+\frac {b \,c^{2} \mathrm {arccosh}\left (c x \right ) d^{3}}{6 e}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\mathrm {arccosh}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \mathrm {arccosh}\left (c x \right ) x^{6}}{6}-\frac {b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 e \sqrt {c^{2} x^{2}-1}}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b c e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{3}}{8}-\frac {b c \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{36}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2}}{4 \sqrt {c^{2} x^{2}-1}}-\frac {3 b d e x \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {5 b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c}-\frac {3 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{16 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {5 b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{3}}-\frac {5 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{4} \sqrt {c^{2} x^{2}-1}}}{c^{2}}\)	\(435\)
default	\(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{3} a}{6 c^{4} e}+\frac {b \,c^{2} \mathrm {arccosh}\left (c x \right ) d^{3}}{6 e}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\mathrm {arccosh}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \mathrm {arccosh}\left (c x \right ) x^{6}}{6}-\frac {b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 e \sqrt {c^{2} x^{2}-1}}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b c e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{3}}{8}-\frac {b c \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{36}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2}}{4 \sqrt {c^{2} x^{2}-1}}-\frac {3 b d e x \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {5 b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{144 c}-\frac {3 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{16 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {5 b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{96 c^{3}}-\frac {5 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{96 c^{4} \sqrt {c^{2} x^{2}-1}}}{c^{2}}\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(1/6*(c^2*e*x^2+c^2*d)^3*a/c^4/e+1/6*b*c^2/e*arccosh(c*x)*d^3+1/2*d^2*b*arccosh(c*x)*c^2*x^2+1/2*b*c^2*e
*arccosh(c*x)*d*x^4+1/6*b*c^2*e^2*arccosh(c*x)*x^6-1/6*b*c^2/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d
^3*ln(c*x+(c^2*x^2-1)^(1/2))-1/4*b*c*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/8*b*c*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d
*x^3-1/36*b*c*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^5-1/4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+(
c^2*x^2-1)^(1/2))*d^2-3/16*b*d*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-5/144*b*e^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
-3/16*b/c^2*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2))*d-5/96*b*e^2*x*(c*x-1)^(
1/2)*(c*x+1)^(1/2)/c^3-5/96*b/c^4*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2)))

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Maxima [A]
time = 0.26, size = 273, normalized size = 1.01 \begin {gather*} \frac {1}{6} \, a x^{6} e^{2} + \frac {1}{2} \, a d x^{4} e + \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*x^6*e^2 + 1/2*a*d*x^4*e + 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

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Fricas [A]
time = 0.36, size = 373, normalized size = 1.39 \begin {gather*} \frac {48 \, a c^{6} x^{6} \cosh \left (1\right )^{2} + 48 \, a c^{6} x^{6} \sinh \left (1\right )^{2} + 144 \, a c^{6} d x^{4} \cosh \left (1\right ) + 144 \, a c^{6} d^{2} x^{2} + 3 \, {\left (48 \, b c^{6} d^{2} x^{2} - 24 \, b c^{4} d^{2} + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \cosh \left (1\right )^{2} + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \sinh \left (1\right )^{2} + 6 \, {\left (8 \, b c^{6} d x^{4} - 3 \, b c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (24 \, b c^{6} d x^{4} - 9 \, b c^{2} d + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 48 \, {\left (2 \, a c^{6} x^{6} \cosh \left (1\right ) + 3 \, a c^{6} d x^{4}\right )} \sinh \left (1\right ) - {\left (72 \, b c^{5} d^{2} x + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \cosh \left (1\right )^{2} + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sinh \left (1\right )^{2} + 18 \, {\left (2 \, b c^{5} d x^{3} + 3 \, b c^{3} d x\right )} \cosh \left (1\right ) + 2 \, {\left (18 \, b c^{5} d x^{3} + 27 \, b c^{3} d x + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{288 \, c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(48*a*c^6*x^6*cosh(1)^2 + 48*a*c^6*x^6*sinh(1)^2 + 144*a*c^6*d*x^4*cosh(1) + 144*a*c^6*d^2*x^2 + 3*(48*b
*c^6*d^2*x^2 - 24*b*c^4*d^2 + (16*b*c^6*x^6 - 5*b)*cosh(1)^2 + (16*b*c^6*x^6 - 5*b)*sinh(1)^2 + 6*(8*b*c^6*d*x
^4 - 3*b*c^2*d)*cosh(1) + 2*(24*b*c^6*d*x^4 - 9*b*c^2*d + (16*b*c^6*x^6 - 5*b)*cosh(1))*sinh(1))*log(c*x + sqr
t(c^2*x^2 - 1)) + 48*(2*a*c^6*x^6*cosh(1) + 3*a*c^6*d*x^4)*sinh(1) - (72*b*c^5*d^2*x + (8*b*c^5*x^5 + 10*b*c^3
*x^3 + 15*b*c*x)*cosh(1)^2 + (8*b*c^5*x^5 + 10*b*c^3*x^3 + 15*b*c*x)*sinh(1)^2 + 18*(2*b*c^5*d*x^3 + 3*b*c^3*d
*x)*cosh(1) + 2*(18*b*c^5*d*x^3 + 27*b*c^3*d*x + (8*b*c^5*x^5 + 10*b*c^3*x^3 + 15*b*c*x)*cosh(1))*sinh(1))*sqr
t(c^2*x^2 - 1))/c^6

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Sympy [C] Result contains complex when optimal does not.
time = 0.53, size = 306, normalized size = 1.14 \begin {gather*} \begin {cases} \frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {acosh}{\left (c x \right )}}{6} - \frac {b d^{2} x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {b d e x^{3} \sqrt {c^{2} x^{2} - 1}}{8 c} - \frac {b e^{2} x^{5} \sqrt {c^{2} x^{2} - 1}}{36 c} - \frac {b d^{2} \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} - \frac {3 b d e x \sqrt {c^{2} x^{2} - 1}}{16 c^{3}} - \frac {5 b e^{2} x^{3} \sqrt {c^{2} x^{2} - 1}}{144 c^{3}} - \frac {3 b d e \operatorname {acosh}{\left (c x \right )}}{16 c^{4}} - \frac {5 b e^{2} x \sqrt {c^{2} x^{2} - 1}}{96 c^{5}} - \frac {5 b e^{2} \operatorname {acosh}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*acosh(c*x)/2 + b*d*e*x**4*acosh(c*x)/2 +
b*e**2*x**6*acosh(c*x)/6 - b*d**2*x*sqrt(c**2*x**2 - 1)/(4*c) - b*d*e*x**3*sqrt(c**2*x**2 - 1)/(8*c) - b*e**2
*x**5*sqrt(c**2*x**2 - 1)/(36*c) - b*d**2*acosh(c*x)/(4*c**2) - 3*b*d*e*x*sqrt(c**2*x**2 - 1)/(16*c**3) - 5*b*
e**2*x**3*sqrt(c**2*x**2 - 1)/(144*c**3) - 3*b*d*e*acosh(c*x)/(16*c**4) - 5*b*e**2*x*sqrt(c**2*x**2 - 1)/(96*c
**5) - 5*b*e**2*acosh(c*x)/(96*c**6), Ne(c, 0)), ((a + I*pi*b/2)*(d**2*x**2/2 + d*e*x**4/2 + e**2*x**6/6), Tru
e))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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