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

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

[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])

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Rubi [A]  time = 0.248621, antiderivative size = 269, normalized size of antiderivative = 1., 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 = {5788, 902, 416, 528, 388, 217, 206} \[ \frac{\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 e}+\frac{b x \left (1-c^2 x^2\right ) \left (44 c^4 d^2+44 c^2 d e+15 e^2\right )}{288 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (2 c^2 d+e\right ) \left (8 c^4 d^2+8 c^2 d e+5 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{96 c^6 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{36 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )}{144 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

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 5788

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]

Rule 902

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 416

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 528

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 388

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 217

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 206

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

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 ) \operatorname{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*}

Mathematica [A]  time = 0.306075, size = 183, normalized size = 0.68 \[ \frac{c x \left (48 a c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )-b \sqrt{c x-1} \sqrt{c x+1} \left (4 c^4 \left (18 d^2+9 d e x^2+2 e^2 x^4\right )+2 c^2 e \left (27 d+5 e x^2\right )+15 e^2\right )\right )+48 b c^6 x^2 \cosh ^{-1}(c x) \left (3 d^2+3 d e x^2+e^2 x^4\right )-6 b \left (24 c^4 d^2+18 c^2 d e+5 e^2\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{288 c^6} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [A]  time = 0.016, size = 363, normalized size = 1.4 \begin{align*}{\frac{a{e}^{2}{x}^{6}}{6}}+{\frac{ade{x}^{4}}{2}}+{\frac{a{x}^{2}{d}^{2}}{2}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{2}{x}^{6}}{6}}+{\frac{b{\rm arccosh} \left (cx\right )de{x}^{4}}{2}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{2}{d}^{2}}{2}}-{\frac{b{e}^{2}{x}^{5}}{36\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bde{x}^{3}}{8\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{2}x}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{2}}{4\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{5\,b{e}^{2}{x}^{3}}{144\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,bdex}{16\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,bde}{16\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{5\,b{e}^{2}x}{96\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,b{e}^{2}}{96\,{c}^{6}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13256, size = 405, normalized size = 1.51 \begin{align*} \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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b e^{2} \end{align*}

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*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)*sqrt(c^2))/(sqrt(c^2)*c^2)))*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)*sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d
*e + 1/288*(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*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^6))*c)*b*e^2

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Fricas [A]  time = 2.574, size = 436, normalized size = 1.62 \begin{align*} \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}} \end{align*}

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*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

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Sympy [A]  time = 11.7074, size = 306, normalized size = 1.14 \begin{align*} \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{align*}

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 [A]  time = 1.45162, size = 387, normalized size = 1.44 \begin{align*} \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b d^{2} + \frac{1}{288} \,{\left (48 \, a x^{6} +{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b\right )} e^{2} + \frac{1}{16} \,{\left (8 \, a d x^{4} +{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b d\right )} e \end{align*}

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]

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