3.482 \(\int x (d+e x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=358 \[ \frac{\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}+\frac{b x \left (1-c^2 x^2\right ) \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (288 c^4 d^2 e^2+256 c^6 d^3 e+128 c^8 d^4+160 c^2 d e^3+35 e^4\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{1024 c^8 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[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])

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Rubi [A]  time = 0.364305, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 e}+\frac{b x \left (1-c^2 x^2\right ) \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (288 c^4 d^2 e^2+256 c^6 d^3 e+128 c^8 d^4+160 c^2 d e^3+35 e^4\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{1024 c^8 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{64 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b x \left (1-c^2 x^2\right ) \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[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 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 )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 \left (a+b \cosh ^{-1}(c x)\right )}{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 ) \operatorname{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 \left (a+b \cosh ^{-1}(c x)\right )}{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} \tanh ^{-1}\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]  time = 0.384862, size = 256, normalized size = 0.72 \[ \frac{c x \left (384 a c^7 x \left (6 d^2 e x^2+4 d^3+4 d e^2 x^4+e^3 x^6\right )-b \sqrt{c x-1} \sqrt{c x+1} \left (16 c^6 \left (36 d^2 e x^2+48 d^3+16 d e^2 x^4+3 e^3 x^6\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+10 c^2 e^2 \left (48 d+7 e x^2\right )+105 e^3\right )\right )+384 b c^8 x^2 \cosh ^{-1}(c x) \left (6 d^2 e x^2+4 d^3+4 d e^2 x^4+e^3 x^6\right )-6 b \left (288 c^4 d^2 e+256 c^6 d^3+160 c^2 d e^2+35 e^3\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{3072 c^8} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [A]  time = 0.017, size = 553, normalized size = 1.5 \begin{align*}{\frac{a{e}^{3}{x}^{8}}{8}}+{\frac{ad{e}^{2}{x}^{6}}{2}}+{\frac{3\,a{d}^{2}e{x}^{4}}{4}}+{\frac{a{x}^{2}{d}^{3}}{2}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{3}{x}^{8}}{8}}+{\frac{b{\rm arccosh} \left (cx\right )d{e}^{2}{x}^{6}}{2}}+{\frac{3\,b{\rm arccosh} \left (cx\right ){d}^{2}e{x}^{4}}{4}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{2}{d}^{3}}{2}}-{\frac{b{e}^{3}{x}^{7}}{64\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{x}^{5}d{e}^{2}}{12\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{x}^{3}{d}^{2}e}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{3}x}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{3}}{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{7\,b{e}^{3}{x}^{5}}{384\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,b{x}^{3}d{e}^{2}}{48\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{9\,bx{d}^{2}e}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{9\,b{d}^{2}e}{32\,{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{35\,b{e}^{3}{x}^{3}}{1536\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bxd{e}^{2}}{32\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bd{e}^{2}}{32\,{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}}}}-{\frac{35\,b{e}^{3}x}{1024\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{35\,b{e}^{3}}{1024\,{c}^{8}}\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)^3*(a+b*arccosh(c*x)),x)

[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*x^2*d^3+1/8*b*arccosh(c*x)*e^3*x^8+1/2*b*arccosh(c*x)*d*e^
2*x^6+3/4*b*arccosh(c*x)*d^2*e*x^4+1/2*b*arccosh(c*x)*x^2*d^3-1/64/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^7-1/1
2/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^5*d*e^2-3/16/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3*d^2*e-1/4*b*d^3*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)*ln(c*x+(c^2*x^2-1)^(1/2))*d^3-7
/384/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^5-5/48/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3*d*e^2-9/32/c^3*b*(c*
x-1)^(1/2)*(c*x+1)^(1/2)*x*d^2*e-9/32/c^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*e-35/1536/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^3-5/32/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*d*e^2-5
/32/c^6*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*e^2-35/1024/c^7*b*(c*x-1)^
(1/2)*(c*x+1)^(1/2)*e^3*x-35/1024/c^8*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*e^3*ln(c*x+(c^2*x^2-1)^(
1/2))

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Maxima [A]  time = 1.17718, size = 601, normalized size = 1.68 \begin{align*} \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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*sqrt(c^2))/(sqrt(c^2)*c^2)))*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)*sqrt(c^2))/(sq
rt(c^2)*c^4))*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*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^6))*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)*sqrt(c^2))/(sqrt(c^2)*c
^8))*c)*b*e^3

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Fricas [A]  time = 2.45161, size = 657, normalized size = 1.84 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 19.4456, size = 490, normalized size = 1.37 \begin{align*} \begin{cases} \frac{a d^{3} x^{2}}{2} + \frac{3 a d^{2} e x^{4}}{4} + \frac{a d e^{2} x^{6}}{2} + \frac{a e^{3} x^{8}}{8} + \frac{b d^{3} x^{2} \operatorname{acosh}{\left (c x \right )}}{2} + \frac{3 b d^{2} e x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{b d e^{2} x^{6} \operatorname{acosh}{\left (c x \right )}}{2} + \frac{b e^{3} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} - \frac{b d^{3} x \sqrt{c^{2} x^{2} - 1}}{4 c} - \frac{3 b d^{2} e x^{3} \sqrt{c^{2} x^{2} - 1}}{16 c} - \frac{b d e^{2} x^{5} \sqrt{c^{2} x^{2} - 1}}{12 c} - \frac{b e^{3} x^{7} \sqrt{c^{2} x^{2} - 1}}{64 c} - \frac{b d^{3} \operatorname{acosh}{\left (c x \right )}}{4 c^{2}} - \frac{9 b d^{2} e x \sqrt{c^{2} x^{2} - 1}}{32 c^{3}} - \frac{5 b d e^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{48 c^{3}} - \frac{7 b e^{3} x^{5} \sqrt{c^{2} x^{2} - 1}}{384 c^{3}} - \frac{9 b d^{2} e \operatorname{acosh}{\left (c x \right )}}{32 c^{4}} - \frac{5 b d e^{2} x \sqrt{c^{2} x^{2} - 1}}{32 c^{5}} - \frac{35 b e^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{1536 c^{5}} - \frac{5 b d e^{2} \operatorname{acosh}{\left (c x \right )}}{32 c^{6}} - \frac{35 b e^{3} x \sqrt{c^{2} x^{2} - 1}}{1024 c^{7}} - \frac{35 b e^{3} \operatorname{acosh}{\left (c x \right )}}{1024 c^{8}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{3} x^{2}}{2} + \frac{3 d^{2} e x^{4}}{4} + \frac{d e^{2} x^{6}}{2} + \frac{e^{3} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x**2/2 + 3*a*d**2*e*x**4/4 + a*d*e**2*x**6/2 + a*e**3*x**8/8 + b*d**3*x**2*acosh(c*x)/2 + 3*
b*d**2*e*x**4*acosh(c*x)/4 + b*d*e**2*x**6*acosh(c*x)/2 + b*e**3*x**8*acosh(c*x)/8 - b*d**3*x*sqrt(c**2*x**2 -
1)/(4*c) - 3*b*d**2*e*x**3*sqrt(c**2*x**2 - 1)/(16*c) - b*d*e**2*x**5*sqrt(c**2*x**2 - 1)/(12*c) - b*e**3*x**
7*sqrt(c**2*x**2 - 1)/(64*c) - b*d**3*acosh(c*x)/(4*c**2) - 9*b*d**2*e*x*sqrt(c**2*x**2 - 1)/(32*c**3) - 5*b*d
*e**2*x**3*sqrt(c**2*x**2 - 1)/(48*c**3) - 7*b*e**3*x**5*sqrt(c**2*x**2 - 1)/(384*c**3) - 9*b*d**2*e*acosh(c*x
)/(32*c**4) - 5*b*d*e**2*x*sqrt(c**2*x**2 - 1)/(32*c**5) - 35*b*e**3*x**3*sqrt(c**2*x**2 - 1)/(1536*c**5) - 5*
b*d*e**2*acosh(c*x)/(32*c**6) - 35*b*e**3*x*sqrt(c**2*x**2 - 1)/(1024*c**7) - 35*b*e**3*acosh(c*x)/(1024*c**8)
, Ne(c, 0)), ((a + I*pi*b/2)*(d**3*x**2/2 + 3*d**2*e*x**4/4 + d*e**2*x**6/2 + e**3*x**8/8), True))

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Giac [A]  time = 1.58357, size = 552, normalized size = 1.54 \begin{align*} \frac{1}{2} \, a d^{3} 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^{3} + \frac{1}{3072} \,{\left (384 \, a x^{8} +{\left (384 \, x^{8} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \, x^{2}{\left (\frac{6 \, x^{2}}{c^{2}} + \frac{7}{c^{4}}\right )} + \frac{35}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x - \frac{105 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{8}{\left | c \right |}}\right )} c\right )} b\right )} e^{3} + \frac{1}{96} \,{\left (48 \, a d 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 d\right )} e^{2} + \frac{3}{32} \,{\left (8 \, a d^{2} 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^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*d^3*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^3 + 1/3072*(384*a*x^8 + (384*x^8*log(c*x + sqrt(c^2*x^2 - 1)) - (sqrt(c^2*
x^2 - 1)*(2*(4*x^2*(6*x^2/c^2 + 7/c^4) + 35/c^6)*x^2 + 105/c^8)*x - 105*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1))
)/(c^8*abs(c)))*c)*b)*e^3 + 1/96*(48*a*d*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*d)*e^2 + 3/3
2*(8*a*d^2*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^2)*e