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

Optimal. Leaf size=287 \[ d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2+5 e^3\right )}{35 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 \left (1-c^2 x^2\right )^3 \left (7 c^2 d+5 e\right )}{175 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*(1 - c^2*x^2))/(35*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
b*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2)^2)/(105*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*e^2*(7*c
^2*d + 5*e)*(1 - c^2*x^2)^3)/(175*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(1 - c^2*x^2)^4)/(49*c^7*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) + d^2*e*x^3*(a + b*ArcCosh[c*x]) + (3*d*e^2*x^5*(a + b*ArcC
osh[c*x]))/5 + (e^3*x^7*(a + b*ArcCosh[c*x]))/7

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Rubi [A]  time = 0.37592, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {194, 5705, 12, 1610, 1799, 1850} \[ d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2+5 e^3\right )}{35 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 \left (1-c^2 x^2\right )^3 \left (7 c^2 d+5 e\right )}{175 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*(1 - c^2*x^2))/(35*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
b*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2)^2)/(105*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*e^2*(7*c
^2*d + 5*e)*(1 - c^2*x^2)^3)/(175*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(1 - c^2*x^2)^4)/(49*c^7*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) + d^2*e*x^3*(a + b*ArcCosh[c*x]) + (3*d*e^2*x^5*(a + b*ArcC
osh[c*x]))/5 + (e^3*x^7*(a + b*ArcCosh[c*x]))/7

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5705

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{35} (b c) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{35 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{70 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3}{c^6 \sqrt{-1+c^2 x}}+\frac{e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt{-1+c^2 x}}{c^6}+\frac{3 e^2 \left (7 c^2 d+5 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac{5 e^3 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{70 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \left (1-c^2 x^2\right )}{35 c^7 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^2}{105 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.271764, size = 193, normalized size = 0.67 \[ a \left (d^2 e x^3+d^3 x+\frac{3}{5} d e^2 x^5+\frac{e^3 x^7}{7}\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^6 \left (1225 d^2 e x^2+3675 d^3+441 d e^2 x^4+75 e^3 x^6\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )+240 e^3\right )}{3675 c^7}+\frac{1}{35} b x \cosh ^{-1}(c x) \left (35 d^2 e x^2+35 d^3+21 d e^2 x^4+5 e^3 x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*
(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*
x^4 + 75*e^3*x^6)))/(3675*c^7) + (b*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcCosh[c*x])/35

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Maple [A]  time = 0.012, size = 235, normalized size = 0.8 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{7}d{e}^{2}{x}^{5}}{5}}+{c}^{7}{d}^{2}e{x}^{3}+x{c}^{7}{d}^{3} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{3}{c}^{7}{x}^{7}}{7}}+{\frac{3\,{\rm arccosh} \left (cx\right )d{e}^{2}{c}^{7}{x}^{5}}{5}}+{\rm arccosh} \left (cx\right ){c}^{7}{d}^{2}e{x}^{3}+{\rm arccosh} \left (cx\right ){c}^{7}x{d}^{3}-{\frac{75\,{c}^{6}{e}^{3}{x}^{6}+441\,{c}^{6}d{e}^{2}{x}^{4}+1225\,{c}^{6}{d}^{2}e{x}^{2}+90\,{c}^{4}{e}^{3}{x}^{4}+3675\,{d}^{3}{c}^{6}+588\,{c}^{4}d{e}^{2}{x}^{2}+2450\,{c}^{4}{d}^{2}e+120\,{c}^{2}{e}^{3}{x}^{2}+1176\,d{e}^{2}{c}^{2}+240\,{e}^{3}}{3675}\sqrt{cx-1}\sqrt{cx+1}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^6*(1/7*e^3*c^7*x^7+3/5*c^7*d*e^2*x^5+c^7*d^2*e*x^3+x*c^7*d^3)+b/c^6*(1/7*arccosh(c*x)*e^3*c^7*x^7+3/5
*arccosh(c*x)*d*e^2*c^7*x^5+arccosh(c*x)*c^7*d^2*e*x^3+arccosh(c*x)*c^7*x*d^3-1/3675*(c*x-1)^(1/2)*(c*x+1)^(1/
2)*(75*c^6*e^3*x^6+441*c^6*d*e^2*x^4+1225*c^6*d^2*e*x^2+90*c^4*e^3*x^4+3675*c^6*d^3+588*c^4*d*e^2*x^2+2450*c^4
*d^2*e+120*c^2*e^3*x^2+1176*c^2*d*e^2+240*e^3)))

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Maxima [A]  time = 1.13003, size = 387, normalized size = 1.35 \begin{align*} \frac{1}{7} \, a e^{3} x^{7} + \frac{3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac{1}{3} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac{1}{25} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac{1}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d^{3}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqr
t(c^2*x^2 - 1)/c^4))*b*d^2*e + 1/25*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*
x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*d*e^2 + 1/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1)*x^6/c^2 + 6*
sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*e^3 + a*d^3*x + (c*x*
arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3/c

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Fricas [A]  time = 2.35321, size = 559, normalized size = 1.95 \begin{align*} \frac{525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \,{\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} + 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \,{\left (49 \, b c^{6} d e^{2} + 10 \, b c^{4} e^{3}\right )} x^{4} + 240 \, b e^{3} +{\left (1225 \, b c^{6} d^{2} e + 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{3675 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*
x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^2*e*x^3 + 35*b*c^7*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*e^3*x
^6 + 3675*b*c^6*d^3 + 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^2 + 10*b*c^4*e^3)*x^4 + 240*b*e^3
+ (1225*b*c^6*d^2*e + 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))/c^7

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Sympy [A]  time = 13.3653, size = 396, normalized size = 1.38 \begin{align*} \begin{cases} a d^{3} x + a d^{2} e x^{3} + \frac{3 a d e^{2} x^{5}}{5} + \frac{a e^{3} x^{7}}{7} + b d^{3} x \operatorname{acosh}{\left (c x \right )} + b d^{2} e x^{3} \operatorname{acosh}{\left (c x \right )} + \frac{3 b d e^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{b e^{3} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} - \frac{b d^{3} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{b d^{2} e x^{2} \sqrt{c^{2} x^{2} - 1}}{3 c} - \frac{3 b d e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{b e^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{2 b d^{2} e \sqrt{c^{2} x^{2} - 1}}{3 c^{3}} - \frac{4 b d e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{25 c^{3}} - \frac{6 b e^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{8 b d e^{2} \sqrt{c^{2} x^{2} - 1}}{25 c^{5}} - \frac{8 b e^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b e^{3} \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (d^{3} x + d^{2} e x^{3} + \frac{3 d e^{2} x^{5}}{5} + \frac{e^{3} x^{7}}{7}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x + a*d**2*e*x**3 + 3*a*d*e**2*x**5/5 + a*e**3*x**7/7 + b*d**3*x*acosh(c*x) + b*d**2*e*x**3*
acosh(c*x) + 3*b*d*e**2*x**5*acosh(c*x)/5 + b*e**3*x**7*acosh(c*x)/7 - b*d**3*sqrt(c**2*x**2 - 1)/c - b*d**2*e
*x**2*sqrt(c**2*x**2 - 1)/(3*c) - 3*b*d*e**2*x**4*sqrt(c**2*x**2 - 1)/(25*c) - b*e**3*x**6*sqrt(c**2*x**2 - 1)
/(49*c) - 2*b*d**2*e*sqrt(c**2*x**2 - 1)/(3*c**3) - 4*b*d*e**2*x**2*sqrt(c**2*x**2 - 1)/(25*c**3) - 6*b*e**3*x
**4*sqrt(c**2*x**2 - 1)/(245*c**3) - 8*b*d*e**2*sqrt(c**2*x**2 - 1)/(25*c**5) - 8*b*e**3*x**2*sqrt(c**2*x**2 -
1)/(245*c**5) - 16*b*e**3*sqrt(c**2*x**2 - 1)/(245*c**7), Ne(c, 0)), ((a + I*pi*b/2)*(d**3*x + d**2*e*x**3 +
3*d*e**2*x**5/5 + e**3*x**7/7), True))

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Giac [A]  time = 1.37013, size = 396, normalized size = 1.38 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d^{3} + a d^{3} x + \frac{1}{245} \,{\left (35 \, a x^{7} +{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b\right )} e^{3} + \frac{1}{25} \,{\left (15 \, a d x^{5} +{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b d\right )} e^{2} + \frac{1}{3} \,{\left (3 \, a d^{2} x^{3} +{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b d^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log(c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2*x^2 - 1)/c)*b*d^3 + a*d^3*x + 1/245*(35*a*x^7 + (35*x^7*log(c*x + s
qrt(c^2*x^2 - 1)) - (5*(c^2*x^2 - 1)^(7/2) + 21*(c^2*x^2 - 1)^(5/2) + 35*(c^2*x^2 - 1)^(3/2) + 35*sqrt(c^2*x^2
- 1))/c^7)*b)*e^3 + 1/25*(15*a*d*x^5 + (15*x^5*log(c*x + sqrt(c^2*x^2 - 1)) - (3*(c^2*x^2 - 1)^(5/2) + 10*(c^
2*x^2 - 1)^(3/2) + 15*sqrt(c^2*x^2 - 1))/c^5)*b*d)*e^2 + 1/3*(3*a*d^2*x^3 + (3*x^3*log(c*x + sqrt(c^2*x^2 - 1)
) - ((c^2*x^2 - 1)^(3/2) + 3*sqrt(c^2*x^2 - 1))/c^3)*b*d^2)*e