3.1.21 \(\int x^2 (d-c^2 d x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\) [21]

Optimal. Leaf size=227 \[ -\frac {16 b d^3 \sqrt {-1+c x} \sqrt {1+c x}}{315 c^3}+\frac {8 b d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}{945 c^3}-\frac {2 b d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{525 c^3}+\frac {b d^3 (-1+c x)^{7/2} (1+c x)^{7/2}}{441 c^3}+\frac {b d^3 (-1+c x)^{9/2} (1+c x)^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right ) \]

[Out]

8/945*b*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^3-2/525*b*d^3*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^3+1/441*b*d^3*(c*x-1)^(7
/2)*(c*x+1)^(7/2)/c^3+1/81*b*d^3*(c*x-1)^(9/2)*(c*x+1)^(9/2)/c^3+1/3*d^3*x^3*(a+b*arccosh(c*x))-3/5*c^2*d^3*x^
5*(a+b*arccosh(c*x))+3/7*c^4*d^3*x^7*(a+b*arccosh(c*x))-1/9*c^6*d^3*x^9*(a+b*arccosh(c*x))-16/315*b*d^3*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/c^3

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Rubi [A]
time = 0.27, antiderivative size = 285, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {276, 5921, 12, 1624, 1813, 1634} \begin {gather*} -\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b d^3 \left (1-c^2 x^2\right )^5}{81 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^3 \left (1-c^2 x^2\right )^4}{441 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^3 \left (1-c^2 x^2\right )^3}{525 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{945 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b d^3 \left (1-c^2 x^2\right )}{315 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b*d^3*(1 - c^2*x^2))/(315*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(945*c^3*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (2*b*d^3*(1 - c^2*x^2)^3)/(525*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)
^4)/(441*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^3*(1 - c^2*x^2)^5)/(81*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
d^3*x^3*(a + b*ArcCosh[c*x]))/3 - (3*c^2*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcCosh[c*x])
)/7 - (c^6*d^3*x^9*(a + b*ArcCosh[c*x]))/9

Rule 12

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

Rule 276

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

Rule 1624

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 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1813

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 5921

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(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, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{315 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^3\right ) \int \frac {x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x \left (105-189 c^2 x+135 c^4 x^2-35 c^6 x^3\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {16}{c^2 \sqrt {-1+c^2 x}}-\frac {8 \sqrt {-1+c^2 x}}{c^2}+\frac {6 \left (-1+c^2 x\right )^{3/2}}{c^2}-\frac {5 \left (-1+c^2 x\right )^{5/2}}{c^2}-\frac {35 \left (-1+c^2 x\right )^{7/2}}{c^2}\right ) \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {16 b d^3 \left (1-c^2 x^2\right )}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{945 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^3 \left (1-c^2 x^2\right )^3}{525 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \left (1-c^2 x^2\right )^4}{441 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^3 \left (1-c^2 x^2\right )^5}{81 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 139, normalized size = 0.61 \begin {gather*} -\frac {d^3 \left (315 a c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (5258+2629 c^2 x^2-6297 c^4 x^4+4675 c^6 x^6-1225 c^8 x^8\right )+315 b c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right ) \cosh ^{-1}(c x)\right )}{99225 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/99225*(d^3*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*
(5258 + 2629*c^2*x^2 - 6297*c^4*x^4 + 4675*c^6*x^6 - 1225*c^8*x^8) + 315*b*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c
^4*x^4 + 35*c^6*x^6)*ArcCosh[c*x]))/c^3

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Maple [A]
time = 2.69, size = 150, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \,\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\mathrm {arccosh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-4675 x^{6} c^{6}+6297 c^{4} x^{4}-2629 c^{2} x^{2}-5258\right )}{99225}\right )}{c^{3}}\)	\(150\)
default	\(\frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \,\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\mathrm {arccosh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-4675 x^{6} c^{6}+6297 c^{4} x^{4}-2629 c^{2} x^{2}-5258\right )}{99225}\right )}{c^{3}}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-d^3*a*(1/9*c^9*x^9-3/7*c^7*x^7+3/5*c^5*x^5-1/3*c^3*x^3)-d^3*b*(1/9*arccosh(c*x)*c^9*x^9-3/7*arccosh(c*
x)*c^7*x^7+3/5*arccosh(c*x)*c^5*x^5-1/3*arccosh(c*x)*c^3*x^3-1/99225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*x^8
-4675*c^6*x^6+6297*c^4*x^4-2629*c^2*x^2-5258)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (189) = 378\).
time = 0.27, size = 388, normalized size = 1.71 \begin {gather*} -\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} - \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} - \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{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 c^{4} d^{3} - \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 c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\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^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 - 1/2835*(315*x^9*arccosh(c*x) - (35*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqr
t(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x^2 - 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c
^10)*c)*b*c^6*d^3 - 3/5*a*c^2*d^3*x^5 + 3/245*(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*c^4*d^3 - 1/25*(15*x^5*arccos
h(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*c^2*d^3 +
1/3*a*d^3*x^3 + 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d^3

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Fricas [A]
time = 0.35, size = 189, normalized size = 0.83 \begin {gather*} -\frac {11025 \, a c^{9} d^{3} x^{9} - 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} - 33075 \, a c^{3} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} d^{3} x^{9} - 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} - 105 \, b c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} d^{3} x^{8} - 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} - 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/99225*(11025*a*c^9*d^3*x^9 - 42525*a*c^7*d^3*x^7 + 59535*a*c^5*d^3*x^5 - 33075*a*c^3*d^3*x^3 + 315*(35*b*c^
9*d^3*x^9 - 135*b*c^7*d^3*x^7 + 189*b*c^5*d^3*x^5 - 105*b*c^3*d^3*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*
c^8*d^3*x^8 - 4675*b*c^6*d^3*x^6 + 6297*b*c^4*d^3*x^4 - 2629*b*c^2*d^3*x^2 - 5258*b*d^3)*sqrt(c^2*x^2 - 1))/c^
3

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Sympy [C] Result contains complex when optimal does not.
time = 1.47, size = 272, normalized size = 1.20 \begin {gather*} \begin {cases} - \frac {a c^{6} d^{3} x^{9}}{9} + \frac {3 a c^{4} d^{3} x^{7}}{7} - \frac {3 a c^{2} d^{3} x^{5}}{5} + \frac {a d^{3} x^{3}}{3} - \frac {b c^{6} d^{3} x^{9} \operatorname {acosh}{\left (c x \right )}}{9} + \frac {b c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} - 1}}{81} + \frac {3 b c^{4} d^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {187 b c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{3969} - \frac {3 b c^{2} d^{3} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {2099 b c d^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{33075} + \frac {b d^{3} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {2629 b d^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{99225 c} - \frac {5258 b d^{3} \sqrt {c^{2} x^{2} - 1}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {d^{3} x^{3} \left (a + \frac {i \pi b}{2}\right )}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**6*d**3*x**9/9 + 3*a*c**4*d**3*x**7/7 - 3*a*c**2*d**3*x**5/5 + a*d**3*x**3/3 - b*c**6*d**3*x**
9*acosh(c*x)/9 + b*c**5*d**3*x**8*sqrt(c**2*x**2 - 1)/81 + 3*b*c**4*d**3*x**7*acosh(c*x)/7 - 187*b*c**3*d**3*x
**6*sqrt(c**2*x**2 - 1)/3969 - 3*b*c**2*d**3*x**5*acosh(c*x)/5 + 2099*b*c*d**3*x**4*sqrt(c**2*x**2 - 1)/33075
+ b*d**3*x**3*acosh(c*x)/3 - 2629*b*d**3*x**2*sqrt(c**2*x**2 - 1)/(99225*c) - 5258*b*d**3*sqrt(c**2*x**2 - 1)/
(99225*c**3), Ne(c, 0)), (d**3*x**3*(a + I*pi*b/2)/3, True))

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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^2*(-c^2*d*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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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