3.27 \(\int \frac {(d-c^2 d x^2)^3 (a+b \cosh ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=195 \[ -\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {1}{9} b c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}-\frac {8}{3} b c^3 d^3 \sqrt {c x-1} \sqrt {c x+1}-\frac {17}{6} b c^3 d^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b c d^3 \sqrt {c x-1} \sqrt {c x+1}}{6 x^2} \]

[Out]

1/9*b*c^3*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)-1/3*d^3*(a+b*arccosh(c*x))/x^3+3*c^2*d^3*(a+b*arccosh(c*x))/x+3*c^4*
d^3*x*(a+b*arccosh(c*x))-1/3*c^6*d^3*x^3*(a+b*arccosh(c*x))-17/6*b*c^3*d^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))
-8/3*b*c^3*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/6*b*c*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x^2

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Rubi [A]  time = 0.39, antiderivative size = 252, normalized size of antiderivative = 1.29, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {270, 5731, 12, 1610, 1799, 1621, 897, 1153, 205} \[ -\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {17 b c^3 d^3 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]

[Out]

(8*b*c^3*d^3*(1 - c^2*x^2))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) + (b*c^3*d^3*(1 - c^2*x^2)^2)/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/(3*x
^3) + (3*c^2*d^3*(a + b*ArcCosh[c*x]))/x + 3*c^4*d^3*x*(a + b*ArcCosh[c*x]) - (c^6*d^3*x^3*(a + b*ArcCosh[c*x]
))/3 - (17*b*c^3*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 270

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 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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 1621

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*
(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expo
n[Px, x], 2]

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 5731

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 \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^3 \left (-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6\right )}{3 x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^3\right ) \int \frac {-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {-1+9 c^2 x^2+9 c^4 x^4-c^6 x^6}{x^3 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-1+9 c^2 x+9 c^4 x^2-c^6 x^3}{x^2 \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {17 c^2}{2}+9 c^4 x-c^6 x^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {33 c^2}{2}+7 c^2 x^2-c^2 x^4}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (8 c^4-c^4 x^2+\frac {17 c^2}{2 \left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (17 b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b c^3 d^3 \left (1-c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{6 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^3 \left (1-c^2 x^2\right )^2}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )}{x}+3 c^4 d^3 x \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{3} c^6 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {17 b c^3 d^3 \sqrt {-1+c^2 x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 142, normalized size = 0.73 \[ \frac {d^3 \left (-6 a c^6 x^6+54 a c^4 x^4+54 a c^2 x^2-6 a+51 b c^3 x^3 \tan ^{-1}\left (\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )+b c x \sqrt {c x-1} \sqrt {c x+1} \left (2 c^4 x^4-50 c^2 x^2+3\right )-6 b \left (c^6 x^6-9 c^4 x^4-9 c^2 x^2+1\right ) \cosh ^{-1}(c x)\right )}{18 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]

[Out]

(d^3*(-6*a + 54*a*c^2*x^2 + 54*a*c^4*x^4 - 6*a*c^6*x^6 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - 50*c^2*x^2 +
2*c^4*x^4) - 6*b*(1 - 9*c^2*x^2 - 9*c^4*x^4 + c^6*x^6)*ArcCosh[c*x] + 51*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sq
rt[1 + c*x])]))/(18*x^3)

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fricas [A]  time = 0.66, size = 253, normalized size = 1.30 \[ -\frac {6 \, a c^{6} d^{3} x^{6} - 54 \, a c^{4} d^{3} x^{4} + 102 \, b c^{3} d^{3} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 54 \, a c^{2} d^{3} x^{2} - 6 \, {\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, a d^{3} + 6 \, {\left (b c^{6} d^{3} x^{6} - 9 \, b c^{4} d^{3} x^{4} - 9 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} + b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d^{3} x^{5} - 50 \, b c^{3} d^{3} x^{3} + 3 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="fricas")

[Out]

-1/18*(6*a*c^6*d^3*x^6 - 54*a*c^4*d^3*x^4 + 102*b*c^3*d^3*x^3*arctan(-c*x + sqrt(c^2*x^2 - 1)) - 54*a*c^2*d^3*
x^2 - 6*(b*c^6 - 9*b*c^4 - 9*b*c^2 + b)*d^3*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) + 6*a*d^3 + 6*(b*c^6*d^3*x^6 - 9
*b*c^4*d^3*x^4 - 9*b*c^2*d^3*x^2 - (b*c^6 - 9*b*c^4 - 9*b*c^2 + b)*d^3*x^3 + b*d^3)*log(c*x + sqrt(c^2*x^2 - 1
)) - (2*b*c^5*d^3*x^5 - 50*b*c^3*d^3*x^3 + 3*b*c*d^3*x)*sqrt(c^2*x^2 - 1))/x^3

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.02, size = 223, normalized size = 1.14 \[ -\frac {c^{6} d^{3} a \,x^{3}}{3}+3 c^{4} d^{3} a x +\frac {3 c^{2} d^{3} a}{x}-\frac {d^{3} a}{3 x^{3}}-\frac {c^{6} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{3}}{3}+3 c^{4} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x +\frac {3 c^{2} d^{3} b \,\mathrm {arccosh}\left (c x \right )}{x}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right )}{3 x^{3}}+\frac {c^{5} d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{9}-\frac {25 b \,c^{3} d^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{9}+\frac {17 c^{3} d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 \sqrt {c^{2} x^{2}-1}}+\frac {b c \,d^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x)

[Out]

-1/3*c^6*d^3*a*x^3+3*c^4*d^3*a*x+3*c^2*d^3*a/x-1/3*d^3*a/x^3-1/3*c^6*d^3*b*arccosh(c*x)*x^3+3*c^4*d^3*b*arccos
h(c*x)*x+3*c^2*d^3*b*arccosh(c*x)/x-1/3*d^3*b*arccosh(c*x)/x^3+1/9*c^5*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^2-2
5/9*b*c^3*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+17/6*c^3*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*arctan(
1/(c^2*x^2-1)^(1/2))+1/6*b*c*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x^2

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maxima [A]  time = 0.56, size = 208, normalized size = 1.07 \[ -\frac {1}{3} \, a c^{6} d^{3} x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{6} d^{3} + 3 \, a c^{4} d^{3} x + 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c^{3} d^{3} + 3 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{3} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{3} + \frac {3 \, a c^{2} d^{3}}{x} - \frac {a d^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="maxima")

[Out]

-1/3*a*c^6*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*c^6*
d^3 + 3*a*c^4*d^3*x + 3*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*c^3*d^3 + 3*(c*arcsin(1/(c*abs(x))) + arccosh
(c*x)/x)*b*c^2*d^3 - 1/6*((c^2*arcsin(1/(c*abs(x))) - sqrt(c^2*x^2 - 1)/x^2)*c + 2*arccosh(c*x)/x^3)*b*d^3 + 3
*a*c^2*d^3/x - 1/3*a*d^3/x^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^3)/x^4,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^3)/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - d^{3} \left (\int \left (- 3 a c^{4}\right )\, dx + \int \left (- \frac {a}{x^{4}}\right )\, dx + \int \frac {3 a c^{2}}{x^{2}}\, dx + \int a c^{6} x^{2}\, dx + \int \left (- 3 b c^{4} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \frac {3 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x^{2}}\, dx + \int b c^{6} x^{2} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**3*(a+b*acosh(c*x))/x**4,x)

[Out]

-d**3*(Integral(-3*a*c**4, x) + Integral(-a/x**4, x) + Integral(3*a*c**2/x**2, x) + Integral(a*c**6*x**2, x) +
Integral(-3*b*c**4*acosh(c*x), x) + Integral(-b*acosh(c*x)/x**4, x) + Integral(3*b*c**2*acosh(c*x)/x**2, x) +
Integral(b*c**6*x**2*acosh(c*x), x))

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