3.60 \(\int (f+g x) (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=398 \[ \frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d g (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

3/8*d*f*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+1/4*d*f*x*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)
^(1/2)-1/5*d*g*(-c*x+1)^2*(c*x+1)^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/5*b*d*g*x*(-c^2*d*x^2+d)^(1/
2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/16*b*c*d*f*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/15*b*c*d*
g*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16*b*c^3*d*f*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(
c*x+1)^(1/2)+1/25*b*c^3*d*g*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/16*d*f*(a+b*arccosh(c*x))^2
*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.76, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {5836, 5822, 5685, 5683, 5676, 30, 14, 5718, 194} \[ \frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d g (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*
f*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]) + (3*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 + (d*f*x*(1 - c*x)*(1 + c*x)*Sqr
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/4 - (d*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x
]))/(5*c^2) - (3*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5683

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5685

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Simp[(x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d1*d2*p)/(2*p + 1),
Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)
*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)]
&& GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p - 1/2]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5822

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x
)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m,
0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] |
| EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rule 5836

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f + g*x
)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} (f+g x) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+g x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d f \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d g \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}+\frac {\left (3 d f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {\left (3 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d f \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d g \sqrt {d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d f x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d g (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2}-\frac {3 d f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 1.61, size = 432, normalized size = 1.09 \[ \frac {-10800 a c d^{3/2} f \sqrt {\frac {c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-720 a d \sqrt {\frac {c x-1}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} \left (5 c^2 f x \left (2 c^2 x^2-5\right )+8 g \left (c^2 x^2-1\right )^2\right )-3600 b c d f \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+225 b c d f \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+800 b d g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-8 b d g \sqrt {d-c^2 d x^2} \left (450 c x-450 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-25 \cosh \left (3 \cosh ^{-1}(c x)\right )-9 \cosh \left (5 \cosh ^{-1}(c x)\right )+75 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )+45 \cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )}{28800 c^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-720*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c
^2*x^2)) - 10800*a*c*d^(3/2)*f*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*
(-1 + c^2*x^2))] + 800*b*d*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[
c*x] - Cosh[3*ArcCosh[c*x]]) - 3600*b*c*d*f*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCos
h[c*x] - Sinh[2*ArcCosh[c*x]])) + 225*b*c*d*f*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4
*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 8*b*d*g*Sqrt[d - c^2*d*x^2]*(450*c*x - 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1
+ c*x)*ArcCosh[c*x] - 25*Cosh[3*ArcCosh[c*x]] - 9*Cosh[5*ArcCosh[c*x]] + 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]]
+ 45*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]))/(28800*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d g x^{3} + a c^{2} d f x^{2} - a d g x - a d f + {\left (b c^{2} d g x^{3} + b c^{2} d f x^{2} - b d g x - b d f\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a*c^2*d*g*x^3 + a*c^2*d*f*x^2 - a*d*g*x - a*d*f + (b*c^2*d*g*x^3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)
*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c^2*d*x^2+d)^(3/2)*(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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maple [A]  time = 0.77, size = 656, normalized size = 1.65 \[ -\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d x}{5 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f d \,c^{3} x^{4}}{16 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f d c \,x^{2}}{16 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f d \,c^{4} \mathrm {arccosh}\left (c x \right ) x^{5}}{4 \left (c x +1\right ) \left (c x -1\right )}+\frac {7 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f d \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{8 \left (c x +1\right ) \left (c x -1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f d \,\mathrm {arccosh}\left (c x \right ) x}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {17 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f d}{128 \sqrt {c x +1}\, \sqrt {c x -1}\, c}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d \,c^{4} \mathrm {arccosh}\left (c x \right ) x^{6}}{5 \left (c x +1\right ) \left (c x -1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{4}}{5 \left (c x +1\right ) \left (c x -1\right )}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d \,\mathrm {arccosh}\left (c x \right ) x^{2}}{5 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d \,c^{3} x^{5}}{25 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d c \,x^{3}}{15 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \mathrm {arccosh}\left (c x \right )^{2} d}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g d \,\mathrm {arccosh}\left (c x \right )}{5 \left (c x +1\right ) c^{2} \left (c x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a*g/c^2/d*(-c^2*d*x^2+d)^(5/2)+1/4*a*f*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f*d^
2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)^(1/2)/c/
(c*x-1)^(1/2)*x+1/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3*x^4-5/16*b*(-d*(c^2*x^2-1))^
(1/2)*f*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c*x^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f*d/(c*x+1)/(c*x-1)*c^4*arccosh(c*x)*
x^5+7/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d/(c*x+1)/(c*x-1)*c^2*arccosh(c*x)*x^3-5/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d/(c*
x+1)/(c*x-1)*arccosh(c*x)*x+17/128*b*(-d*(c^2*x^2-1))^(1/2)*f*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)/c-1/5*b*(-d*(c^2*x
^2-1))^(1/2)*g*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^6+3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)*c^2/(c*x-1)*arc
cosh(c*x)*x^4-3/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x^2+1/25*b*(-d*(c^2*x^2-1))^(1/2)*
g*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^5-2/15*b*(-d*(c^2*x^2-1))^(1/2)*g*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^3-3/
16*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*f*arccosh(c*x)^2*d+1/5*b*(-d*(c^2*x^2-1))^(1/2)*g*d/
(c*x+1)/c^2/(c*x-1)*arccosh(c*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a f - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a g}{5 \, c^{2} d} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b f \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*arcsin(c*x)/c)*a*f - 1/5*(-c^2*d*x^2
+ d)^(5/2)*a*g/(c^2*d) + integrate((-c^2*d*x^2 + d)^(3/2)*b*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (-c^2
*d*x^2 + d)^(3/2)*b*f*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))*(f + g*x), x)

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