3.67 \(\int \frac {(f+g x)^2 (a+b \cosh ^{-1}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=288 \[ \frac {f^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}} \]

[Out]

-2*f*g*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)-1/2*g^2*x*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*
x))/c^2/(-c^2*d*x^2+d)^(1/2)-2*b*f*g*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-1/4*b*g^2*x^2*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/2*f^2*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(-c^
2*d*x^2+d)^(1/2)+1/4*g^2*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c^3/(-c^2*d*x^2+d)^(1/2)

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Rubi [A]  time = 0.91, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5836, 5822, 5676, 5718, 8, 5759, 30} \[ \frac {f^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b f g x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*f*g*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2]) - (b*g^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(
4*c*Sqrt[d - c^2*d*x^2]) - (2*f*g*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x
*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) + (f^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a
+ b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) + (g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(
4*b*c^3*Sqrt[d - c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

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 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 5759

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

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

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Mathematica [A]  time = 1.59, size = 267, normalized size = 0.93 \[ \frac {-\frac {4 a \left (2 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{\sqrt {d}}-\frac {4 a c g \sqrt {d-c^2 d x^2} (4 f+g x)}{d}+\frac {4 b c^2 f^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{\sqrt {d-c^2 d x^2}}+\frac {16 b c f g \sqrt {d-c^2 d x^2} \left (\frac {c x}{\sqrt {c x-1} \sqrt {c x+1}}-\cosh ^{-1}(c x)\right )}{d}+\frac {b g^2 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+\sinh \left (2 \cosh ^{-1}(c x)\right )\right )-\cosh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt {d-c^2 d x^2}}}{8 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )}}{c^{2} d x^{2} - d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.82, size = 559, normalized size = 1.94 \[ -\frac {a \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}-\frac {2 a f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a \,f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{4 d c \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \,\mathrm {arccosh}\left (c x \right ) x^{2}}{\left (c^{2} x^{2}-1\right ) d}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \sqrt {c x +1}\, \sqrt {c x -1}\, x}{c \left (c^{2} x^{2}-1\right ) d}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2} f^{2}}{2 d c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2} g^{2}}{4 d \,c^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{2 d \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \mathrm {arccosh}\left (c x \right ) x}{2 d \,c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{8 d \,c^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \,\mathrm {arccosh}\left (c x \right )}{c^{2} \left (c^{2} x^{2}-1\right ) d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)

[Out]

-1/2*a*g^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a*g^2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2
))-2*a*f*g/c^2/d*(-c^2*d*x^2+d)^(1/2)+a*f^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/4*b*(
-d*(c^2*x^2-1))^(1/2)*g^2/d/c/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-2*b*(-d*(c^2*x^2-1))^(1/2)*f*g/(c^2*
x^2-1)/d*arccosh(c*x)*x^2+2*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c/(c^2*x^2-1)/d*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x-1/2*b*(
-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c/(c^2*x^2-1)*arccosh(c*x)^2*f^2-1/4*b*(-d*(c^2*x^2-1))^(1
/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/c^3/(c^2*x^2-1)*arccosh(c*x)^2*g^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)*g^2/d/(c^2*x
^2-1)*arccosh(c*x)*x^3+1/2*b*(-d*(c^2*x^2-1))^(1/2)*g^2/d/c^2/(c^2*x^2-1)*arccosh(c*x)*x-1/8*b*(-d*(c^2*x^2-1)
)^(1/2)*g^2/d/c^3/(c^2*x^2-1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c^2/(c^2*x^2-1)/d*arc
cosh(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

-1/2*a*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + a*f^2*arcsin(c*x)/(c*sqrt(d)) - 2*sq
rt(-c^2*d*x^2 + d)*a*f*g/(c^2*d) + integrate(b*g^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2
+ d) + 2*b*f*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d) + b*f^2*log(c*x + sqrt(c*x + 1)*s
qrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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