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

Optimal. Leaf size=156 \[ -\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d}-\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}} \]

[Out]

(-2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*
Sqrt[d - c^2*d*x^2]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x]))/(3*c^2*d)

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Rubi [A]  time = 0.49645, antiderivative size = 172, normalized size of antiderivative = 1.1, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {5798, 5759, 5718, 8, 30} \[ -\frac{x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*
Sqrt[d - c^2*d*x^2]) - (2*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(3*c^4*Sqrt[d - c^2*d*x^2]) - (x^2*(1 - c*
x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(3*c^2*Sqrt[d - c^2*d*x^2])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(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*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, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

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 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 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]

Rubi steps

\begin{align*} \int \frac{x^3 \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{x^3 \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{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 \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}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^2 \, dx}{3 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int 1 \, dx}{3 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.212323, size = 113, normalized size = 0.72 \[ \frac{\sqrt{d-c^2 d x^2} \left (-3 a \left (c^4 x^4+c^2 x^2-2\right )+b c x \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2+6\right )-3 b \left (c^4 x^4+c^2 x^2-2\right ) \cosh ^{-1}(c x)\right )}{9 c^4 d (c x-1) (c x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.234, size = 382, normalized size = 2.5 \begin{align*} a \left ( -{\frac{{x}^{2}}{3\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{2}{3\,d{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}} \right ) +b \left ( -{\frac{-1+3\,{\rm arccosh} \left (cx\right )}{72\,d{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 4\,{c}^{4}{x}^{4}-5\,{c}^{2}{x}^{2}+4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}-3\,\sqrt{cx+1}\sqrt{cx-1}xc+1 \right ) }-{\frac{-3+3\,{\rm arccosh} \left (cx\right )}{8\,d{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{3+3\,{\rm arccosh} \left (cx\right )}{8\,d{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{1+3\,{\rm arccosh} \left (cx\right )}{72\,d{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( -4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}+4\,{c}^{4}{x}^{4}+3\,\sqrt{cx+1}\sqrt{cx-1}xc-5\,{c}^{2}{x}^{2}+1 \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)

[Out]

a*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+b*(-1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*
x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))/c
^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))/c^4/
d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))/c^4/d/(
c^2*x^2-1)-1/72*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-
1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))/c^4/d/(c^2*x^2-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.46026, size = 308, normalized size = 1.97 \begin{align*} -\frac{3 \,{\left (b c^{4} x^{4} + b c^{2} x^{2} - 2 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (b c^{3} x^{3} + 6 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 3 \,{\left (a c^{4} x^{4} + a c^{2} x^{2} - 2 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{9 \,{\left (c^{6} d x^{2} - c^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

-1/9*(3*(b*c^4*x^4 + b*c^2*x^2 - 2*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^3*x^3 + 6*b*c*x
)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 3*(a*c^4*x^4 + a*c^2*x^2 - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^6*d*x^2 -
c^4*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{3}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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