3.275 \(\int e^{\cosh ^{-1}(a+b x)} x^3 \, dx\)

Optimal. Leaf size=165 \[ -\frac{\left (4 a^2+3\right ) a e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac{\left (4 a^2+3\right ) a \cosh ^{-1}(a+b x)}{8 b^4}+\frac{\left (6 a^2+1\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}+\frac{\left (6 a^2+1\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac{3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}-\frac{3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac{e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}+\frac{e^{5 \cosh ^{-1}(a+b x)}}{80 b^4} \]

[Out]

1/(48*b^4*E^(3*ArcCosh[a + b*x])) - (3*a)/(16*b^4*E^(2*ArcCosh[a + b*x])) + (1 + 6*a^2)/(8*b^4*E^ArcCosh[a + b
*x]) - (a*(3 + 4*a^2)*E^(2*ArcCosh[a + b*x]))/(16*b^4) + ((1 + 6*a^2)*E^(3*ArcCosh[a + b*x]))/(24*b^4) - (3*a*
E^(4*ArcCosh[a + b*x]))/(32*b^4) + E^(5*ArcCosh[a + b*x])/(80*b^4) + (a*(3 + 4*a^2)*ArcCosh[a + b*x])/(8*b^4)

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Rubi [A]  time = 0.16092, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5899, 2282, 12, 1628} \[ -\frac{\left (4 a^2+3\right ) a e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac{\left (4 a^2+3\right ) a \cosh ^{-1}(a+b x)}{8 b^4}+\frac{\left (6 a^2+1\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}+\frac{\left (6 a^2+1\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac{3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}-\frac{3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac{e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}+\frac{e^{5 \cosh ^{-1}(a+b x)}}{80 b^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x]*x^3,x]

[Out]

1/(48*b^4*E^(3*ArcCosh[a + b*x])) - (3*a)/(16*b^4*E^(2*ArcCosh[a + b*x])) + (1 + 6*a^2)/(8*b^4*E^ArcCosh[a + b
*x]) - (a*(3 + 4*a^2)*E^(2*ArcCosh[a + b*x]))/(16*b^4) + ((1 + 6*a^2)*E^(3*ArcCosh[a + b*x]))/(24*b^4) - (3*a*
E^(4*ArcCosh[a + b*x]))/(32*b^4) + E^(5*ArcCosh[a + b*x])/(80*b^4) + (a*(3 + 4*a^2)*ArcCosh[a + b*x])/(8*b^4)

Rule 5899

Int[(f_)^(ArcCosh[(a_.) + (b_.)*(x_)]^(n_.)*(c_.))*(x_)^(m_.), x_Symbol] :> Dist[1/b, Subst[Int[(-(a/b) + Cosh
[x]/b)^m*f^(c*x^n)*Sinh[x], x], x, ArcCosh[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int e^{\cosh ^{-1}(a+b x)} x^3 \, dx &=\frac{\operatorname{Subst}\left (\int e^x \left (-\frac{a}{b}+\frac{\cosh (x)}{b}\right )^3 \sinh (x) \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right ) \left (1-2 a x+x^2\right )^3}{16 b^3 x^4} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right ) \left (1-2 a x+x^2\right )^3}{x^4} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{16 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{x^4}+\frac{6 a}{x^3}-\frac{2 \left (1+6 a^2\right )}{x^2}+\frac{2 a \left (3+4 a^2\right )}{x}-2 a \left (3+4 a^2\right ) x+2 \left (1+6 a^2\right ) x^2-6 a x^3+x^4\right ) \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{16 b^4}\\ &=\frac{e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}-\frac{3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac{\left (1+6 a^2\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}-\frac{a \left (3+4 a^2\right ) e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac{\left (1+6 a^2\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac{3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac{e^{5 \cosh ^{-1}(a+b x)}}{80 b^4}+\frac{a \left (3+4 a^2\right ) \cosh ^{-1}(a+b x)}{8 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0885515, size = 138, normalized size = 0.84 \[ \frac{\sqrt{a+b x-1} \sqrt{a+b x+1} \left (-2 \left (3 a^2+4\right ) b^2 x^2+\left (6 a^2+29\right ) a b x-6 a^4-83 a^2+6 a b^3 x^3+24 b^4 x^4-16\right )+15 a \left (4 a^2+3\right ) \log \left (\sqrt{a+b x-1} \sqrt{a+b x+1}+a+b x\right )+30 a b^4 x^4+24 b^5 x^5}{120 b^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCosh[a + b*x]*x^3,x]

[Out]

(30*a*b^4*x^4 + 24*b^5*x^5 + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-16 - 83*a^2 - 6*a^4 + a*(29 + 6*a^2)*b*x -
2*(4 + 3*a^2)*b^2*x^2 + 6*a*b^3*x^3 + 24*b^4*x^4) + 15*a*(3 + 4*a^2)*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1
+ a + b*x]])/(120*b^4)

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Maple [C]  time = 0.047, size = 376, normalized size = 2.3 \begin{align*}{\frac{{\it csgn} \left ( b \right ) }{120\,{b}^{4}}\sqrt{bx+a-1}\sqrt{bx+a+1} \left ( 24\,{\it csgn} \left ( b \right ){x}^{4}{b}^{4}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+6\,{\it csgn} \left ( b \right ){x}^{3}a{b}^{3}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}-6\,{\it csgn} \left ( b \right ){x}^{2}{a}^{2}{b}^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}+6\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ) x{a}^{3}b-8\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ){x}^{2}{b}^{2}-6\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ){a}^{4}+29\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ) xab-83\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ){a}^{2}+60\,\ln \left ( \left ( \sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ) +bx+a \right ){\it csgn} \left ( b \right ) \right ){a}^{3}-16\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ) +45\,\ln \left ( \left ( \sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}{\it csgn} \left ( b \right ) +bx+a \right ){\it csgn} \left ( b \right ) \right ) a \right ){\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}-1}}}}+{\frac{b{x}^{5}}{5}}+{\frac{{x}^{4}a}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(24*csgn(b)*x^4*b^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+6*csgn(b)*x^3*a*b^3*(b
^2*x^2+2*a*b*x+a^2-1)^(1/2)-6*csgn(b)*x^2*a^2*b^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2
)*csgn(b)*x*a^3*b-8*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*x^2*b^2-6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*a^4+
29*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*x*a*b-83*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)*a^2+60*ln(((b^2*x^2+2*
a*b*x+a^2-1)^(1/2)*csgn(b)+b*x+a)*csgn(b))*a^3-16*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*csgn(b)+45*ln(((b^2*x^2+2*a*b*
x+a^2-1)^(1/2)*csgn(b)+b*x+a)*csgn(b))*a)*csgn(b)/b^4/(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+1/5*b*x^5+1/4*x^4*a

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89125, size = 327, normalized size = 1.98 \begin{align*} \frac{24 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} +{\left (24 \, b^{4} x^{4} + 6 \, a b^{3} x^{3} - 2 \,{\left (3 \, a^{2} + 4\right )} b^{2} x^{2} - 6 \, a^{4} +{\left (6 \, a^{3} + 29 \, a\right )} b x - 83 \, a^{2} - 16\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} - 15 \,{\left (4 \, a^{3} + 3 \, a\right )} \log \left (-b x + \sqrt{b x + a + 1} \sqrt{b x + a - 1} - a\right )}{120 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(24*b^5*x^5 + 30*a*b^4*x^4 + (24*b^4*x^4 + 6*a*b^3*x^3 - 2*(3*a^2 + 4)*b^2*x^2 - 6*a^4 + (6*a^3 + 29*a)*
b*x - 83*a^2 - 16)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - 15*(4*a^3 + 3*a)*log(-b*x + sqrt(b*x + a + 1)*sqrt(b*
x + a - 1) - a))/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1)), x)

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Giac [A]  time = 1.27554, size = 348, normalized size = 2.11 \begin{align*} \frac{{\left ({\left (2 \,{\left (b x + a + 1\right )}{\left (3 \,{\left (b x + a + 1\right )}{\left (\frac{4 \,{\left (b x + a + 1\right )}}{b^{3}} - \frac{15 \, a b^{12} + 16 \, b^{12}}{b^{15}}\right )} + \frac{60 \, a^{2} b^{12} + 135 \, a b^{12} + 68 \, b^{12}}{b^{15}}\right )} - \frac{5 \,{\left (12 \, a^{3} b^{12} + 48 \, a^{2} b^{12} + 45 \, a b^{12} + 16 \, b^{12}\right )}}{b^{15}}\right )}{\left (b x + a + 1\right )} + \frac{15 \,{\left (4 \, a^{3} b^{12} + 3 \, a b^{12}\right )}}{b^{15}}\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} + 30 \,{\left (b x^{4} - \frac{a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1}{b^{3}}\right )} a + 24 \,{\left (b x^{5} + \frac{a^{5} + 5 \, a^{4} + 10 \, a^{3} + 10 \, a^{2} + 5 \, a + 1}{b^{4}}\right )} b - \frac{30 \,{\left (4 \, a^{3} + 3 \, a\right )} \log \left ({\left | -\sqrt{b x + a + 1} + \sqrt{b x + a - 1} \right |}\right )}{b^{3}}}{120 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(((2*(b*x + a + 1)*(3*(b*x + a + 1)*(4*(b*x + a + 1)/b^3 - (15*a*b^12 + 16*b^12)/b^15) + (60*a^2*b^12 +
135*a*b^12 + 68*b^12)/b^15) - 5*(12*a^3*b^12 + 48*a^2*b^12 + 45*a*b^12 + 16*b^12)/b^15)*(b*x + a + 1) + 15*(4*
a^3*b^12 + 3*a*b^12)/b^15)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 30*(b*x^4 - (a^4 + 4*a^3 + 6*a^2 + 4*a + 1)/b
^3)*a + 24*(b*x^5 + (a^5 + 5*a^4 + 10*a^3 + 10*a^2 + 5*a + 1)/b^4)*b - 30*(4*a^3 + 3*a)*log(abs(-sqrt(b*x + a
+ 1) + sqrt(b*x + a - 1)))/b^3)/b