3.876 \(\int e^{n \tanh ^{-1}(a+b x)} x^3 \, dx\)

Optimal. Leaf size=206 \[ \frac{2^{\frac{n}{2}-2} \left (-36 a^2 n+24 a^3+12 a \left (n^2+2\right )-n \left (n^2+8\right )\right ) (-a-b x+1)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (-a-b x+1)\right )}{3 b^4 (2-n)}-\frac{(a+b x+1)^{\frac{n+2}{2}} \left (18 a^2-2 b x (6 a-n)-10 a n+n^2+6\right ) (-a-b x+1)^{1-\frac{n}{2}}}{24 b^4}-\frac{x^2 (a+b x+1)^{\frac{n+2}{2}} (-a-b x+1)^{1-\frac{n}{2}}}{4 b^2} \]

[Out]

-(x^2*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(4*b^2) - ((1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2
+ n)/2)*(6 + 18*a^2 - 10*a*n + n^2 - 2*b*(6*a - n)*x))/(24*b^4) + (2^(-2 + n/2)*(24*a^3 - 36*a^2*n + 12*a*(2 +
n^2) - n*(8 + n^2))*(1 - a - b*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -n/2, 2 - n/2, (1 - a - b*x)/2])/(3*b^
4*(2 - n))

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Rubi [A]  time = 0.182662, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6163, 100, 147, 69} \[ \frac{2^{\frac{n}{2}-2} \left (-36 a^2 n+24 a^3+12 a \left (n^2+2\right )-n \left (n^2+8\right )\right ) (-a-b x+1)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (-a-b x+1)\right )}{3 b^4 (2-n)}-\frac{(a+b x+1)^{\frac{n+2}{2}} \left (18 a^2-2 b x (6 a-n)-10 a n+n^2+6\right ) (-a-b x+1)^{1-\frac{n}{2}}}{24 b^4}-\frac{x^2 (a+b x+1)^{\frac{n+2}{2}} (-a-b x+1)^{1-\frac{n}{2}}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a + b*x])*x^3,x]

[Out]

-(x^2*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(4*b^2) - ((1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2
+ n)/2)*(6 + 18*a^2 - 10*a*n + n^2 - 2*b*(6*a - n)*x))/(24*b^4) + (2^(-2 + n/2)*(24*a^3 - 36*a^2*n + 12*a*(2 +
n^2) - n*(8 + n^2))*(1 - a - b*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -n/2, 2 - n/2, (1 - a - b*x)/2])/(3*b^
4*(2 - n))

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rule 100

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int e^{n \tanh ^{-1}(a+b x)} x^3 \, dx &=\int x^3 (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx\\ &=-\frac{x^2 (1-a-b x)^{1-\frac{n}{2}} (1+a+b x)^{\frac{2+n}{2}}}{4 b^2}-\frac{\int x (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \left (-2 \left (1-a^2\right )+b (6 a-n) x\right ) \, dx}{4 b^2}\\ &=-\frac{x^2 (1-a-b x)^{1-\frac{n}{2}} (1+a+b x)^{\frac{2+n}{2}}}{4 b^2}-\frac{(1-a-b x)^{1-\frac{n}{2}} (1+a+b x)^{\frac{2+n}{2}} \left (6+18 a^2-10 a n+n^2-2 b (6 a-n) x\right )}{24 b^4}-\frac{\left (24 a^3-36 a^2 n+12 a \left (2+n^2\right )-n \left (8+n^2\right )\right ) \int (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx}{24 b^3}\\ &=-\frac{x^2 (1-a-b x)^{1-\frac{n}{2}} (1+a+b x)^{\frac{2+n}{2}}}{4 b^2}-\frac{(1-a-b x)^{1-\frac{n}{2}} (1+a+b x)^{\frac{2+n}{2}} \left (6+18 a^2-10 a n+n^2-2 b (6 a-n) x\right )}{24 b^4}+\frac{2^{-2+\frac{n}{2}} \left (24 a^3-36 a^2 n+12 a \left (2+n^2\right )-n \left (8+n^2\right )\right ) (1-a-b x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a-b x)\right )}{3 b^4 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.251091, size = 220, normalized size = 1.07 \[ \frac{(-a-b x+1)^{1-\frac{n}{2}} \left (-2^{\frac{n}{2}+3} (n-6 a) \text{Hypergeometric2F1}\left (-\frac{n}{2}-2,1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (-a-b x+1)\right )-(a+1) 2^{\frac{n}{2}+3} (5 a-n+1) \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (-a-b x+1)\right )+(a+1)^2 2^{\frac{n}{2}+1} (4 a-n+2) \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (-a-b x+1)\right )+b^2 (n-2) x^2 (a+b x+1)^{\frac{n}{2}+1}\right )}{4 b^4 (2-n)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcTanh[a + b*x])*x^3,x]

[Out]

((1 - a - b*x)^(1 - n/2)*(b^2*(-2 + n)*x^2*(1 + a + b*x)^(1 + n/2) - 2^(3 + n/2)*(-6*a + n)*Hypergeometric2F1[
-2 - n/2, 1 - n/2, 2 - n/2, (1 - a - b*x)/2] - 2^(3 + n/2)*(1 + a)*(1 + 5*a - n)*Hypergeometric2F1[-1 - n/2, 1
- n/2, 2 - n/2, (1 - a - b*x)/2] + 2^(1 + n/2)*(1 + a)^2*(2 + 4*a - n)*Hypergeometric2F1[1 - n/2, -n/2, 2 - n
/2, (1 - a - b*x)/2]))/(4*b^4*(2 - n))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( bx+a \right ) }}{x}^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(b*x+a))*x^3,x)

[Out]

int(exp(n*arctanh(b*x+a))*x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (\frac{b x + a + 1}{b x + a - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(b*x+a))*x^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \left (\frac{b x + a + 1}{b x + a - 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(b*x+a))*x^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(b*x+a))*x**3,x)

[Out]

Integral(x**3*exp(n*atanh(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (\frac{b x + a + 1}{b x + a - 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(b*x+a))*x^3,x, algorithm="giac")

[Out]

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