Integrand size = 14, antiderivative size = 317 \[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{1+a x}}+\frac {475 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}+\frac {17 x^2 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^2}+\frac {(521-452 a x) (1-a x)^{5/4} (1+a x)^{3/4}}{96 a^4}+\frac {475 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4} \]
-4*x^3*(-a*x+1)^(5/4)/a/(a*x+1)^(1/4)+475/64*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/ a^4+17/4*x^2*(-a*x+1)^(5/4)*(a*x+1)^(3/4)/a^2+1/96*(-452*a*x+521)*(-a*x+1) ^(5/4)*(a*x+1)^(3/4)/a^4-475/128*arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^ (1/4))/a^4*2^(1/2)-475/128*arctan(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/ a^4*2^(1/2)+475/256*ln(1-(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/ 2)/(a*x+1)^(1/2))/a^4*2^(1/2)-475/256*ln(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^ (1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^4*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.25 \[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=-\frac {(1-a x)^{9/4} \left (3 \left (-59-5 a x+6 a^2 x^2\right )+95\ 2^{3/4} \sqrt [4]{1+a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-a x)\right )\right )}{72 a^4 \sqrt [4]{1+a x}} \]
-1/72*((1 - a*x)^(9/4)*(3*(-59 - 5*a*x + 6*a^2*x^2) + 95*2^(3/4)*(1 + a*x) ^(1/4)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - a*x)/2]))/(a^4*(1 + a*x)^(1/ 4))
Time = 0.52 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.99, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {6676, 108, 27, 170, 27, 164, 60, 73, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 e^{-\frac {5}{2} \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6676 |
| \(\displaystyle \int \frac {x^3 (1-a x)^{5/4}}{(a x+1)^{5/4}}dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {4 \int \frac {x^2 (12-17 a x) \sqrt [4]{1-a x}}{4 \sqrt [4]{a x+1}}dx}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 (12-17 a x) \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}dx}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {\int \frac {a x (68-113 a x) \sqrt [4]{1-a x}}{2 \sqrt [4]{a x+1}}dx}{4 a^2}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {\int \frac {x (68-113 a x) \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}dx}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}dx}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {1}{2} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{a x+1}}dx+\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \int \frac {1}{\sqrt [4]{a x+1}}d\sqrt [4]{1-a x}}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \int \frac {1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\right )}{8 a}-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {17 x^2 (1-a x)^{5/4} (a x+1)^{3/4}}{4 a}-\frac {-\frac {(521-452 a x) (a x+1)^{3/4} (1-a x)^{5/4}}{12 a^2}-\frac {475 \left (\frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\right )}{8 a}}{8 a}}{a}-\frac {4 x^3 (1-a x)^{5/4}}{a \sqrt [4]{a x+1}}\) |
(-4*x^3*(1 - a*x)^(5/4))/(a*(1 + a*x)^(1/4)) + ((17*x^2*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(4*a) - (-1/12*((521 - 452*a*x)*(1 - a*x)^(5/4)*(1 + a*x)^( 3/4))/a^2 - (475*(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a - (2*((-(ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]* (1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[1 - a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - a *x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/(8*a ))/(8*a))/a
3.2.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> 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] + Simp[(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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) ^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] && !Int egerQ[(n - 1)/2]
\[\int \frac {x^{3}}{{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}}d x\]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.97 \[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\frac {1425 \, {\left (a^{5} x + a^{4}\right )} \left (-\frac {1}{a^{16}}\right )^{\frac {1}{4}} \log \left (a^{12} \left (-\frac {1}{a^{16}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 1425 \, {\left (i \, a^{5} x + i \, a^{4}\right )} \left (-\frac {1}{a^{16}}\right )^{\frac {1}{4}} \log \left (i \, a^{12} \left (-\frac {1}{a^{16}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 1425 \, {\left (-i \, a^{5} x - i \, a^{4}\right )} \left (-\frac {1}{a^{16}}\right )^{\frac {1}{4}} \log \left (-i \, a^{12} \left (-\frac {1}{a^{16}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 1425 \, {\left (a^{5} x + a^{4}\right )} \left (-\frac {1}{a^{16}}\right )^{\frac {1}{4}} \log \left (-a^{12} \left (-\frac {1}{a^{16}}\right )^{\frac {3}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 2 \, {\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 226 \, a^{2} x^{2} - 521 \, a x - 2467\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{384 \, {\left (a^{5} x + a^{4}\right )}} \]
1/384*(1425*(a^5*x + a^4)*(-1/a^16)^(1/4)*log(a^12*(-1/a^16)^(3/4) + sqrt( -sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 1425*(I*a^5*x + I*a^4)*(-1/a^16)^(1/4)*l og(I*a^12*(-1/a^16)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 1425*(- I*a^5*x - I*a^4)*(-1/a^16)^(1/4)*log(-I*a^12*(-1/a^16)^(3/4) + sqrt(-sqrt( -a^2*x^2 + 1)/(a*x - 1))) - 1425*(a^5*x + a^4)*(-1/a^16)^(1/4)*log(-a^12*( -1/a^16)^(3/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 2*(48*a^4*x^4 - 13 6*a^3*x^3 + 226*a^2*x^2 - 521*a*x - 2467)*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a ^2*x^2 + 1)/(a*x - 1)))/(a^5*x + a^4)
\[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\int \frac {x^{3}}{\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\, dx \]
\[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\int { \frac {x^{3}}{\left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{-\frac {5}{2} \text {arctanh}(a x)} x^3 \, dx=\int \frac {x^3}{{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}} \,d x \]