Integrand size = 18, antiderivative size = 131 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=-\frac {2 c^2 (1-a x)^4}{a \sqrt {1-a^2 x^2}}-\frac {35 c^2 \sqrt {1-a^2 x^2}}{2 a}-\frac {35 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}-\frac {7 c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {35 c^2 \arcsin (a x)}{2 a} \]
-35/2*c^2*arcsin(a*x)/a-2*c^2*(-a*x+1)^4/a/(-a^2*x^2+1)^(1/2)-35/2*c^2*(-a ^2*x^2+1)^(1/2)/a-35/6*c^2*(-a*x+1)*(-a^2*x^2+1)^(1/2)/a-7/3*c^2*(-a*x+1)^ 2*(-a^2*x^2+1)^(1/2)/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.34 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=-\frac {c^2 (1-a x)^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{2},\frac {11}{2},\frac {1}{2} (1-a x)\right )}{9 \sqrt {2} a} \]
Time = 0.39 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6677, 27, 462, 2346, 27, 2346, 27, 455, 223}
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.
| \(\displaystyle \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx\) |
| \(\Big \downarrow \) 6677 |
| \(\displaystyle \frac {\int \frac {c^5 (1-a x)^5}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int \frac {(1-a x)^5}{\left (1-a^2 x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 462 |
| \(\displaystyle c^2 \left (-\int \frac {-a^3 x^3+5 a^2 x^2-11 a x+15}{\sqrt {1-a^2 x^2}}dx-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle c^2 \left (\frac {\int -\frac {5 \left (3 x^2 a^4-7 x a^3+9 a^2\right )}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \left (-\frac {5 \int \frac {3 x^2 a^4-7 x a^3+9 a^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle c^2 \left (-\frac {5 \left (-\frac {\int -\frac {7 a^4 (3-2 a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}\right )}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \left (-\frac {5 \left (\frac {7}{2} a^2 \int \frac {3-2 a x}{\sqrt {1-a^2 x^2}}dx-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}\right )}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c^2 \left (-\frac {5 \left (\frac {7}{2} a^2 \left (3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {2 \sqrt {1-a^2 x^2}}{a}\right )-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}\right )}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^2 \left (-\frac {5 \left (\frac {7}{2} a^2 \left (\frac {2 \sqrt {1-a^2 x^2}}{a}+\frac {3 \arcsin (a x)}{a}\right )-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}\right )}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}-\frac {16 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
c^2*((-16*(1 - a*x))/(a*Sqrt[1 - a^2*x^2]) - (a*x^2*Sqrt[1 - a^2*x^2])/3 - (5*((-3*a^2*x*Sqrt[1 - a^2*x^2])/2 + (7*a^2*((2*Sqrt[1 - a^2*x^2])/a + (3 *ArcSin[a*x])/a))/2))/(3*a^2))
3.3.18.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp [(-2^(n - 1))*d*c^(n - 2)*((c + d*x)/(b*Sqrt[a + b*x^2])), x] + Simp[d^2/b Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(n - 1)*c^(n - 1) - (c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[n, 2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86
| method | result | size |
| risch | \(\frac {\left (2 a^{2} x^{2}-15 a x +70\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{6 a \sqrt {-a^{2} x^{2}+1}}-\left (\frac {35 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+\frac {16 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{2}\) | \(113\) |
| default | \(c^{2} \left (\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{a}-\frac {4 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}+\frac {-\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-8 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{3}}\right )\) | \(462\) |
1/6*(2*a^2*x^2-15*a*x+70)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^2-(35/2/(a^2) ^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+16/a^2/(x+1/a)*(-a^2*(x+1/ a)^2+2*a*(x+1/a))^(1/2))*c^2
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=-\frac {166 \, a c^{2} x + 166 \, c^{2} - 210 \, {\left (a c^{2} x + c^{2}\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} c^{2} x^{3} - 13 \, a^{2} c^{2} x^{2} + 55 \, a c^{2} x + 166 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} x + a\right )}} \]
-1/6*(166*a*c^2*x + 166*c^2 - 210*(a*c^2*x + c^2)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3*c^2*x^3 - 13*a^2*c^2*x^2 + 55*a*c^2*x + 166*c^2)*s qrt(-a^2*x^2 + 1))/(a^2*x + a)
\[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=c^{2} \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx\right ) \]
c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a* x + 1), x) + Integral(2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2 *x**2 + 3*a*x + 1), x) + Integral(-a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x* *3 + 3*a**2*x**2 + 3*a*x + 1), x))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.50 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=\frac {1}{2} \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2} x + \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{a^{2} x + a} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3 \, a} - \frac {i \, c^{2} \arcsin \left (a x + 2\right )}{2 \, a} - \frac {18 \, c^{2} \arcsin \left (a x\right )}{a} - \frac {24 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2} x + a} + \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2}}{a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a} \]
1/2*sqrt(a^2*x^2 + 4*a*x + 3)*c^2*x + 4*(-a^2*x^2 + 1)^(3/2)*c^2/(a^3*x^2 + 2*a^2*x + a) - 2*(-a^2*x^2 + 1)^(3/2)*c^2/(a^2*x + a) + 1/3*(-a^2*x^2 + 1)^(3/2)*c^2/a - 1/2*I*c^2*arcsin(a*x + 2)/a - 18*c^2*arcsin(a*x)/a - 24*s qrt(-a^2*x^2 + 1)*c^2/(a^2*x + a) + sqrt(a^2*x^2 + 4*a*x + 3)*c^2/a - 6*sq rt(-a^2*x^2 + 1)*c^2/a
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.69 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=-\frac {35 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c^{2} x - 15 \, c^{2}\right )} x + \frac {70 \, c^{2}}{a}\right )} + \frac {32 \, c^{2}}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
-35/2*c^2*arcsin(a*x)*sgn(a)/abs(a) - 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c^2*x - 15*c^2)*x + 70*c^2/a) + 32*c^2/(((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))
Time = 3.91 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.15 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^2 \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {11\,a\,c^2}{\sqrt {-a^2}}+\frac {5\,c^2\,x\,\sqrt {-a^2}}{2}-\frac {2\,a^3\,c^2}{3\,{\left (-a^2\right )}^{3/2}}-\frac {a^5\,c^2\,x^2}{3\,{\left (-a^2\right )}^{3/2}}\right )}{\sqrt {-a^2}}-\frac {35\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {16\,c^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]