Integrand size = 23, antiderivative size = 153 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=-\frac {34 c \sqrt {1-a^2 x^2}}{15 a^4}-\frac {23 c x \sqrt {1-a^2 x^2}}{16 a^3}-\frac {17 c x^2 \sqrt {1-a^2 x^2}}{15 a^2}-\frac {23 c x^3 \sqrt {1-a^2 x^2}}{24 a}-\frac {3}{5} c x^4 \sqrt {1-a^2 x^2}-\frac {1}{6} a c x^5 \sqrt {1-a^2 x^2}+\frac {23 c \arcsin (a x)}{16 a^4} \] Output:
-34/15*c*(-a^2*x^2+1)^(1/2)/a^4-23/16*c*x*(-a^2*x^2+1)^(1/2)/a^3-17/15*c*x ^2*(-a^2*x^2+1)^(1/2)/a^2-23/24*c*x^3*(-a^2*x^2+1)^(1/2)/a-3/5*c*x^4*(-a^2 *x^2+1)^(1/2)-1/6*a*c*x^5*(-a^2*x^2+1)^(1/2)+23/16*c*arcsin(a*x)/a^4
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.46 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=\frac {-c \sqrt {1-a^2 x^2} \left (544+345 a x+272 a^2 x^2+230 a^3 x^3+144 a^4 x^4+40 a^5 x^5\right )+345 c \arcsin (a x)}{240 a^4} \] Input:
Integrate[E^(3*ArcTanh[a*x])*x^3*(c - a^2*c*x^2),x]
Output:
(-(c*Sqrt[1 - a^2*x^2]*(544 + 345*a*x + 272*a^2*x^2 + 230*a^3*x^3 + 144*a^ 4*x^4 + 40*a^5*x^5)) + 345*c*ArcSin[a*x])/(240*a^4)
Time = 0.86 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6698, 541, 25, 2340, 25, 27, 533, 27, 533, 27, 533, 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.
| \(\displaystyle \int x^3 e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle c \int \frac {x^3 (a x+1)^3}{\sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c \left (-\frac {\int -\frac {x^3 \left (18 x^2 a^4+23 x a^3+6 a^2\right )}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\int \frac {x^3 \left (18 x^2 a^4+23 x a^3+6 a^2\right )}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle c \left (\frac {-\frac {\int -\frac {a^4 x^3 (115 a x+102)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\frac {\int \frac {a^4 x^3 (115 a x+102)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \int \frac {x^3 (115 a x+102)}{\sqrt {1-a^2 x^2}}dx-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {\int \frac {3 a x^2 (136 a x+115)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \int \frac {x^2 (136 a x+115)}{\sqrt {1-a^2 x^2}}dx}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \left (\frac {\int \frac {a x (345 a x+272)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {136 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \left (\frac {\int \frac {x (345 a x+272)}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {136 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \left (\frac {\frac {\int \frac {a (544 a x+345)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {345 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {136 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \left (\frac {\frac {\int \frac {544 a x+345}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {345 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {136 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \left (\frac {\frac {345 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {544 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {345 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {136 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c \left (\frac {\frac {1}{5} a^2 \left (\frac {3 \left (\frac {\frac {\frac {345 \arcsin (a x)}{a}-\frac {544 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {345 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}-\frac {136 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )}{4 a}-\frac {115 x^3 \sqrt {1-a^2 x^2}}{4 a}\right )-\frac {18}{5} a^2 x^4 \sqrt {1-a^2 x^2}}{6 a^2}-\frac {1}{6} a x^5 \sqrt {1-a^2 x^2}\right )\) |
Input:
Int[E^(3*ArcTanh[a*x])*x^3*(c - a^2*c*x^2),x]
Output:
c*(-1/6*(a*x^5*Sqrt[1 - a^2*x^2]) + ((-18*a^2*x^4*Sqrt[1 - a^2*x^2])/5 + ( a^2*((-115*x^3*Sqrt[1 - a^2*x^2])/(4*a) + (3*((-136*x^2*Sqrt[1 - a^2*x^2]) /(3*a) + ((-345*x*Sqrt[1 - a^2*x^2])/(2*a) + ((-544*Sqrt[1 - a^2*x^2])/a + (345*ArcSin[a*x])/a)/(2*a))/(3*a)))/(4*a)))/5)/(6*a^2))
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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {\left (40 a^{5} x^{5}+144 a^{4} x^{4}+230 a^{3} x^{3}+272 a^{2} x^{2}+345 a x +544\right ) \left (a^{2} x^{2}-1\right ) c}{240 a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {23 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{16 a^{3} \sqrt {a^{2}}}\) | \(98\) |
| meijerg | \(-\frac {2 c \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {\pi }}+\frac {c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {\pi }}-\frac {c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {9}{2}} \left (-24 x^{6} a^{6}-42 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{144 a^{8} \sqrt {-a^{2} x^{2}+1}}-\frac {35 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {9}{2}} \arcsin \left (a x \right )}{16 a^{9}}\right )}{a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (-8 x^{6} a^{6}-16 a^{4} x^{4}-64 a^{2} x^{2}+128\right )}{40 \sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {\pi }}+\frac {3 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{a^{3} \sqrt {\pi }\, \sqrt {-a^{2}}}\) | \(397\) |
| default | \(-c \left (a^{5} \left (-\frac {x^{7}}{6 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {7 x^{5}}{24 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {7 \left (-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}\right )}{6 a^{2}}}{a^{2}}\right )+\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}-3 a \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-2 a^{2} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+2 a^{3} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )+3 a^{4} \left (-\frac {x^{6}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}}{a^{2}}\right )\right )\) | \(517\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c),x,method=_RETURNVERBOS E)
Output:
1/240*(40*a^5*x^5+144*a^4*x^4+230*a^3*x^3+272*a^2*x^2+345*a*x+544)*(a^2*x^ 2-1)/a^4/(-a^2*x^2+1)^(1/2)*c+23/16/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/( -a^2*x^2+1)^(1/2))*c
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.58 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=-\frac {690 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (40 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} + 230 \, a^{3} c x^{3} + 272 \, a^{2} c x^{2} + 345 \, a c x + 544 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{4}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c),x, algorithm="fr icas")
Output:
-1/240*(690*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (40*a^5*c*x^5 + 144 *a^4*c*x^4 + 230*a^3*c*x^3 + 272*a^2*c*x^2 + 345*a*c*x + 544*c)*sqrt(-a^2* x^2 + 1))/a^4
Time = 5.84 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.02 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a c x^{5}}{6} - \frac {3 c x^{4}}{5} - \frac {23 c x^{3}}{24 a} - \frac {17 c x^{2}}{15 a^{2}} - \frac {23 c x}{16 a^{3}} - \frac {34 c}{15 a^{4}}\right ) + \frac {23 c \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 a^{3} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {a^{3} c x^{7}}{7} + \frac {a^{2} c x^{6}}{2} + \frac {3 a c x^{5}}{5} + \frac {c x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(-a**2*c*x**2+c),x)
Output:
Piecewise((sqrt(-a**2*x**2 + 1)*(-a*c*x**5/6 - 3*c*x**4/5 - 23*c*x**3/(24* a) - 17*c*x**2/(15*a**2) - 23*c*x/(16*a**3) - 34*c/(15*a**4)) + 23*c*log(- 2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(16*a**3*sqrt(-a**2)), Ne(a **2, 0)), (a**3*c*x**7/7 + a**2*c*x**6/2 + 3*a*c*x**5/5 + c*x**4/4, True))
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=\frac {a^{3} c x^{7}}{6 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {19 \, a c x^{5}}{24 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, c x^{4}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {23 \, c x^{3}}{48 \, \sqrt {-a^{2} x^{2} + 1} a} + \frac {17 \, c x^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {23 \, c x}{16 \, \sqrt {-a^{2} x^{2} + 1} a^{3}} + \frac {23 \, c \arcsin \left (a x\right )}{16 \, a^{4}} - \frac {34 \, c}{15 \, \sqrt {-a^{2} x^{2} + 1} a^{4}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c),x, algorithm="ma xima")
Output:
1/6*a^3*c*x^7/sqrt(-a^2*x^2 + 1) + 3/5*a^2*c*x^6/sqrt(-a^2*x^2 + 1) + 19/2 4*a*c*x^5/sqrt(-a^2*x^2 + 1) + 8/15*c*x^4/sqrt(-a^2*x^2 + 1) + 23/48*c*x^3 /(sqrt(-a^2*x^2 + 1)*a) + 17/15*c*x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 23/16*c*x /(sqrt(-a^2*x^2 + 1)*a^3) + 23/16*c*arcsin(a*x)/a^4 - 34/15*c/(sqrt(-a^2*x ^2 + 1)*a^4)
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.51 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, a c x + 18 \, c\right )} x + \frac {115 \, c}{a}\right )} x + \frac {136 \, c}{a^{2}}\right )} x + \frac {345 \, c}{a^{3}}\right )} x + \frac {544 \, c}{a^{4}}\right )} + \frac {23 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, a^{3} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c),x, algorithm="gi ac")
Output:
-1/240*sqrt(-a^2*x^2 + 1)*((2*((4*(5*a*c*x + 18*c)*x + 115*c/a)*x + 136*c/ a^2)*x + 345*c/a^3)*x + 544*c/a^4) + 23/16*c*arcsin(a*x)*sgn(a)/(a^3*abs(a ))
Time = 23.78 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=\frac {23\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a^3\,\sqrt {-a^2}}-\frac {3\,c\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {23\,c\,x^3\,\sqrt {1-a^2\,x^2}}{24\,a}-\frac {17\,c\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a^2}-\frac {23\,c\,x\,\sqrt {1-a^2\,x^2}}{16\,a^3}-\frac {a\,c\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {34\,c\,\sqrt {1-a^2\,x^2}}{15\,a^4} \] Input:
int((x^3*(c - a^2*c*x^2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
(23*c*asinh(x*(-a^2)^(1/2)))/(16*a^3*(-a^2)^(1/2)) - (3*c*x^4*(1 - a^2*x^2 )^(1/2))/5 - (23*c*x^3*(1 - a^2*x^2)^(1/2))/(24*a) - (17*c*x^2*(1 - a^2*x^ 2)^(1/2))/(15*a^2) - (23*c*x*(1 - a^2*x^2)^(1/2))/(16*a^3) - (a*c*x^5*(1 - a^2*x^2)^(1/2))/6 - (34*c*(1 - a^2*x^2)^(1/2))/(15*a^4)
Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.77 \[ \int e^{3 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (345 \mathit {asin} \left (a x \right )-40 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-144 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-230 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-272 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-345 \sqrt {-a^{2} x^{2}+1}\, a x -544 \sqrt {-a^{2} x^{2}+1}+544\right )}{240 a^{4}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(-a^2*c*x^2+c),x)
Output:
(c*(345*asin(a*x) - 40*sqrt( - a**2*x**2 + 1)*a**5*x**5 - 144*sqrt( - a**2 *x**2 + 1)*a**4*x**4 - 230*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 272*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 345*sqrt( - a**2*x**2 + 1)*a*x - 544*sqrt( - a* *2*x**2 + 1) + 544))/(240*a**4)