Integrand size = 19, antiderivative size = 146 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {13 \sqrt {1-a^2 x^2}}{3 a^5 c}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}-\frac {27 \arcsin (a x)}{8 a^5 c} \]
-27/8*arcsin(a*x)/a^5/c+(a*x+1)^2/a^5/c/(-a^2*x^2+1)^(1/2)+13/3*(-a^2*x^2+ 1)^(1/2)/a^5/c+11/8*x*(-a^2*x^2+1)^(1/2)/a^4/c+2/3*x^2*(-a^2*x^2+1)^(1/2)/ a^3/c+1/4*x^3*(-a^2*x^2+1)^(1/2)/a^2/c
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {-\frac {\sqrt {1+a x} \left (-128+47 a x+17 a^2 x^2+10 a^3 x^3+6 a^4 x^4\right )}{\sqrt {1-a x}}+162 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{24 a^5 c} \]
(-((Sqrt[1 + a*x]*(-128 + 47*a*x + 17*a^2*x^2 + 10*a^3*x^3 + 6*a^4*x^4))/S qrt[1 - a*x]) + 162*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])/(24*a^5*c)
Time = 0.54 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {6678, 27, 563, 25, 2346, 25, 2346, 25, 2346, 25, 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.
\(\displaystyle \int \frac {x^4 e^{\text {arctanh}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {x^4 \sqrt {1-a^2 x^2}}{c^2 (1-a x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {x^4 \sqrt {1-a^2 x^2}}{(1-a x)^2}dx}{c}\) |
\(\Big \downarrow \) 563 |
\(\displaystyle \frac {\frac {\int -\frac {a^4 x^4+2 a^3 x^3+2 a^2 x^2+2 a x+2}{\sqrt {1-a^2 x^2}}dx}{a^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\int \frac {a^4 x^4+2 a^3 x^3+2 a^2 x^2+2 a x+2}{\sqrt {1-a^2 x^2}}dx}{a^4}}{c}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {-\frac {\int -\frac {8 x^3 a^5+11 x^2 a^4+8 x a^3+8 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\int \frac {8 x^3 a^5+11 x^2 a^4+8 x a^3+8 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {-\frac {\int -\frac {33 x^2 a^6+40 x a^5+24 a^4}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\int \frac {33 x^2 a^6+40 x a^5+24 a^4}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {-\frac {\int -\frac {a^6 (80 a x+81)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {\int \frac {a^6 (80 a x+81)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {1}{2} a^4 \int \frac {80 a x+81}{\sqrt {1-a^2 x^2}}dx-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {1}{2} a^4 \left (81 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {80 \sqrt {1-a^2 x^2}}{a}\right )-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {1}{2} a^4 \left (\frac {81 \arcsin (a x)}{a}-\frac {80 \sqrt {1-a^2 x^2}}{a}\right )-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
((2*Sqrt[1 - a^2*x^2])/(a^5*(1 - a*x)) - (-1/4*(a^2*x^3*Sqrt[1 - a^2*x^2]) + ((-8*a^3*x^2*Sqrt[1 - a^2*x^2])/3 + ((-33*a^4*x*Sqrt[1 - a^2*x^2])/2 + (a^4*((-80*Sqrt[1 - a^2*x^2])/a + (81*ArcSin[a*x])/a))/2)/(3*a^2))/(4*a^2) )/a^4)/c
3.4.27.3.1 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[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-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[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
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_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.15 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\left (6 a^{3} x^{3}+16 a^{2} x^{2}+33 a x +80\right ) \left (a^{2} x^{2}-1\right )}{24 a^{5} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {27 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\) | \(130\) |
default | \(-\frac {-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {11 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {-\frac {2 x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {4 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\) | \(212\) |
-1/24*(6*a^3*x^3+16*a^2*x^2+33*a*x+80)*(a^2*x^2-1)/a^5/(-a^2*x^2+1)^(1/2)/ c-(27/8/a^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/a^6/(x- 1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2))/c
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {128 \, a x + 162 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (6 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 47 \, a x - 128\right )} \sqrt {-a^{2} x^{2} + 1} - 128}{24 \, {\left (a^{6} c x - a^{5} c\right )}} \]
1/24*(128*a*x + 162*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (6* a^4*x^4 + 10*a^3*x^3 + 17*a^2*x^2 + 47*a*x - 128)*sqrt(-a^2*x^2 + 1) - 128 )/(a^6*c*x - a^5*c)
\[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=- \frac {\int \frac {x^{4}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
-(Integral(x**4/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + In tegral(a*x**5/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{4 \, a^{2} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c x - a^{5} c} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{3} c} + \frac {11 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{4} c} - \frac {27 \, \arcsin \left (a x\right )}{8 \, a^{5} c} + \frac {10 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} c} \]
1/4*sqrt(-a^2*x^2 + 1)*x^3/(a^2*c) - 2*sqrt(-a^2*x^2 + 1)/(a^6*c*x - a^5*c ) + 2/3*sqrt(-a^2*x^2 + 1)*x^2/(a^3*c) + 11/8*sqrt(-a^2*x^2 + 1)*x/(a^4*c) - 27/8*arcsin(a*x)/(a^5*c) + 10/3*sqrt(-a^2*x^2 + 1)/(a^5*c)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\text {Exception raised: TypeError} \]
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
Time = 3.41 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {10\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c}-\frac {2\,\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}\,\left (a^3\,c\,\sqrt {-a^2}-a^4\,c\,x\,\sqrt {-a^2}\right )}+\frac {11\,x\,\sqrt {1-a^2\,x^2}}{8\,a^4\,c}-\frac {27\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^4\,c\,\sqrt {-a^2}}+\frac {x^3\,\sqrt {1-a^2\,x^2}}{4\,a^2\,c}+\frac {2\,x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c} \]