Integrand size = 19, antiderivative size = 88 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=\frac {a c^3 (1-a x) \sqrt {1-a^2 x^2}}{x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-a^3 c^3 \arcsin (a x)-a^3 c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
-1/3*c^3*(-a^2*x^2+1)^(3/2)/x^3-a^3*c^3*arcsin(a*x)-a^3*c^3*arctanh((-a^2* x^2+1)^(1/2))+a*c^3*(-a*x+1)*(-a^2*x^2+1)^(1/2)/x^2
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.77 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=\frac {c^3 \left (-2+6 a x-2 a^2 x^2-6 a^3 x^3+4 a^4 x^4+3 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)+18 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-6 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{6 x^3 \sqrt {1-a^2 x^2}} \]
(c^3*(-2 + 6*a*x - 2*a^2*x^2 - 6*a^3*x^3 + 4*a^4*x^4 + 3*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 18*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/S qrt[2]] - 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(6*x^3* Sqrt[1 - a^2*x^2])
Time = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6678, 27, 540, 27, 537, 27, 538, 223, 243, 73, 221}
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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{x^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^3 \int \frac {(1-a x)^2 \sqrt {1-a^2 x^2}}{x^4}dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle c^3 \left (-\frac {1}{3} \int \frac {3 a (2-a x) \sqrt {1-a^2 x^2}}{x^3}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^3 \left (-a \int \frac {(2-a x) \sqrt {1-a^2 x^2}}{x^3}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle c^3 \left (-a \left (\frac {1}{2} a^2 \int -\frac {2 (1-a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^3 \left (-a \left (a^2 \left (-\int \frac {1-a x}{x \sqrt {1-a^2 x^2}}dx\right )-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle c^3 \left (-a \left (-\left (a^2 \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx-a \int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )\right )-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^3 \left (-a \left (-\left (a^2 \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\arcsin (a x)\right )\right )-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c^3 \left (-a \left (-\left (a^2 \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\arcsin (a x)\right )\right )-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c^3 \left (-a \left (-\left (a^2 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-\arcsin (a x)\right )\right )-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c^3 \left (-a \left (-\left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\arcsin (a x)\right )\right )-\frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
c^3*(-1/3*(1 - a^2*x^2)^(3/2)/x^3 - a*(-(((1 - a*x)*Sqrt[1 - a^2*x^2])/x^2 ) - a^2*(-ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]])))
3.4.13.3.1 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] :> 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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 = 104, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {\left (2 a^{4} x^{4}-3 a^{3} x^{3}-a^{2} x^{2}+3 a x -1\right ) c^{3}}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}-\left (\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) c^{3}\) | \(104\) |
| default | \(-c^{3} \left (\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )\right )\) | \(130\) |
| meijerg | \(-a^{3} c^{3} \arcsin \left (a x \right )+\frac {a^{3} c^{3} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{\sqrt {\pi }}+\frac {a^{3} c^{3} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{\sqrt {\pi }}-\frac {c^{3} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\) | \(202\) |
1/3*(2*a^4*x^4-3*a^3*x^3-a^2*x^2+3*a*x-1)/x^3/(-a^2*x^2+1)^(1/2)*c^3-(a^4/ (a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+a^3*arctanh(1/(-a^2*x ^2+1)^(1/2)))*c^3
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=\frac {6 \, a^{3} c^{3} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a^{3} c^{3} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (2 \, a^{2} c^{3} x^{2} - 3 \, a c^{3} x + c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, x^{3}} \]
1/3*(6*a^3*c^3*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^3*x^3* log((sqrt(-a^2*x^2 + 1) - 1)/x) - (2*a^2*c^3*x^2 - 3*a*c^3*x + c^3)*sqrt(- a^2*x^2 + 1))/x^3
Result contains complex when optimal does not.
Time = 2.84 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.12 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=- a^{4} c^{3} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) \]
-a**4*c**3*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/ sqrt(-a**2), Ne(a**2, 0)), (x, True)) + 2*a**3*c**3*Piecewise((-acosh(1/(a *x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - 2*a*c**3*Piecewise ((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3* sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + c**3*Piecewise((-2*I*a**2*sq rt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=-a^{3} c^{3} \arcsin \left (a x\right ) - a^{3} c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3}}{3 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{3}}{x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{3 \, x^{3}} \]
-a^3*c^3*arcsin(a*x) - a^3*c^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2/3*sqrt(-a^2*x^2 + 1)*a^2*c^3/x + sqrt(-a^2*x^2 + 1)*a*c^3/x^2 - 1/3*s qrt(-a^2*x^2 + 1)*c^3/x^3
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.84 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=-\frac {a^{4} c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a^{4} c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {{\left (a^{4} c^{3} - \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{3}}{x} - \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{3}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \]
-a^4*c^3*arcsin(a*x)*sgn(a)/abs(a) - a^4*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/24*(a^4*c^3 - 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^3/x + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2) *a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - 1/24*(9*(sqrt(-a^2*x ^2 + 1)*abs(a) + a)*a^4*c^3/x - 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^ 3/x^2 + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/x^3)/(a^2*abs(a))
Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.30 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^4} \, dx=\frac {a\,c^3\,\sqrt {1-a^2\,x^2}}{x^2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {a^4\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {2\,a^2\,c^3\,\sqrt {1-a^2\,x^2}}{3\,x}+a^3\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]