Integrand size = 19, antiderivative size = 75 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=\frac {a c^2 \sqrt {1-a^2 x^2}}{2 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac {1}{2} a^3 c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
-1/3*c^2*(-a^2*x^2+1)^(3/2)/x^3-1/2*a^3*c^2*arctanh((-a^2*x^2+1)^(1/2))+1/ 2*a*c^2*(-a^2*x^2+1)^(1/2)/x^2
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=-\frac {c^2 \left (2-3 a x-4 a^2 x^2+3 a^3 x^3+2 a^4 x^4+3 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{6 x^3 \sqrt {1-a^2 x^2}} \]
-1/6*(c^2*(2 - 3*a*x - 4*a^2*x^2 + 3*a^3*x^3 + 2*a^4*x^4 + 3*a^3*x^3*Sqrt[ 1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(x^3*Sqrt[1 - a^2*x^2])
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6678, 27, 534, 243, 51, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {c (1-a x) \sqrt {1-a^2 x^2}}{x^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^2 \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^4}dx\) |
\(\Big \downarrow \) 534 |
\(\displaystyle c^2 \left (-a \int \frac {\sqrt {1-a^2 x^2}}{x^3}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c^2 \left (-\frac {1}{2} a \int \frac {\sqrt {1-a^2 x^2}}{x^4}dx^2-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle c^2 \left (-\frac {1}{2} a \left (-\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c^2 \left (-\frac {1}{2} a \left (\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^2 \left (-\frac {1}{2} a \left (a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 x^3}\right )\) |
c^2*(-1/3*(1 - a^2*x^2)^(3/2)/x^3 - (a*(-(Sqrt[1 - a^2*x^2]/x^2) + a^2*Arc Tanh[Sqrt[1 - a^2*x^2]]))/2)
3.4.2.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[(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[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
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 = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {\left (2 a^{4} x^{4}+3 a^{3} x^{3}-4 a^{2} x^{2}-3 a x +2\right ) c^{2}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{2}\) | \(73\) |
default | \(c^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )\right )\) | \(100\) |
meijerg | \(\frac {a^{3} c^{2} \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}+\frac {a^{2} c^{2} \sqrt {-a^{2} x^{2}+1}}{x}+\frac {a^{3} c^{2} \left (-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{2} a^{2}}\right )}{2 \sqrt {\pi }}-\frac {c^{2} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\) | \(214\) |
-1/6*(2*a^4*x^4+3*a^3*x^3-4*a^2*x^2-3*a*x+2)/x^3/(-a^2*x^2+1)^(1/2)*c^2-1/ 2*a^3*arctanh(1/(-a^2*x^2+1)^(1/2))*c^2
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=\frac {3 \, a^{3} c^{2} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (2 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, x^{3}} \]
1/6*(3*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (2*a^2*c^2*x^2 + 3*a* c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/x^3
Result contains complex when optimal does not.
Time = 3.04 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.59 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=a^{3} c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) \]
a**3*c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a* x)), True)) - a**2*c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2 ) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) - a*c**2*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**2*Piecewise((-2*I*a**2*sqrt(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*s qrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=-\frac {1}{2} \, a^{3} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}}{3 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{2 \, x^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{3 \, x^{3}} \]
-1/2*a^3*c^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 1/3*sqrt(-a^2*x ^2 + 1)*a^2*c^2/x + 1/2*sqrt(-a^2*x^2 + 1)*a*c^2/x^2 - 1/3*sqrt(-a^2*x^2 + 1)*c^2/x^3
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (63) = 126\).
Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.11 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=\frac {{\left (a^{4} c^{2} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{2}}{x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {a^{4} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{2}}{x^{2}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \]
1/24*(a^4*c^2 - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2*c^2/x - 3*(sqrt(-a^2 *x^2 + 1)*abs(a) + a)^2*c^2/x^2)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^ 3*abs(a)) - 1/2*a^4*c^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a ^2*abs(x)))/abs(a) + 1/24*(3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^2/x + 3 *(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^2/x^2 - (sqrt(-a^2*x^2 + 1)*abs(a ) + a)^3*c^2/x^3)/(a^2*abs(a))
Time = 3.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^4} \, dx=\frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,x^3}+\frac {a^2\,c^2\,\sqrt {1-a^2\,x^2}}{3\,x}+\frac {a^3\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]