Integrand size = 19, antiderivative size = 75 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=c^3 (1-a x) \sqrt {1-a^2 x^2}-\frac {1}{3} c^3 \left (1-a^2 x^2\right )^{3/2}-c^3 \arcsin (a x)-c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
c^3*(-a*x+1)*(-a^2*x^2+1)^(1/2)-1/3*c^3*(-a^2*x^2+1)^(3/2)-c^3*arcsin(a*x) -c^3*arctanh((-a^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=\frac {c^3 \left (4-6 a x-2 a^2 x^2+6 a^3 x^3-2 a^4 x^4+3 \sqrt {1-a^2 x^2} \arcsin (a x)+18 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-6 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{6 \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x,x]
Output:
(c^3*(4 - 6*a*x - 2*a^2*x^2 + 6*a^3*x^3 - 2*a^4*x^4 + 3*Sqrt[1 - a^2*x^2]* ArcSin[a*x] + 18*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 6*Sqrt[ 1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(6*Sqrt[1 - a^2*x^2])
Time = 0.35 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6678, 27, 541, 27, 535, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^3 \int \frac {(1-a x)^2 \sqrt {1-a^2 x^2}}{x}dx\) |
\(\Big \downarrow \) 541 |
\(\displaystyle c^3 \left (-\frac {\int -\frac {3 a^2 (1-2 a x) \sqrt {1-a^2 x^2}}{x}dx}{3 a^2}-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^3 \left (\int \frac {(1-2 a x) \sqrt {1-a^2 x^2}}{x}dx-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}\right )\) |
\(\Big \downarrow \) 535 |
\(\displaystyle c^3 \left (\frac {1}{2} \int \frac {2 (1-a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^3 \left (\int \frac {1-a x}{x \sqrt {1-a^2 x^2}}dx-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c^3 \left (-a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^3 \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}-\arcsin (a x)\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c^3 \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}-\arcsin (a x)\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c^3 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}-\arcsin (a x)\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^3 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{3} \left (1-a^2 x^2\right )^{3/2}+(1-a x) \sqrt {1-a^2 x^2}-\arcsin (a x)\right )\) |
Input:
Int[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x,x]
Output:
c^3*((1 - a*x)*Sqrt[1 - a^2*x^2] - (1 - a^2*x^2)^(3/2)/3 - ArcSin[a*x] - A rcTanh[Sqrt[1 - a^2*x^2]])
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[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p + 1) Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free Q[{a, b, c, d}, x] && GtQ[p, 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_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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(67)=134\).
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.92
method | result | size |
default | \(-c^{3} \left (a^{4} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+\frac {2 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 a^{3} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )\right )\) | \(144\) |
meijerg | \(-\frac {c^{3} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 \sqrt {\pi }}-\frac {a \,c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-2 c^{3} \arcsin \left (a x \right )+\frac {c^{3} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}\) | \(165\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x,x,method=_RETURNVERBOSE)
Output:
-c^3*(a^4*(-1/3*x^2/a^2*(-a^2*x^2+1)^(1/2)-2/3*(-a^2*x^2+1)^(1/2)/a^4)+2*a /(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+arctanh(1/(-a^2*x^2+ 1)^(1/2))-2*a^3*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2/a^2/(a^2)^(1/2)*arctan( (a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=2 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \frac {1}{3} \, {\left (a^{2} c^{3} x^{2} - 3 \, a c^{3} x + 2 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x,x, algorithm="fricas")
Output:
2*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) + 1/3*(a^2*c^3*x^2 - 3*a*c^3*x + 2*c^3)*sqrt(-a^2*x^2 + 1)
Result contains complex when optimal does not.
Time = 4.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.79 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=- a^{4} c^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{2 a^{2}} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x,x)
Output:
-a**4*c**3*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x **2 + 1)/(3*a**4), Ne(a**2, 0)), (x**4/4, True)) + 2*a**3*c**3*Piecewise(( -x*sqrt(-a**2*x**2 + 1)/(2*a**2) + log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a** 2*x**2 + 1))/(2*a**2*sqrt(-a**2)), Ne(a**2, 0)), (x**3/3, True)) - 2*a*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)) + c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2 *x**2) > 1), (I*asin(1/(a*x)), True))
Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=\frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{2} - \sqrt {-a^{2} x^{2} + 1} a c^{3} x - c^{3} \arcsin \left (a x\right ) - c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {2}{3} \, \sqrt {-a^{2} x^{2} + 1} c^{3} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x,x, algorithm="maxima")
Output:
1/3*sqrt(-a^2*x^2 + 1)*a^2*c^3*x^2 - sqrt(-a^2*x^2 + 1)*a*c^3*x - c^3*arcs in(a*x) - c^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 2/3*sqrt(-a^2* x^2 + 1)*c^3
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=-\frac {a c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, c^{3} + {\left (a^{2} c^{3} x - 3 \, a c^{3}\right )} x\right )} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x,x, algorithm="giac")
Output:
-a*c^3*arcsin(a*x)*sgn(a)/abs(a) - a*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1) *abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/3*sqrt(-a^2*x^2 + 1)*(2*c^3 + (a^2 *c^3*x - 3*a*c^3)*x)
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a^4\,c^3}{3\,{\left (-a^2\right )}^{3/2}}+\frac {a^6\,c^3\,x^2}{3\,{\left (-a^2\right )}^{3/2}}-a\,c^3\,x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-c^3\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}} \] Input:
int(((c - a*c*x)^3*(a*x + 1))/(x*(1 - a^2*x^2)^(1/2)),x)
Output:
((1 - a^2*x^2)^(1/2)*((2*a^4*c^3)/(3*(-a^2)^(3/2)) + (a^6*c^3*x^2)/(3*(-a^ 2)^(3/2)) - a*c^3*x*(-a^2)^(1/2)))/(-a^2)^(1/2) - c^3*atanh((1 - a^2*x^2)^ (1/2)) - (a*c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2)
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x} \, dx=\frac {c^{3} \left (-3 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-3 \sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )-2\right )}{3} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x,x)
Output:
(c**3*( - 3*asin(a*x) + sqrt( - a**2*x**2 + 1)*a**2*x**2 - 3*sqrt( - a**2* x**2 + 1)*a*x + 2*sqrt( - a**2*x**2 + 1) + 3*log(tan(asin(a*x)/2)) - 2))/3