Integrand size = 19, antiderivative size = 83 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=-\frac {1}{2} a c^3 (4+a x) \sqrt {1-a^2 x^2}-\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{x}-\frac {1}{2} a c^3 \arcsin (a x)+2 a c^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-1/2*a*c^3*(a*x+4)*(-a^2*x^2+1)^(1/2)-c^3*(-a^2*x^2+1)^(3/2)/x-1/2*a*c^3*a rcsin(a*x)+2*a*c^3*arctanh((-a^2*x^2+1)^(1/2))
Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.72 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=-\frac {c^3 \left (2+4 a x-3 a^2 x^2-4 a^3 x^3+a^4 x^4+2 a x \sqrt {1-a^2 x^2} \arcsin (a x)+2 a x \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-4 a x \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{2 x \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^2,x]
Output:
-1/2*(c^3*(2 + 4*a*x - 3*a^2*x^2 - 4*a^3*x^3 + a^4*x^4 + 2*a*x*Sqrt[1 - a^ 2*x^2]*ArcSin[a*x] + 2*a*x*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 4*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(x*Sqrt[1 - a^2*x^ 2])
Time = 0.55 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {6678, 27, 540, 27, 535, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^3 \int \frac {(1-a x)^2 \sqrt {1-a^2 x^2}}{x^2}dx\) |
\(\Big \downarrow \) 540 |
\(\displaystyle c^3 \left (-\int \frac {a (a x+2) \sqrt {1-a^2 x^2}}{x}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^3 \left (-a \int \frac {(a x+2) \sqrt {1-a^2 x^2}}{x}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 535 |
\(\displaystyle c^3 \left (-a \left (\frac {1}{2} \int \frac {a x+4}{x \sqrt {1-a^2 x^2}}dx+\frac {1}{2} \sqrt {1-a^2 x^2} (a x+4)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c^3 \left (-a \left (\frac {1}{2} \left (a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+4 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (a x+4)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^3 \left (-a \left (\frac {1}{2} \left (4 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\arcsin (a x)\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (a x+4)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c^3 \left (-a \left (\frac {1}{2} \left (2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\arcsin (a x)\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (a x+4)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c^3 \left (-a \left (\frac {1}{2} \left (\arcsin (a x)-\frac {4 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (a x+4)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^3 \left (-a \left (\frac {1}{2} \left (\arcsin (a x)-4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (a x+4)\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{x}\right )\) |
Input:
Int[(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^2,x]
Output:
c^3*(-((1 - a^2*x^2)^(3/2)/x) - a*(((4 + a*x)*Sqrt[1 - a^2*x^2])/2 + (ArcS in[a*x] - 4*ArcTanh[Sqrt[1 - a^2*x^2]])/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_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 = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.29
method | result | size |
default | \(-c^{3} \left (a^{4} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{x}-2 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \sqrt {-a^{2} x^{2}+1}\right )\) | \(107\) |
risch | \(\frac {\left (a^{2} x^{2}-1\right ) c^{3}}{x \sqrt {-a^{2} x^{2}+1}}-\left (-\frac {a^{2} x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-2 a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \sqrt {-a^{2} x^{2}+1}\right ) c^{3}\) | \(115\) |
meijerg | \(\frac {a^{2} c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {a \,c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{\sqrt {\pi }}-\frac {a \,c^{3} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{\sqrt {\pi }}-\frac {c^{3} \sqrt {-a^{2} x^{2}+1}}{x}\) | \(170\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^2,x,method=_RETURNVERBOSE)
Output:
-c^3*(a^4*(-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)))+(-a^2*x^2+1)^(1/2)/x-2*a*arctanh(1/(-a^2*x^2+ 1)^(1/2))+2*a*(-a^2*x^2+1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=\frac {2 \, a c^{3} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 4 \, a c^{3} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, a c^{3} x + {\left (a^{2} c^{3} x^{2} - 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^2,x, algorithm="fricas ")
Output:
1/2*(2*a*c^3*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 4*a*c^3*x*log((sqr t(-a^2*x^2 + 1) - 1)/x) - 4*a*c^3*x + (a^2*c^3*x^2 - 4*a*c^3*x - 2*c^3)*sq rt(-a^2*x^2 + 1))/x
Result contains complex when optimal does not.
Time = 2.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.19 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=- a^{4} c^{3} \left (\begin {cases} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{2 a^{2}} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x**2,x)
Output:
-a**4*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**3*c**3*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True)) - 2*a*c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x* *2) > 1), (I*asin(1/(a*x)), True)) + c**3*Piecewise((-I*sqrt(a**2*x**2 - 1 )/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \frac {1}{2} \, a c^{3} \arcsin \left (a x\right ) + 2 \, a c^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} a c^{3} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{x} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^2,x, algorithm="maxima ")
Output:
1/2*sqrt(-a^2*x^2 + 1)*a^2*c^3*x - 1/2*a*c^3*arcsin(a*x) + 2*a*c^3*log(2*s qrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 2*sqrt(-a^2*x^2 + 1)*a*c^3 - sqrt(- a^2*x^2 + 1)*c^3/x
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (73) = 146\).
Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=\frac {a^{4} c^{3} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} + \frac {2 \, a^{2} c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{2 \, x {\left | a \right |}} + \frac {1}{2} \, {\left (a^{2} c^{3} x - 4 \, a c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^2,x, algorithm="giac")
Output:
1/2*a^4*c^3*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - 1/2*a^2*c^3*arcsi n(a*x)*sgn(a)/abs(a) + 2*a^2*c^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(x*a bs(a)) + 1/2*(a^2*c^3*x - 4*a*c^3)*sqrt(-a^2*x^2 + 1)
Time = 23.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.30 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=\frac {a^2\,c^3\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{x}-2\,a\,c^3\,\sqrt {1-a^2\,x^2}-\frac {a^2\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-a\,c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,2{}\mathrm {i} \] Input:
int(((c - a*c*x)^3*(a*x + 1))/(x^2*(1 - a^2*x^2)^(1/2)),x)
Output:
(a^2*c^3*x*(1 - a^2*x^2)^(1/2))/2 - (c^3*(1 - a^2*x^2)^(1/2))/x - a*c^3*at an((1 - a^2*x^2)^(1/2)*1i)*2i - 2*a*c^3*(1 - a^2*x^2)^(1/2) - (a^2*c^3*asi nh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^3}{x^2} \, dx=\frac {c^{3} \left (-\mathit {asin} \left (a x \right ) a x +\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-4 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}-4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x +4 a x \right )}{2 x} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^3/x^2,x)
Output:
(c**3*( - asin(a*x)*a*x + sqrt( - a**2*x**2 + 1)*a**2*x**2 - 4*sqrt( - a** 2*x**2 + 1)*a*x - 2*sqrt( - a**2*x**2 + 1) - 4*log(tan(asin(a*x)/2))*a*x + 4*a*x))/(2*x)