Integrand size = 23, antiderivative size = 66 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=-3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+\frac {7}{2} c \arcsin (a x)-c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
7/2*c*arcsin(a*x)-c*arctanh((-a^2*x^2+1)^(1/2))-3*c*(-a^2*x^2+1)^(1/2)-1/2 *a*c*x*(-a^2*x^2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=-\frac {1}{2} c \left ((6+a x) \sqrt {1-a^2 x^2}-7 \arcsin (a x)+2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right ) \]
Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6698, 541, 25, 2340, 25, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle c \int \frac {(a x+1)^3}{x \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 541 |
\(\displaystyle c \left (-\frac {\int -\frac {6 x^2 a^4+7 x a^3+2 a^2}{x \sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c \left (\frac {\int \frac {6 x^2 a^4+7 x a^3+2 a^2}{x \sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 2340 |
\(\displaystyle c \left (\frac {-\frac {\int -\frac {a^4 (7 a x+2)}{x \sqrt {1-a^2 x^2}}dx}{a^2}-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c \left (\frac {\frac {\int \frac {a^4 (7 a x+2)}{x \sqrt {1-a^2 x^2}}dx}{a^2}-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {a^2 \int \frac {7 a x+2}{x \sqrt {1-a^2 x^2}}dx-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c \left (\frac {a^2 \left (7 a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+2 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx\right )-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c \left (\frac {a^2 \left (2 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+7 \arcsin (a x)\right )-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c \left (\frac {a^2 \left (\int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+7 \arcsin (a x)\right )-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c \left (\frac {a^2 \left (7 \arcsin (a x)-\frac {2 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {a^2 \left (7 \arcsin (a x)-2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-6 a^2 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {1}{2} a x \sqrt {1-a^2 x^2}\right )\) |
c*(-1/2*(a*x*Sqrt[1 - a^2*x^2]) + (-6*a^2*Sqrt[1 - a^2*x^2] + a^2*(7*ArcSi n[a*x] - 2*ArcTanh[Sqrt[1 - a^2*x^2]]))/(2*a^2))
3.12.41.3.1 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[((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[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(56)=112\).
Time = 0.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.36
method | result | size |
default | \(-c \left (-\frac {3 a x}{\sqrt {-a^{2} x^{2}+1}}+a^{5} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-\frac {3}{\sqrt {-a^{2} x^{2}+1}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{4} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+2 a^{3} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )\right )\) | \(222\) |
meijerg | \(-\frac {2 c \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {c \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}-\frac {a c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 a c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {3 a c x}{\sqrt {-a^{2} x^{2}+1}}\) | \(288\) |
-c*(-3*a/(-a^2*x^2+1)^(1/2)*x+a^5*(-1/2*x^3/a^2/(-a^2*x^2+1)^(1/2)+3/2/a^2 *(x/a^2/(-a^2*x^2+1)^(1/2)-1/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^ 2+1)^(1/2))))-3/(-a^2*x^2+1)^(1/2)+arctanh(1/(-a^2*x^2+1)^(1/2))+3*a^4*(-x ^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))+2*a^3*(x/a^2/(-a^2*x^2 +1)^(1/2)-1/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))))
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=-7 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 6 \, c\right )} \]
-7*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1/2*sqrt(-a^2*x^2 + 1)*(a*c*x + 6*c)
Result contains complex when optimal does not.
Time = 5.85 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.70 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=a^{3} c \left (\begin {cases} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{2 a^{2}} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \]
a**3*c*Piecewise((-x*sqrt(-a**2*x**2 + 1)/(2*a**2) + log(-2*a**2*x + 2*sqr t(-a**2)*sqrt(-a**2*x**2 + 1))/(2*a**2*sqrt(-a**2)), Ne(a**2, 0)), (x**3/3 , True)) + 3*a**2*c*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), ( x**2/2, True)) + 3*a*c*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*Piecewise((-acosh(1/ (a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.68 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=\frac {a^{3} c x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{2}}{\sqrt {-a^{2} x^{2} + 1}} - \frac {a c x}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7}{2} \, c \arcsin \left (a x\right ) - c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, c}{\sqrt {-a^{2} x^{2} + 1}} \]
1/2*a^3*c*x^3/sqrt(-a^2*x^2 + 1) + 3*a^2*c*x^2/sqrt(-a^2*x^2 + 1) - 1/2*a* c*x/sqrt(-a^2*x^2 + 1) + 7/2*c*arcsin(a*x) - c*log(2*sqrt(-a^2*x^2 + 1)/ab s(x) + 2/abs(x)) - 3*c/sqrt(-a^2*x^2 + 1)
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=\frac {7 \, a c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {a c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 6 \, c\right )} \]
7/2*a*c*arcsin(a*x)*sgn(a)/abs(a) - a*c*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)*(a*c*x + 6*c)
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx=\frac {7\,a\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,c\,x\,\sqrt {1-a^2\,x^2}}{2}-3\,c\,\sqrt {1-a^2\,x^2} \]