Integrand size = 23, antiderivative size = 144 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{5 x^5}-\frac {3 a c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {19 a^2 c \sqrt {1-a^2 x^2}}{15 x^3}-\frac {13 a^3 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {38 a^4 c \sqrt {1-a^2 x^2}}{15 x}-\frac {13}{8} a^5 c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-1/5*c*(-a^2*x^2+1)^(1/2)/x^5-3/4*a*c*(-a^2*x^2+1)^(1/2)/x^4-19/15*a^2*c*( -a^2*x^2+1)^(1/2)/x^3-13/8*a^3*c*(-a^2*x^2+1)^(1/2)/x^2-38/15*a^4*c*(-a^2* x^2+1)^(1/2)/x-13/8*a^5*c*arctanh((-a^2*x^2+1)^(1/2))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=-\frac {c \sqrt {1-a^2 x^2} \left (6+38 a^2 x^2+15 a^3 x^3+76 a^4 x^4\right )}{30 x^5}-\frac {1}{2} a^5 c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-3 a^5 c \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-a^2 x^2\right ) \] Input:
Integrate[(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2))/x^6,x]
Output:
-1/30*(c*Sqrt[1 - a^2*x^2]*(6 + 38*a^2*x^2 + 15*a^3*x^3 + 76*a^4*x^4))/x^5 - (a^5*c*ArcTanh[Sqrt[1 - a^2*x^2]])/2 - 3*a^5*c*Sqrt[1 - a^2*x^2]*Hyperg eometric2F1[1/2, 3, 3/2, 1 - a^2*x^2]
Time = 0.50 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {6698, 540, 25, 2338, 25, 27, 539, 25, 27, 539, 25, 27, 534, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle c \int \frac {(a x+1)^3}{x^6 \sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle c \left (-\frac {1}{5} \int -\frac {5 x^2 a^3+19 x a^2+15 a}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{5} \int \frac {5 x^2 a^3+19 x a^2+15 a}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle c \left (\frac {1}{5} \left (-\frac {1}{4} \int -\frac {a^2 (65 a x+76)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {a^2 (65 a x+76)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \int \frac {65 a x+76}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (-\frac {1}{3} \int -\frac {a (152 a x+195)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} \int \frac {a (152 a x+195)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \int \frac {152 a x+195}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (-\frac {1}{2} \int -\frac {a (195 a x+304)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (\frac {1}{2} \int \frac {a (195 a x+304)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (\frac {1}{2} a \int \frac {195 a x+304}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (\frac {1}{2} a \left (195 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {304 \sqrt {1-a^2 x^2}}{x}\right )-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (\frac {1}{2} a \left (\frac {195}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {304 \sqrt {1-a^2 x^2}}{x}\right )-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-\frac {195 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {304 \sqrt {1-a^2 x^2}}{x}\right )-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c \left (\frac {1}{5} \left (\frac {1}{4} a^2 \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-195 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {304 \sqrt {1-a^2 x^2}}{x}\right )-\frac {195 \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {76 \sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {15 a \sqrt {1-a^2 x^2}}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
Input:
Int[(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2))/x^6,x]
Output:
c*(-1/5*Sqrt[1 - a^2*x^2]/x^5 + ((-15*a*Sqrt[1 - a^2*x^2])/(4*x^4) + (a^2* ((-76*Sqrt[1 - a^2*x^2])/(3*x^3) + (a*((-195*Sqrt[1 - a^2*x^2])/(2*x^2) + (a*((-304*Sqrt[1 - a^2*x^2])/x - 195*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/3) )/4)/5)
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[(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[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*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*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
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[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 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]
Time = 0.51 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(\frac {\left (304 x^{6} a^{6}+195 a^{5} x^{5}-152 a^{4} x^{4}-105 a^{3} x^{3}-128 a^{2} x^{2}-90 a x -24\right ) c}{120 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {13 a^{5} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{8}\) | \(85\) |
| default | \(-c \left (a^{5} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {1}{5 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {16 a^{2} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )}{5}-3 a \left (-\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {5 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}\right )+2 a^{3} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+3 a^{4} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(292\) |
| meijerg | \(-\frac {2 a^{2} c \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {c \left (-16 x^{6} a^{6}+8 a^{4} x^{4}+2 a^{2} x^{2}+1\right )}{5 x^{5} \sqrt {-a^{2} x^{2}+1}}-\frac {a^{5} c \left (\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{\sqrt {\pi }}+\frac {2 a^{5} c \left (\frac {\sqrt {\pi }}{2 x^{2} a^{2}}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}\right )}{\sqrt {\pi }}+\frac {3 a^{4} c \left (-2 a^{2} x^{2}+1\right )}{x \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{5} c \left (-\frac {\sqrt {\pi }}{4 x^{4} a^{4}}-\frac {3 \sqrt {\pi }}{4 x^{2} a^{2}}+\frac {15 \left (\frac {47}{30}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{16}+\frac {\sqrt {\pi }\, \left (-47 a^{4} x^{4}+24 a^{2} x^{2}+8\right )}{32 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-60 a^{4} x^{4}+20 a^{2} x^{2}+8\right )}{32 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{8}\right )}{\sqrt {\pi }}\) | \(453\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^6,x,method=_RETURNVERBOS E)
Output:
1/120*(304*a^6*x^6+195*a^5*x^5-152*a^4*x^4-105*a^3*x^3-128*a^2*x^2-90*a*x- 24)/x^5/(-a^2*x^2+1)^(1/2)*c-13/8*a^5*arctanh(1/(-a^2*x^2+1)^(1/2))*c
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.58 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=\frac {195 \, a^{5} c x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (304 \, a^{4} c x^{4} + 195 \, a^{3} c x^{3} + 152 \, a^{2} c x^{2} + 90 \, a c x + 24 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, x^{5}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^6,x, algorithm="fr icas")
Output:
1/120*(195*a^5*c*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (304*a^4*c*x^4 + 19 5*a^3*c*x^3 + 152*a^2*c*x^2 + 90*a*c*x + 24*c)*sqrt(-a^2*x^2 + 1))/x^5
Result contains complex when optimal does not.
Time = 21.64 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.59 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=a^{3} c \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x**6,x)
Output:
a**3*c*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)) + 3*a**2*c*Pie cewise((-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*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2* x**2 + 1)/(3*x**3), True)) + 3*a*c*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3 *a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4 *asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sq rt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + c* Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2* x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x** 2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))
Time = 0.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.18 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=\frac {38 \, a^{6} c x}{15 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13}{8} \, a^{5} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {13 \, a^{5} c}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {19 \, a^{4} c}{15 \, \sqrt {-a^{2} x^{2} + 1} x} - \frac {7 \, a^{3} c}{8 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {16 \, a^{2} c}{15 \, \sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {3 \, a c}{4 \, \sqrt {-a^{2} x^{2} + 1} x^{4}} - \frac {c}{5 \, \sqrt {-a^{2} x^{2} + 1} x^{5}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^6,x, algorithm="ma xima")
Output:
38/15*a^6*c*x/sqrt(-a^2*x^2 + 1) - 13/8*a^5*c*log(2*sqrt(-a^2*x^2 + 1)/abs (x) + 2/abs(x)) + 13/8*a^5*c/sqrt(-a^2*x^2 + 1) - 19/15*a^4*c/(sqrt(-a^2*x ^2 + 1)*x) - 7/8*a^3*c/(sqrt(-a^2*x^2 + 1)*x^2) - 16/15*a^2*c/(sqrt(-a^2*x ^2 + 1)*x^3) - 3/4*a*c/(sqrt(-a^2*x^2 + 1)*x^4) - 1/5*c/(sqrt(-a^2*x^2 + 1 )*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (120) = 240\).
Time = 0.14 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.31 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=\frac {{\left (6 \, a^{6} c + \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c}{x} + \frac {170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c}{x^{2}} + \frac {480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{x^{3}} + \frac {1380 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c}{a^{2} x^{4}}\right )} a^{10} x^{5}}{960 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} - \frac {13 \, a^{6} c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {1380 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{8} c}{x} + \frac {480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{6} c}{x^{2}} + \frac {170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a^{4} c}{x^{3}} + \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} a^{2} c}{x^{4}} + \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c}{x^{5}}}{960 \, a^{4} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^6,x, algorithm="gi ac")
Output:
1/960*(6*a^6*c + 45*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c/x + 170*(sqrt(-a ^2*x^2 + 1)*abs(a) + a)^2*a^2*c/x^2 + 480*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^ 3*c/x^3 + 1380*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c/(a^2*x^4))*a^10*x^5/((s qrt(-a^2*x^2 + 1)*abs(a) + a)^5*abs(a)) - 13/8*a^6*c*log(1/2*abs(-2*sqrt(- a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/960*(1380*(sqrt(-a^2*x ^2 + 1)*abs(a) + a)*a^8*c/x + 480*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^6*c/ x^2 + 170*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a^4*c/x^3 + 45*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*a^2*c/x^4 + 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c/x^5)/ (a^4*abs(a))
Time = 0.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.86 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=-\frac {c\,\sqrt {1-a^2\,x^2}}{5\,x^5}-\frac {19\,a^2\,c\,\sqrt {1-a^2\,x^2}}{15\,x^3}-\frac {13\,a^3\,c\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {38\,a^4\,c\,\sqrt {1-a^2\,x^2}}{15\,x}-\frac {3\,a\,c\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^5\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,13{}\mathrm {i}}{8} \] Input:
int(((c - a^2*c*x^2)*(a*x + 1)^3)/(x^6*(1 - a^2*x^2)^(3/2)),x)
Output:
(a^5*c*atan((1 - a^2*x^2)^(1/2)*1i)*13i)/8 - (c*(1 - a^2*x^2)^(1/2))/(5*x^ 5) - (19*a^2*c*(1 - a^2*x^2)^(1/2))/(15*x^3) - (13*a^3*c*(1 - a^2*x^2)^(1/ 2))/(8*x^2) - (38*a^4*c*(1 - a^2*x^2)^(1/2))/(15*x) - (3*a*c*(1 - a^2*x^2) ^(1/2))/(4*x^4)
Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.75 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )}{x^6} \, dx=\frac {c \left (-304 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-195 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-152 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-90 \sqrt {-a^{2} x^{2}+1}\, a x -24 \sqrt {-a^{2} x^{2}+1}+195 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{5} x^{5}\right )}{120 x^{5}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)/x^6,x)
Output:
(c*( - 304*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 195*sqrt( - a**2*x**2 + 1)*a **3*x**3 - 152*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 90*sqrt( - a**2*x**2 + 1 )*a*x - 24*sqrt( - a**2*x**2 + 1) + 195*log(tan(asin(a*x)/2))*a**5*x**5))/ (120*x**5)