Integrand size = 25, antiderivative size = 45 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\arcsin (a x)-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
4*(a*x+1)/(-a^2*x^2+1)^(1/2)-arcsin(a*x)-arctanh((-a^2*x^2+1)^(1/2))
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.53 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {4 \sqrt {1-a^2 x^2}}{-1+a x}-2 \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )-\log (x)+\log \left (-1+\sqrt {1-a^2 x^2}\right ) \] Input:
Integrate[(1 + a*x)^3/(x*(1 - a^2*x^2)^(3/2)),x]
Output:
(-4*Sqrt[1 - a^2*x^2])/(-1 + a*x) - 2*ArcTan[(a*x)/(-1 + Sqrt[1 - a^2*x^2] )] - Log[x] + Log[-1 + Sqrt[1 - a^2*x^2]]
Time = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {528, 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.
\(\displaystyle \int \frac {(a x+1)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 528 |
\(\displaystyle \int \frac {1-a x}{x \sqrt {1-a^2 x^2}}dx+\frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 538 |
\(\displaystyle -a \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\arcsin (a x)\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\arcsin (a x)\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}+\frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\arcsin (a x)\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\arcsin (a x)\) |
Input:
Int[(1 + a*x)^3/(x*(1 - a^2*x^2)^(3/2)),x]
Output:
(4*(1 + a*x))/Sqrt[1 - a^2*x^2] - ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]
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[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(41)=82\).
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.20
method | result | size |
default | \(\frac {4}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+a^{3} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {3 a x}{\sqrt {-a^{2} x^{2}+1}}\) | \(99\) |
meijerg | \(\frac {\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{\sqrt {\pi }}-\frac {a \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {3 a x}{\sqrt {-a^{2} x^{2}+1}}\) | \(173\) |
Input:
int((a*x+1)^3/x/(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
4/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2))+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)))+3*a*x/( -a^2*x^2+1)^(1/2)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.82 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {4 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} - 4}{a x - 1} \] Input:
integrate((a*x+1)^3/x/(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
(4*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a*x - 1)*lo g((sqrt(-a^2*x^2 + 1) - 1)/x) - 4*sqrt(-a^2*x^2 + 1) - 4)/(a*x - 1)
\[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/x/(-a**2*x**2+1)**(3/2),x)
Output:
Integral((a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {4 \, a x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4}{\sqrt {-a^{2} x^{2} + 1}} - \arcsin \left (a x\right ) - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate((a*x+1)^3/x/(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
4*a*x/sqrt(-a^2*x^2 + 1) + 4/sqrt(-a^2*x^2 + 1) - arcsin(a*x) - log(2*sqrt (-a^2*x^2 + 1)/abs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (41) = 82\).
Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.93 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {a \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {8 \, a}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/x/(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
-a*arcsin(a*x)*sgn(a)/abs(a) - a*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 8*a/(((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x ) - 1)*abs(a))
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.82 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {4\,a\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right ) \] Input:
int((a*x + 1)^3/(x*(1 - a^2*x^2)^(3/2)),x)
Output:
(4*a*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2)) - (a*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - atanh((1 - a^2*x^2)^(1/2))
Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.44 \[ \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {-\mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+\mathit {asin} \left (a x \right )+\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )-8 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )}{\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1} \] Input:
int((a*x+1)^3/x/(-a^2*x^2+1)^(3/2),x)
Output:
( - asin(a*x)*tan(asin(a*x)/2) + asin(a*x) + log(tan(asin(a*x)/2))*tan(asi n(a*x)/2) - log(tan(asin(a*x)/2)) - 8*tan(asin(a*x)/2))/(tan(asin(a*x)/2) - 1)