Integrand size = 16, antiderivative size = 48 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\frac {1}{2 a \sqrt {a-b x^4}}-\frac {\text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Output:
1/2/a/(-b*x^4+a)^(1/2)-1/2*arctanh((-b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\frac {1}{2 a \sqrt {a-b x^4}}-\frac {\text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Input:
Integrate[1/(x*(a - b*x^4)^(3/2)),x]
Output:
1/(2*a*Sqrt[a - b*x^4]) - ArcTanh[Sqrt[a - b*x^4]/Sqrt[a]]/(2*a^(3/2))
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {798, 61, 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.
\(\displaystyle \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {1}{x^4 \left (a-b x^4\right )^{3/2}}dx^4\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {1}{x^4 \sqrt {a-b x^4}}dx^4}{a}+\frac {2}{a \sqrt {a-b x^4}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {2}{a \sqrt {a-b x^4}}-\frac {2 \int \frac {1}{\frac {a}{b}-\frac {x^8}{b}}d\sqrt {a-b x^4}}{a b}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {2}{a \sqrt {a-b x^4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{a^{3/2}}\right )\) |
Input:
Int[1/(x*(a - b*x^4)^(3/2)),x]
Output:
(2/(a*Sqrt[a - b*x^4]) - (2*ArcTanh[Sqrt[a - b*x^4]/Sqrt[a]])/a^(3/2))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.58 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(\frac {1}{2 a \sqrt {-b \,x^{4}+a}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {-b \,x^{4}+a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}\) | \(37\) |
default | \(\frac {1}{2 a \sqrt {-b \,x^{4}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\) | \(46\) |
elliptic | \(\frac {1}{2 a \sqrt {-b \,x^{4}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}\) | \(46\) |
Input:
int(1/x/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2/a/(-b*x^4+a)^(1/2)-1/2*arctanh((-b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (36) = 72\).
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 3.10 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\left [\frac {{\left (b x^{4} - a\right )} \sqrt {a} \log \left (\frac {b x^{4} + 2 \, \sqrt {-b x^{4} + a} \sqrt {a} - 2 \, a}{x^{4}}\right ) - 2 \, \sqrt {-b x^{4} + a} a}{4 \, {\left (a^{2} b x^{4} - a^{3}\right )}}, -\frac {{\left (b x^{4} - a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {-a}}{b x^{4} - a}\right ) + \sqrt {-b x^{4} + a} a}{2 \, {\left (a^{2} b x^{4} - a^{3}\right )}}\right ] \] Input:
integrate(1/x/(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
[1/4*((b*x^4 - a)*sqrt(a)*log((b*x^4 + 2*sqrt(-b*x^4 + a)*sqrt(a) - 2*a)/x ^4) - 2*sqrt(-b*x^4 + a)*a)/(a^2*b*x^4 - a^3), -1/2*((b*x^4 - a)*sqrt(-a)* arctan(sqrt(-b*x^4 + a)*sqrt(-a)/(b*x^4 - a)) + sqrt(-b*x^4 + a)*a)/(a^2*b *x^4 - a^3)]
Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 520, normalized size of antiderivative = 10.83 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\begin {cases} - \frac {2 i a^{3} \sqrt {-1 + \frac {b x^{4}}{a}}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {a^{3} \log {\left (\frac {b x^{4}}{a} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {2 a^{3} \log {\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 i a^{3} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{2} b x^{4} \log {\left (\frac {b x^{4}}{a} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{2} b x^{4} \log {\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {2 i a^{2} b x^{4} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} & \text {for}\: \left |{\frac {b x^{4}}{a}}\right | > 1 \\- \frac {2 a^{3} \sqrt {1 - \frac {b x^{4}}{a}}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {a^{3} \log {\left (\frac {b x^{4}}{a} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {2 a^{3} \log {\left (\sqrt {1 - \frac {b x^{4}}{a}} + 1 \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {i \pi a^{3}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{2} b x^{4} \log {\left (\frac {b x^{4}}{a} \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{2} b x^{4} \log {\left (\sqrt {1 - \frac {b x^{4}}{a}} + 1 \right )}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {i \pi a^{2} b x^{4}}{- 4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x/(-b*x**4+a)**(3/2),x)
Output:
Piecewise((-2*I*a**3*sqrt(-1 + b*x**4/a)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) - a**3*log(b*x**4/a)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) + 2*a**3*log(sqrt( b)*x**2/sqrt(a))/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) - 2*I*a**3*asin(sqrt(a) /(sqrt(b)*x**2))/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) + a**2*b*x**4*log(b*x** 4/a)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) - 2*a**2*b*x**4*log(sqrt(b)*x**2/sq rt(a))/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) + 2*I*a**2*b*x**4*asin(sqrt(a)/(s qrt(b)*x**2))/(-4*a**(9/2) + 4*a**(7/2)*b*x**4), Abs(b*x**4/a) > 1), (-2*a **3*sqrt(1 - b*x**4/a)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) - a**3*log(b*x**4 /a)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) + 2*a**3*log(sqrt(1 - b*x**4/a) + 1) /(-4*a**(9/2) + 4*a**(7/2)*b*x**4) - I*pi*a**3/(-4*a**(9/2) + 4*a**(7/2)*b *x**4) + a**2*b*x**4*log(b*x**4/a)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) - 2*a **2*b*x**4*log(sqrt(1 - b*x**4/a) + 1)/(-4*a**(9/2) + 4*a**(7/2)*b*x**4) + I*pi*a**2*b*x**4/(-4*a**(9/2) + 4*a**(7/2)*b*x**4), True))
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\frac {\log \left (\frac {\sqrt {-b x^{4} + a} - \sqrt {a}}{\sqrt {-b x^{4} + a} + \sqrt {a}}\right )}{4 \, a^{\frac {3}{2}}} + \frac {1}{2 \, \sqrt {-b x^{4} + a} a} \] Input:
integrate(1/x/(-b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
1/4*log((sqrt(-b*x^4 + a) - sqrt(a))/(sqrt(-b*x^4 + a) + sqrt(a)))/a^(3/2) + 1/2/(sqrt(-b*x^4 + a)*a)
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a} + \frac {1}{2 \, \sqrt {-b x^{4} + a} a} \] Input:
integrate(1/x/(-b*x^4+a)^(3/2),x, algorithm="giac")
Output:
1/2*arctan(sqrt(-b*x^4 + a)/sqrt(-a))/(sqrt(-a)*a) + 1/2/(sqrt(-b*x^4 + a) *a)
Time = 0.47 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\frac {1}{2\,a\,\sqrt {a-b\,x^4}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {a-b\,x^4}}{\sqrt {a}}\right )}{2\,a^{3/2}} \] Input:
int(1/(x*(a - b*x^4)^(3/2)),x)
Output:
1/(2*a*(a - b*x^4)^(1/2)) - atanh((a - b*x^4)^(1/2)/a^(1/2))/(2*a^(3/2))
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x \left (a-b x^4\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \sqrt {-b \,x^{4}+a}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}\right )}{2}\right )\right )-\sqrt {a}\, \sqrt {-b \,x^{4}+a}+a}{2 \sqrt {-b \,x^{4}+a}\, a^{2}} \] Input:
int(1/x/(-b*x^4+a)^(3/2),x)
Output:
(sqrt(a)*sqrt(a - b*x**4)*log(tan(asin((sqrt(b)*x**2)/sqrt(a))/2)) - sqrt( a)*sqrt(a - b*x**4) + a)/(2*sqrt(a - b*x**4)*a**2)