Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=-\frac {\sqrt {1-x^3}}{6 x^6}-\frac {\sqrt {1-x^3}}{4 x^3}-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^3}\right ) \] Output:
-1/6*(-x^3+1)^(1/2)/x^6-1/4*(-x^3+1)^(1/2)/x^3-1/4*arctanh((-x^3+1)^(1/2))
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=\frac {\left (-2-3 x^3\right ) \sqrt {1-x^3}}{12 x^6}-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^3}\right ) \] Input:
Integrate[1/(x^7*Sqrt[1 - x^3]),x]
Output:
((-2 - 3*x^3)*Sqrt[1 - x^3])/(12*x^6) - ArcTanh[Sqrt[1 - x^3]]/4
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 52, 52, 73, 219}
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.
\(\displaystyle \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^9 \sqrt {1-x^3}}dx^3\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \int \frac {1}{x^6 \sqrt {1-x^3}}dx^3-\frac {\sqrt {1-x^3}}{2 x^6}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{x^3 \sqrt {1-x^3}}dx^3-\frac {\sqrt {1-x^3}}{x^3}\right )-\frac {\sqrt {1-x^3}}{2 x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \left (-\int \frac {1}{1-x^6}d\sqrt {1-x^3}-\frac {\sqrt {1-x^3}}{x^3}\right )-\frac {\sqrt {1-x^3}}{2 x^6}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{4} \left (-\text {arctanh}\left (\sqrt {1-x^3}\right )-\frac {\sqrt {1-x^3}}{x^3}\right )-\frac {\sqrt {1-x^3}}{2 x^6}\right )\) |
Input:
Int[1/(x^7*Sqrt[1 - x^3]),x]
Output:
(-1/2*Sqrt[1 - x^3]/x^6 + (3*(-(Sqrt[1 - x^3]/x^3) - ArcTanh[Sqrt[1 - x^3] ]))/4)/3
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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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 = 0.90 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {3 x^{6}-x^{3}-2}{12 x^{6} \sqrt {-x^{3}+1}}-\frac {\operatorname {arctanh}\left (\sqrt {-x^{3}+1}\right )}{4}\) | \(40\) |
default | \(-\frac {\sqrt {-x^{3}+1}}{6 x^{6}}-\frac {\sqrt {-x^{3}+1}}{4 x^{3}}-\frac {\operatorname {arctanh}\left (\sqrt {-x^{3}+1}\right )}{4}\) | \(42\) |
elliptic | \(-\frac {\sqrt {-x^{3}+1}}{6 x^{6}}-\frac {\sqrt {-x^{3}+1}}{4 x^{3}}-\frac {\operatorname {arctanh}\left (\sqrt {-x^{3}+1}\right )}{4}\) | \(42\) |
trager | \(-\frac {\left (3 x^{3}+2\right ) \sqrt {-x^{3}+1}}{12 x^{6}}+\frac {\ln \left (-\frac {x^{3}+2 \sqrt {-x^{3}+1}-2}{x^{3}}\right )}{8}\) | \(47\) |
pseudoelliptic | \(\frac {3 \ln \left (-1+\sqrt {-x^{3}+1}\right ) x^{6}-3 \ln \left (1+\sqrt {-x^{3}+1}\right ) x^{6}-6 x^{3} \sqrt {-x^{3}+1}-4 \sqrt {-x^{3}+1}}{24 \left (-1+\sqrt {-x^{3}+1}\right )^{2} \left (1+\sqrt {-x^{3}+1}\right )^{2}}\) | \(89\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }}{2 x^{6}}-\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}+8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (12 x^{3}+8\right ) \sqrt {-x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{4}}{3 \sqrt {\pi }}\) | \(105\) |
Input:
int(1/x^7/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/12*(3*x^6-x^3-2)/x^6/(-x^3+1)^(1/2)-1/4*arctanh((-x^3+1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=-\frac {3 \, x^{6} \log \left (\sqrt {-x^{3} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt {-x^{3} + 1} - 1\right ) + 2 \, {\left (3 \, x^{3} + 2\right )} \sqrt {-x^{3} + 1}}{24 \, x^{6}} \] Input:
integrate(1/x^7/(-x^3+1)^(1/2),x, algorithm="fricas")
Output:
-1/24*(3*x^6*log(sqrt(-x^3 + 1) + 1) - 3*x^6*log(sqrt(-x^3 + 1) - 1) + 2*( 3*x^3 + 2)*sqrt(-x^3 + 1))/x^6
Result contains complex when optimal does not.
Time = 2.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} + \frac {1}{4 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {1}{12 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} - \frac {i}{4 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} + \frac {i}{12 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} + \frac {i}{6 x^{\frac {15}{2}} \sqrt {1 - \frac {1}{x^{3}}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**7/(-x**3+1)**(1/2),x)
Output:
Piecewise((-acosh(x**(-3/2))/4 + 1/(4*x**(3/2)*sqrt(-1 + x**(-3))) - 1/(12 *x**(9/2)*sqrt(-1 + x**(-3))) - 1/(6*x**(15/2)*sqrt(-1 + x**(-3))), 1/Abs( x**3) > 1), (I*asin(x**(-3/2))/4 - I/(4*x**(3/2)*sqrt(1 - 1/x**3)) + I/(12 *x**(9/2)*sqrt(1 - 1/x**3)) + I/(6*x**(15/2)*sqrt(1 - 1/x**3)), True))
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=\frac {3 \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-x^{3} + 1}}{12 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {1}{8} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {-x^{3} + 1} - 1\right ) \] Input:
integrate(1/x^7/(-x^3+1)^(1/2),x, algorithm="maxima")
Output:
1/12*(3*(-x^3 + 1)^(3/2) - 5*sqrt(-x^3 + 1))/(2*x^3 + (x^3 - 1)^2 - 1) - 1 /8*log(sqrt(-x^3 + 1) + 1) + 1/8*log(sqrt(-x^3 + 1) - 1)
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=\frac {3 \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-x^{3} + 1}}{12 \, x^{6}} - \frac {1}{8} \, \log \left (\sqrt {-x^{3} + 1} + 1\right ) + \frac {1}{8} \, \log \left ({\left | \sqrt {-x^{3} + 1} - 1 \right |}\right ) \] Input:
integrate(1/x^7/(-x^3+1)^(1/2),x, algorithm="giac")
Output:
1/12*(3*(-x^3 + 1)^(3/2) - 5*sqrt(-x^3 + 1))/x^6 - 1/8*log(sqrt(-x^3 + 1) + 1) + 1/8*log(abs(sqrt(-x^3 + 1) - 1))
Time = 0.05 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.94 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=-\frac {\sqrt {1-x^3}}{4\,x^3}-\frac {\sqrt {1-x^3}}{6\,x^6}-\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(1/(x^7*(1 - x^3)^(1/2)),x)
Output:
- (1 - x^3)^(1/2)/(4*x^3) - (1 - x^3)^(1/2)/(6*x^6) - (3*((3^(1/2)*1i)/2 + 3/2)*(x^3 - 1)^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2)) ^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1) /((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i)/2 + 3/2, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i) /2 - 3/2)))/(4*(1 - x^3)^(1/2)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1 /2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^7 \sqrt {1-x^3}} \, dx=\frac {-6 \sqrt {-x^{3}+1}\, x^{3}-4 \sqrt {-x^{3}+1}+3 \,\mathrm {log}\left (\sqrt {-x^{3}+1}-1\right ) x^{6}-3 \,\mathrm {log}\left (\sqrt {-x^{3}+1}+1\right ) x^{6}}{24 x^{6}} \] Input:
int(1/x^7/(-x^3+1)^(1/2),x)
Output:
( - 6*sqrt( - x**3 + 1)*x**3 - 4*sqrt( - x**3 + 1) + 3*log(sqrt( - x**3 + 1) - 1)*x**6 - 3*log(sqrt( - x**3 + 1) + 1)*x**6)/(24*x**6)