Integrand size = 13, antiderivative size = 47 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {\sqrt {1+x^3}}{6 x^6}+\frac {\sqrt {1+x^3}}{4 x^3}-\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^3}\right ) \] 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 = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {\sqrt {1+x^3} \left (-2+3 x^3\right )}{12 x^6}-\frac {1}{4} \text {arctanh}\left (\sqrt {1+x^3}\right ) \] Input:
Integrate[1/(x^7*Sqrt[1 + x^3]),x]
Output:
(Sqrt[1 + x^3]*(-2 + 3*x^3))/(12*x^6) - ArcTanh[Sqrt[1 + x^3]]/4
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 52, 52, 73, 220}
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.
\(\displaystyle \int \frac {1}{x^7 \sqrt {x^3+1}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^9 \sqrt {x^3+1}}dx^3\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{4} \int \frac {1}{x^6 \sqrt {x^3+1}}dx^3-\frac {\sqrt {x^3+1}}{2 x^6}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {1}{x^3 \sqrt {x^3+1}}dx^3-\frac {\sqrt {x^3+1}}{x^3}\right )-\frac {\sqrt {x^3+1}}{2 x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{4} \left (-\int \frac {1}{x^6-1}d\sqrt {x^3+1}-\frac {\sqrt {x^3+1}}{x^3}\right )-\frac {\sqrt {x^3+1}}{2 x^6}\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{4} \left (\text {arctanh}\left (\sqrt {x^3+1}\right )-\frac {\sqrt {x^3+1}}{x^3}\right )-\frac {\sqrt {x^3+1}}{2 x^6}\right )\) |
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[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 = 1.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {3 x^{6}+x^{3}-2}{12 x^{6} \sqrt {x^{3}+1}}-\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{4}\) | \(34\) |
default | \(-\frac {\sqrt {x^{3}+1}}{6 x^{6}}+\frac {\sqrt {x^{3}+1}}{4 x^{3}}-\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{4}\) | \(36\) |
elliptic | \(-\frac {\sqrt {x^{3}+1}}{6 x^{6}}+\frac {\sqrt {x^{3}+1}}{4 x^{3}}-\frac {\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{4}\) | \(36\) |
trager | \(\frac {\left (3 x^{3}-2\right ) \sqrt {x^{3}+1}}{12 x^{6}}+\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{8}\) | \(45\) |
pseudoelliptic | \(\frac {3 \ln \left (-1+\sqrt {x^{3}+1}\right ) x^{6}-3 \ln \left (1+\sqrt {x^{3}+1}\right ) x^{6}+6 x^{3} \sqrt {x^{3}+1}-4 \sqrt {x^{3}+1}}{24 \left (-1+\sqrt {x^{3}+1}\right )^{2} \left (1+\sqrt {x^{3}+1}\right )^{2}}\) | \(77\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }}{2 x^{6}}+\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}-8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (-12 x^{3}+8\right ) \sqrt {x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )}{4}}{3 \sqrt {\pi }}\) | \(97\) |
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 = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {3 \, x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) - 2 \, {\left (3 \, x^{3} - 2\right )} \sqrt {x^{3} + 1}}{24 \, x^{6}} \] 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
Time = 2.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=- \frac {\operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} + \frac {1}{4 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {1}{12 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {1}{6 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \] Input:
integrate(1/x**7/(x**3+1)**(1/2),x)
Output:
-asinh(x**(-3/2))/4 + 1/(4*x**(3/2)*sqrt(1 + x**(-3))) + 1/(12*x**(9/2)*sq rt(1 + x**(-3))) - 1/(6*x**(15/2)*sqrt(1 + x**(-3)))
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=-\frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{3} + 1}}{12 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} - \frac {1}{8} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{3} + 1} - 1\right ) \] 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 = 50, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {3 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{3} + 1}}{12 \, x^{6}} - \frac {1}{8} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) + \frac {1}{8} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \] 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 = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.02 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {\sqrt {x^3+1}}{4\,x^3}-\frac {\sqrt {x^3+1}}{6\,x^6}-\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(1/(x^7*(x^3 + 1)^(1/2)),x)
Output:
(x^3 + 1)^(1/2)/(4*x^3) - (x^3 + 1)^(1/2)/(6*x^6) - (3*((3^(1/2)*1i)/2 + 3 /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)*(((3^(1/2)*1i)/2 - x + 1/2)/((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*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))
Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx=\frac {6 \sqrt {x^{3}+1}\, x^{3}-4 \sqrt {x^{3}+1}+3 \,\mathrm {log}\left (\sqrt {x^{3}+1}-1\right ) x^{6}-3 \,\mathrm {log}\left (\sqrt {x^{3}+1}+1\right ) x^{6}}{24 x^{6}} \] 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)