3.8.11 \(\int \frac {1+x^{7/2}}{1-x^2} \, dx\) [711]

Optimal. Leaf size=43 \[ -2 \sqrt {x}-\frac {2 x^{5/2}}{5}+\tan ^{-1}\left (\sqrt {x}\right )-\log \left (1-\sqrt {x}\right )+\frac {1}{2} \log (1+x) \]

[Out]

-2/5*x^(5/2)+arctan(x^(1/2))+1/2*ln(1+x)-ln(1-x^(1/2))-2*x^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 31, normalized size of antiderivative = 0.72, number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1847, 281, 212, 308, 218, 209} \begin {gather*} \text {ArcTan}\left (\sqrt {x}\right )-\frac {2 x^{5/2}}{5}-2 \sqrt {x}+\tanh ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^(7/2))/(1 - x^2),x]

[Out]

-2*Sqrt[x] - (2*x^(5/2))/5 + ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]] + ArcTanh[x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 1847

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps

\begin {align*} \int \frac {1+x^{7/2}}{1-x^2} \, dx &=2 \text {Subst}\left (\int \frac {x \left (1+x^7\right )}{1-x^4} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {x}{1-x^4}+\frac {x^8}{1-x^4}\right ) \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \frac {x}{1-x^4} \, dx,x,\sqrt {x}\right )+2 \text {Subst}\left (\int \frac {x^8}{1-x^4} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (-1-x^4+\frac {1}{1-x^4}\right ) \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,x\right )\\ &=-2 \sqrt {x}-\frac {2 x^{5/2}}{5}+\tanh ^{-1}(x)+2 \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}-\frac {2 x^{5/2}}{5}+\tanh ^{-1}(x)+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {x}-\frac {2 x^{5/2}}{5}+\tan ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}\left (\sqrt {x}\right )+\tanh ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.91 \begin {gather*} -\frac {2}{5} \sqrt {x} \left (5+x^2\right )+\tan ^{-1}\left (\sqrt {x}\right )-\log \left (-1+\sqrt {x}\right )+\frac {1}{2} \log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^(7/2))/(1 - x^2),x]

[Out]

(-2*Sqrt[x]*(5 + x^2))/5 + ArcTan[Sqrt[x]] - Log[-1 + Sqrt[x]] + Log[1 + x]/2

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Maple [A]
time = 0.39, size = 34, normalized size = 0.79

method	result	size
derivativedivides	\(-\frac {2 x^{\frac {5}{2}}}{5}-2 \sqrt {x}+\frac {\ln \left (1+x \right )}{2}+\arctan \left (\sqrt {x}\right )-\ln \left (-1+\sqrt {x}\right )\)	\(30\)
default	\(-\frac {2 x^{\frac {5}{2}}}{5}-2 \sqrt {x}-\frac {\ln \left (-1+\sqrt {x}\right )}{2}+\frac {\ln \left (1+\sqrt {x}\right )}{2}+\arctan \left (\sqrt {x}\right )+\arctanh \left (x \right )\)	\(34\)
meijerg	\(\arctanh \left (x \right )-\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {4 \sqrt {x}\, \left (-1\right )^{\frac {1}{4}} \left (9 x^{2}+45\right )}{45}-\frac {\sqrt {x}\, \left (-1\right )^{\frac {1}{4}} \left (\ln \left (1-\left (x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{2}\right )^{\frac {1}{4}}\right )\right )}{\left (x^{2}\right )^{\frac {1}{4}}}\right )}{2}\)	\(67\)
trager	\(\left (-\frac {2 x^{2}}{5}-2\right ) \sqrt {x}-2 \ln \left (-\frac {24 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x -48 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}+16 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \sqrt {x}+2 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +26 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-\sqrt {x}-x -3}{-1+x}\right ) \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+2 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (-\frac {24 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x -48 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-16 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \sqrt {x}-26 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +22 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+7 \sqrt {x}+6 x -2}{-1+x}\right )+\ln \left (-\frac {24 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x -48 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}+16 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \sqrt {x}+2 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +26 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-\sqrt {x}-x -3}{-1+x}\right )\)	\(315\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(7/2))/(-x^2+1),x,method=_RETURNVERBOSE)

[Out]

-2/5*x^(5/2)-2*x^(1/2)-1/2*ln(-1+x^(1/2))+1/2*ln(1+x^(1/2))+arctan(x^(1/2))+arctanh(x)

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Maxima [A]
time = 0.49, size = 29, normalized size = 0.67 \begin {gather*} -\frac {2}{5} \, x^{\frac {5}{2}} - 2 \, \sqrt {x} + \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (x + 1\right ) - \log \left (\sqrt {x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(7/2))/(-x^2+1),x, algorithm="maxima")

[Out]

-2/5*x^(5/2) - 2*sqrt(x) + arctan(sqrt(x)) + 1/2*log(x + 1) - log(sqrt(x) - 1)

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Fricas [A]
time = 0.36, size = 29, normalized size = 0.67 \begin {gather*} -\frac {2}{5} \, {\left (x^{2} + 5\right )} \sqrt {x} + \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (x + 1\right ) - \log \left (\sqrt {x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(7/2))/(-x^2+1),x, algorithm="fricas")

[Out]

-2/5*(x^2 + 5)*sqrt(x) + arctan(sqrt(x)) + 1/2*log(x + 1) - log(sqrt(x) - 1)

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Sympy [A]
time = 0.54, size = 36, normalized size = 0.84 \begin {gather*} - \frac {2 x^{\frac {5}{2}}}{5} - 2 \sqrt {x} - \log {\left (\sqrt {x} - 1 \right )} + \frac {\log {\left (x + 1 \right )}}{2} + \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**(7/2))/(-x**2+1),x)

[Out]

-2*x**(5/2)/5 - 2*sqrt(x) - log(sqrt(x) - 1) + log(x + 1)/2 + atan(sqrt(x))

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Giac [A]
time = 2.57, size = 30, normalized size = 0.70 \begin {gather*} -\frac {2}{5} \, x^{\frac {5}{2}} - 2 \, \sqrt {x} + \arctan \left (\sqrt {x}\right ) + \frac {1}{2} \, \log \left (x + 1\right ) - \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(7/2))/(-x^2+1),x, algorithm="giac")

[Out]

-2/5*x^(5/2) - 2*sqrt(x) + arctan(sqrt(x)) + 1/2*log(x + 1) - log(abs(sqrt(x) - 1))

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Mupad [B]
time = 3.11, size = 53, normalized size = 1.23 \begin {gather*} -\ln \left (10\,\sqrt {x}-10\right )-2\,\sqrt {x}-\frac {2\,x^{5/2}}{5}+\ln \left (1+\sqrt {x}\,\left (-3-\mathrm {i}\right )-3{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\ln \left (1+\sqrt {x}\,\left (-3+1{}\mathrm {i}\right )+3{}\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^(7/2) + 1)/(x^2 - 1),x)

[Out]

log((1 - 3i) - x^(1/2)*(3 + 1i))*(1/2 + 1i/2) - log(10*x^(1/2) - 10) + log((1 + 3i) - x^(1/2)*(3 - 1i))*(1/2 -
1i/2) - 2*x^(1/2) - (2*x^(5/2))/5

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