3.4.93 \(\int \frac {(1+x^2)^2}{a x^6+b (1+x^2)^3} \, dx\) [393]
Optimal. Leaf size=168 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{b}} x}{\sqrt [6]{b}}\right )}{3 \sqrt {\sqrt [3]{a}+\sqrt [3]{b}} b^{5/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} x}{\sqrt [6]{b}}\right )}{3 \sqrt {-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{5/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}} x}{\sqrt [6]{b}}\right )}{3 \sqrt {(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b}} b^{5/6}} \]
[Out]
1/3*arctan(x*(a^(1/3)+b^(1/3))^(1/2)/b^(1/6))/b^(5/6)/(a^(1/3)+b^(1/3))^(1/2)+1/3*arctan(x*(-(-1)^(1/3)*a^(1/3
)+b^(1/3))^(1/2)/b^(1/6))/b^(5/6)/(-(-1)^(1/3)*a^(1/3)+b^(1/3))^(1/2)+1/3*arctan(x*((-1)^(2/3)*a^(1/3)+b^(1/3)
)^(1/2)/b^(1/6))/b^(5/6)/((-1)^(2/3)*a^(1/3)+b^(1/3))^(1/2)
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Rubi [F]
time = 0.28, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]
[Out]
Defer[Int][(a*x^6 + b*(1 + x^2)^3)^(-1), x] + 2*Defer[Int][x^2/(a*x^6 + b*(1 + x^2)^3), x] + Defer[Int][x^4/(a
*x^6 + b*(1 + x^2)^3), x]
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right )^2}{a x^6+b \left (1+x^2\right )^3} \, dx &=\int \left (\frac {1}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6}+\frac {2 x^2}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6}+\frac {x^4}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6}\right ) \, dx\\ &=2 \int \frac {x^2}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6} \, dx+\int \frac {1}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6} \, dx+\int \frac {x^4}{b+3 b x^2+3 b x^4+a \left (1+\frac {b}{a}\right ) x^6} \, dx\\ &=2 \int \frac {x^2}{a x^6+b \left (1+x^2\right )^3} \, dx+\int \frac {1}{a x^6+b \left (1+x^2\right )^3} \, dx+\int \frac {x^4}{a x^6+b \left (1+x^2\right )^3} \, dx\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.04, size = 95, normalized size = 0.57 \begin {gather*} \frac {1}{6} \text {RootSum}\left [b+3 b \text {$\#$1}^2+3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})+2 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 + x^2)^2/(a*x^6 + b*(1 + x^2)^3),x]
[Out]
RootSum[b + 3*b*#1^2 + 3*b*#1^4 + a*#1^6 + b*#1^6 & , (Log[x - #1] + 2*Log[x - #1]*#1^2 + Log[x - #1]*#1^4)/(b
*#1 + 2*b*#1^3 + a*#1^5 + b*#1^5) & ]/6
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.30, size = 67, normalized size = 0.40
| | |
method |
result |
size |
| | |
default |
\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{6}+3 \textit {\_Z}^{4} b +3 \textit {\_Z}^{2} b +b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b +2 \textit {\_R}^{3} b +\textit {\_R} b}\right )}{6}\) |
\(67\) |
risch |
\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{6}+3 \textit {\_Z}^{4} b +3 \textit {\_Z}^{2} b +b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5} a +\textit {\_R}^{5} b +2 \textit {\_R}^{3} b +\textit {\_R} b}\right )}{6}\) |
\(67\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x,method=_RETURNVERBOSE)
[Out]
1/6*sum((_R^4+2*_R^2+1)/(_R^5*a+_R^5*b+2*_R^3*b+_R*b)*ln(x-_R),_R=RootOf((a+b)*_Z^6+3*_Z^4*b+3*_Z^2*b+b))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x, algorithm="maxima")
[Out]
integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b), x)
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Fricas [C] Result contains complex when optimal does not.
time = 1.12, size = 5653, normalized size = 33.65 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x, algorithm="fricas")
[Out]
1/36*sqrt(1/2)*sqrt((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a
*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1
/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5)
)^(1/3) - 72/(a*b + b^2))*log(1/6*sqrt(1/2)*sqrt((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/9331
2/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/
3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^
3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b + b^2))*b + x) - 1/36*sqrt(1/2)*sqrt((-I*sqrt(3) + 1)*(1/(a*b^3
+ b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)
^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*
(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b + b^2))*log(-1/6*sqrt(1/2)*s
qrt((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b
+ b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 +
b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*
b + b^2))*b + x) + 1/72*sqrt(-((a*b + b^2)*((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*
b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) -
1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1
/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b + b^2)) + 3*sqrt(1/3)*(a*b + b^2)*sqrt(-((a^2*b^3 + 2*a*b^4 + b^5)*(
(-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b
^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6)
+ 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b +
b^2))^2 + 144*(a*b^2 + b^3)*((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/
31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3)
+ 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a +
b)^2*b^5))^(1/3) - 72/(a*b + b^2)) + 20736*a + 5184*b)/(a^2*b^3 + 2*a*b^4 + b^5)) + 216)/(a*b + b^2))*log(1/12
*b*sqrt(-((a*b + b^2)*((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/
((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*
(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b
^5))^(1/3) - 72/(a*b + b^2)) + 3*sqrt(1/3)*(a*b + b^2)*sqrt(-((a^2*b^3 + 2*a*b^4 + b^5)*((-I*sqrt(3) + 1)*(1/(
a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b +
b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 +
b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b + b^2))^2 + 144*(a*b^2
+ b^3)*((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*
(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^
5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72
/(a*b + b^2)) + 20736*a + 5184*b)/(a^2*b^3 + 2*a*b^4 + b^5)) + 216)/(a*b + b^2)) + x) - 1/72*sqrt(-((a*b + b^2
)*((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b
+ b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b
^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b
+ b^2)) + 3*sqrt(1/3)*(a*b + b^2)*sqrt(-((a^2*b^3 + 2*a*b^4 + b^5)*((-I*sqrt(3) + 1)*(1/(a*b^3 + b^4) - 1/(a*
b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/
((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) -
1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b + b^2))^2 + 144*(a*b^2 + b^3)*((-I*sqrt(3)
+ 1)*(1/(a*b^3 + b^4) - 1/(a*b + b^2)^2)/(-1/93312/(a*b^5 + b^6) + 1/31104/((a*b^3 + b^4)*(a*b + b^2)) - 1/466
56/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 1296*(I*sqrt(3) + 1)*(-1/93312/(a*b^5 + b^6) + 1/31104/(
(a*b^3 + b^4)*(a*b + b^2)) - 1/46656/(a*b + b^2)^3 + 1/93312*a/((a + b)^2*b^5))^(1/3) - 72/(a*b + b^2)) + 2073
6*a + 5184*b)/(a^2*b^3 + 2*a*b^4 + b^5)) + 216)...
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Sympy [A]
time = 1.08, size = 42, normalized size = 0.25 \begin {gather*} \operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a b^{5} + 46656 b^{6}\right ) + 3888 t^{4} b^{4} + 108 t^{2} b^{2} + 1, \left ( t \mapsto t \log {\left (6 t b + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2+1)**2/(a*x**6+b*(x**2+1)**3),x)
[Out]
RootSum(_t**6*(46656*a*b**5 + 46656*b**6) + 3888*_t**4*b**4 + 108*_t**2*b**2 + 1, Lambda(_t, _t*log(6*_t*b + x
)))
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2+1)^2/(a*x^6+b*(x^2+1)^3),x, algorithm="giac")
[Out]
integrate((x^2 + 1)^2/(a*x^6 + (x^2 + 1)^3*b), x)
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Mupad [B]
time = 3.08, size = 504, normalized size = 3.00 \begin {gather*} \sum _{k=1}^6\ln \left (-a^3\,\left (a+b\right )\,\left (-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^2\,b^2\,60-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^4\,b^4\,864-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^4\,a\,b^3\,864+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^3\,b^3\,x\,504+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^5\,b^5\,x\,7776+\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )\,a\,x\,2+\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )\,b\,x\,8+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^2\,a\,b\,12-{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^3\,a\,b^2\,x\,144+{\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right )}^5\,a\,b^4\,x\,7776-1\right )\,3\right )\,\mathrm {root}\left (46656\,a\,b^5\,z^6+46656\,b^6\,z^6+3888\,b^4\,z^4+108\,b^2\,z^2+1,z,k\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2 + 1)^2/(b*(x^2 + 1)^3 + a*x^6),x)
[Out]
symsum(log(-3*a^3*(a + b)*(504*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^3*
b^3*x - 864*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^4*b^4 - 864*root(4665
6*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^4*a*b^3 - 60*root(46656*a*b^5*z^6 + 46656*
b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^2*b^2 + 7776*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z
^4 + 108*b^2*z^2 + 1, z, k)^5*b^5*x + 2*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1,
z, k)*a*x + 8*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)*b*x + 12*root(4665
6*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^2*a*b - 144*root(46656*a*b^5*z^6 + 46656*b
^6*z^6 + 3888*b^4*z^4 + 108*b^2*z^2 + 1, z, k)^3*a*b^2*x + 7776*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^
4*z^4 + 108*b^2*z^2 + 1, z, k)^5*a*b^4*x - 1))*root(46656*a*b^5*z^6 + 46656*b^6*z^6 + 3888*b^4*z^4 + 108*b^2*z
^2 + 1, z, k), k, 1, 6)
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