3.7.86 \(\int \frac {1}{x^2 (a+c x^4)^3} \, dx\) [686]

Optimal. Leaf size=233 \[ -\frac {45}{32 a^3 x}+\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}+\frac {45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{c} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}} \]

[Out]

-45/32/a^3/x+1/8/a/x/(c*x^4+a)^2+9/32/a^2/x/(c*x^4+a)-45/128*c^(1/4)*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))/a^(1
3/4)*2^(1/2)-45/128*c^(1/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))/a^(13/4)*2^(1/2)-45/256*c^(1/4)*ln(-a^(1/4)*c^
(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))/a^(13/4)*2^(1/2)+45/256*c^(1/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*
c^(1/2))/a^(13/4)*2^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {296, 331, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {45 \sqrt [4]{c} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{c} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}-\frac {45}{32 a^3 x}+\frac {9}{32 a^2 x \left (a+c x^4\right )}+\frac {1}{8 a x \left (a+c x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + c*x^4)^3),x]

[Out]

-45/(32*a^3*x) + 1/(8*a*x*(a + c*x^4)^2) + 9/(32*a^2*x*(a + c*x^4)) + (45*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)])/(64*Sqrt[2]*a^(13/4)) - (45*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(13/4)
) - (45*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(13/4)) + (45*c^(1/4)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(13/4))

Rule 210

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

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+c x^4\right )^3} \, dx &=\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9 \int \frac {1}{x^2 \left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}+\frac {45 \int \frac {1}{x^2 \left (a+c x^4\right )} \, dx}{32 a^2}\\ &=-\frac {45}{32 a^3 x}+\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}-\frac {(45 c) \int \frac {x^2}{a+c x^4} \, dx}{32 a^3}\\ &=-\frac {45}{32 a^3 x}+\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}+\frac {\left (45 \sqrt {c}\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^3}-\frac {\left (45 \sqrt {c}\right ) \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^3}\\ &=-\frac {45}{32 a^3 x}+\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}-\frac {45 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^3}-\frac {45 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^3}-\frac {\left (45 \sqrt [4]{c}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{13/4}}-\frac {\left (45 \sqrt [4]{c}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{13/4}}\\ &=-\frac {45}{32 a^3 x}+\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}-\frac {45 \sqrt [4]{c} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{c} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}-\frac {\left (45 \sqrt [4]{c}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}+\frac {\left (45 \sqrt [4]{c}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}\\ &=-\frac {45}{32 a^3 x}+\frac {1}{8 a x \left (a+c x^4\right )^2}+\frac {9}{32 a^2 x \left (a+c x^4\right )}+\frac {45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{c} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{c} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{13/4}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 216, normalized size = 0.93 \begin {gather*} \frac {-\frac {256 \sqrt [4]{a}}{x}-\frac {32 a^{5/4} c x^3}{\left (a+c x^4\right )^2}-\frac {104 \sqrt [4]{a} c x^3}{a+c x^4}+90 \sqrt {2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-90 \sqrt {2} \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-45 \sqrt {2} \sqrt [4]{c} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+45 \sqrt {2} \sqrt [4]{c} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{256 a^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + c*x^4)^3),x]

[Out]

((-256*a^(1/4))/x - (32*a^(5/4)*c*x^3)/(a + c*x^4)^2 - (104*a^(1/4)*c*x^3)/(a + c*x^4) + 90*Sqrt[2]*c^(1/4)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 90*Sqrt[2]*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 45*Sqrt[2
]*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 45*Sqrt[2]*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(256*a^(13/4))

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Maple [A]
time = 0.15, size = 141, normalized size = 0.61

method	result	size
risch	\(\frac {-\frac {45 c^{2} x^{8}}{32 a^{3}}-\frac {81 c \,x^{4}}{32 a^{2}}-\frac {1}{a}}{x \left (x^{4} c +a \right )^{2}}+\frac {45 \left (\munderset {\textit {\_R} =\RootOf \left (a^{13} \textit {\_Z}^{4}+c \right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4} a^{13}+4 c \right ) x +a^{10} \textit {\_R}^{3}\right )\right )}{128}\)	\(81\)
default	\(-\frac {c \left (\frac {\frac {13}{32} c \,x^{7}+\frac {17}{32} a \,x^{3}}{\left (x^{4} c +a \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{a^{3}}-\frac {1}{a^{3} x}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(c*x^4+a)^3,x,method=_RETURNVERBOSE)

[Out]

-c/a^3*((13/32*c*x^7+17/32*a*x^3)/(c*x^4+a)^2+45/256/c/(1/c*a)^(1/4)*2^(1/2)*(ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+
(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))+2*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+2*arctan(2^(1/
2)/(1/c*a)^(1/4)*x-1)))-1/a^3/x

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Maxima [A]
time = 0.50, size = 225, normalized size = 0.97 \begin {gather*} -\frac {45 \, c^{2} x^{8} + 81 \, a c x^{4} + 32 \, a^{2}}{32 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )}} - \frac {45 \, c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{256 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

-1/32*(45*c^2*x^8 + 81*a*c*x^4 + 32*a^2)/(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x) - 45/256*c*(2*sqrt(2)*arctan(1/2*
sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqr
t(2)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(sqrt(a)*sqrt(c))*
sqrt(c)) - sqrt(2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(1/4)*c^(3/4)) + sqrt(2)*log(sqrt
(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(1/4)*c^(3/4)))/a^3

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Fricas [A]
time = 0.38, size = 253, normalized size = 1.09 \begin {gather*} -\frac {180 \, c^{2} x^{8} + 324 \, a c x^{4} - 180 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {91125 \, a^{3} c x \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} - \sqrt {-8303765625 \, a^{7} c \sqrt {-\frac {c}{a^{13}}} + 8303765625 \, c^{2} x^{2}} a^{3} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}}}{91125 \, c}\right ) + 45 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} \left (-\frac {c}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, c x\right ) - 45 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )} \left (-\frac {c}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} \left (-\frac {c}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, c x\right ) + 128 \, a^{2}}{128 \, {\left (a^{3} c^{2} x^{9} + 2 \, a^{4} c x^{5} + a^{5} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+a)^3,x, algorithm="fricas")

[Out]

-1/128*(180*c^2*x^8 + 324*a*c*x^4 - 180*(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)*(-c/a^13)^(1/4)*arctan(-1/91125*(9
1125*a^3*c*x*(-c/a^13)^(1/4) - sqrt(-8303765625*a^7*c*sqrt(-c/a^13) + 8303765625*c^2*x^2)*a^3*(-c/a^13)^(1/4))
/c) + 45*(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)*(-c/a^13)^(1/4)*log(91125*a^10*(-c/a^13)^(3/4) + 91125*c*x) - 45*
(a^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)*(-c/a^13)^(1/4)*log(-91125*a^10*(-c/a^13)^(3/4) + 91125*c*x) + 128*a^2)/(a
^3*c^2*x^9 + 2*a^4*c*x^5 + a^5*x)

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Sympy [A]
time = 0.26, size = 80, normalized size = 0.34 \begin {gather*} \frac {- 32 a^{2} - 81 a c x^{4} - 45 c^{2} x^{8}}{32 a^{5} x + 64 a^{4} c x^{5} + 32 a^{3} c^{2} x^{9}} + \operatorname {RootSum} {\left (268435456 t^{4} a^{13} + 4100625 c, \left ( t \mapsto t \log {\left (- \frac {2097152 t^{3} a^{10}}{91125 c} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(c*x**4+a)**3,x)

[Out]

(-32*a**2 - 81*a*c*x**4 - 45*c**2*x**8)/(32*a**5*x + 64*a**4*c*x**5 + 32*a**3*c**2*x**9) + RootSum(268435456*_
t**4*a**13 + 4100625*c, Lambda(_t, _t*log(-2097152*_t**3*a**10/(91125*c) + x)))

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Giac [A]
time = 0.54, size = 217, normalized size = 0.93 \begin {gather*} -\frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{4} c^{2}} - \frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{4} c^{2}} + \frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{4} c^{2}} - \frac {45 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{4} c^{2}} - \frac {13 \, c^{2} x^{7} + 17 \, a c x^{3}}{32 \, {\left (c x^{4} + a\right )}^{2} a^{3}} - \frac {1}{a^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+a)^3,x, algorithm="giac")

[Out]

-45/128*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^4*c^2) - 45/128*s
qrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^4*c^2) + 45/256*sqrt(2)*(a
*c^3)^(3/4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^4*c^2) - 45/256*sqrt(2)*(a*c^3)^(3/4)*log(x^2 - sq
rt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^4*c^2) - 1/32*(13*c^2*x^7 + 17*a*c*x^3)/((c*x^4 + a)^2*a^3) - 1/(a^3*x)

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Mupad [B]
time = 0.11, size = 91, normalized size = 0.39 \begin {gather*} \frac {45\,{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{64\,a^{13/4}}-\frac {45\,{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,x}{a^{1/4}}\right )}{64\,a^{13/4}}-\frac {\frac {1}{a}+\frac {81\,c\,x^4}{32\,a^2}+\frac {45\,c^2\,x^8}{32\,a^3}}{a^2\,x+2\,a\,c\,x^5+c^2\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + c*x^4)^3),x)

[Out]

(45*(-c)^(1/4)*atanh(((-c)^(1/4)*x)/a^(1/4)))/(64*a^(13/4)) - (45*(-c)^(1/4)*atan(((-c)^(1/4)*x)/a^(1/4)))/(64
*a^(13/4)) - (1/a + (81*c*x^4)/(32*a^2) + (45*c^2*x^8)/(32*a^3))/(a^2*x + c^2*x^9 + 2*a*c*x^5)

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