3.1.85 \(\int \frac {1}{x (a+b x^4)} \, dx\) [85]

Optimal. Leaf size=21 \[ \frac {\log \left (\frac {x^4}{a+b x^4}\right )}{4 a} \]

[Out]

1/4/a*ln(x^4/(b*x^4+a))

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31} \begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b x^4\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^4)),x]

[Out]

Log[x]/a - Log[a + b*x^4]/(4*a)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^4\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )}{4 a}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^4\right )}{4 a}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b x^4\right )}{4 a}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.05 \begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b x^4\right )}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^4)),x]

[Out]

Log[x]/a - Log[a + b*x^4]/(4*a)

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Maple [A]
time = 0.02, size = 21, normalized size = 1.00

method	result	size
default	\(-\frac {\ln \left (b \,x^{4}+a \right )}{4 a}+\frac {\ln \left (x \right )}{a}\)	\(21\)
norman	\(-\frac {\ln \left (b \,x^{4}+a \right )}{4 a}+\frac {\ln \left (x \right )}{a}\)	\(21\)
risch	\(-\frac {\ln \left (b \,x^{4}+a \right )}{4 a}+\frac {\ln \left (x \right )}{a}\)	\(21\)
parallelrisch	\(\frac {4 \ln \left (x \right )-\ln \left (b \,x^{4}+a \right )}{4 a}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

-1/4/a*ln(b*x^4+a)+1/a*ln(x)

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Maxima [A]
time = 0.32, size = 23, normalized size = 1.10 \begin {gather*} -\frac {\log \left (b x^{4} + a\right )}{4 \, a} + \frac {\log \left (x^{4}\right )}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(b*x^4 + a)/a + 1/4*log(x^4)/a

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Fricas [A]
time = 0.65, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\log \left (b x^{4} + a\right ) - 4 \, \log \left (x\right )}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(log(b*x^4 + a) - 4*log(x))/a

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Sympy [A]
time = 0.11, size = 15, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{4} \right )}}{4 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**4+a),x)

[Out]

log(x)/a - log(a/b + x**4)/(4*a)

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Giac [A]
time = 0.45, size = 24, normalized size = 1.14 \begin {gather*} \frac {\log \left (x^{4}\right )}{4 \, a} - \frac {\log \left ({\left | b x^{4} + a \right |}\right )}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(x^4)/a - 1/4*log(abs(b*x^4 + a))/a

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Mupad [B]
time = 0.11, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\ln \left (b\,x^4+a\right )-4\,\ln \left (x\right )}{4\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^4)),x)

[Out]

-(log(a + b*x^4) - 4*log(x))/(4*a)

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