3.4.1 \(\int \frac {\log (c+d x)}{a+b x^4} \, dx\) [301]

Optimal. Leaf size=497 \[ \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}} \]

[Out]

1/4*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(3/4)/b^(1/4)-1/4*ln(-d*((-a)^(1/4)+b
^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(3/4)/b^(1/4)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)
^(1/4)*d))/(-a)^(3/4)/b^(1/4)+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(1/4)*d))/(-a)^(3/4)/b^(1/4)-1/4*l
n(d*x+c)*ln(-d*(b^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))/b^(1/4)/(-(-a)^(1/2))^(3/2)+
1/4*ln(d*x+c)*ln(d*(-b^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))/b^(1/4)/(-(-a)^(1/2))^(
3/2)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))/b^(1/4)/(-(-a)^(1/2))^(3/2)+1/4*polylog(
2,b^(1/4)*(d*x+c)/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))/b^(1/4)/(-(-a)^(1/2))^(3/2)

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Rubi [A]
time = 0.41, antiderivative size = 497, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2456, 2441, 2440, 2438} \begin {gather*} -\frac {\text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c + d*x]/(a + b*x^4),x]

[Out]

(Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-Sqrt[-a])^(3/2)*b^(
1/4)) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(3/4)*b^(1/4)) - (
Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-Sqrt[-a])^(3/2)*b
^(1/4)) - (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/4)*b^(1/4)
) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) + PolyLog[2,
(b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) - PolyLog[2, (b^(1/4)*(c +
d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*(-a)^(3/4)*b^(1/4)) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(
1/4)*d)]/(4*(-a)^(3/4)*b^(1/4))

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rubi steps

\begin {align*} \int \frac {\log (c+d x)}{a+b x^4} \, dx &=\int \left (\frac {\sqrt {-a} \log (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x^2\right )}+\frac {\sqrt {-a} \log (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x^2\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log (c+d x)}{\sqrt {-a}-\sqrt {b} x^2} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {\log (c+d x)}{\sqrt {-a}+\sqrt {b} x^2} \, dx}{2 \sqrt {-a}}\\ &=-\frac {\int \left (\frac {\sqrt {-\sqrt {-a}} \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}+\frac {\sqrt {-\sqrt {-a}} \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 \sqrt {-a}}-\frac {\int \left (\frac {\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac {\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 \sqrt {-a}}\\ &=-\frac {\int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 \left (-\sqrt {-a}\right )^{3/2}}-\frac {\int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 \left (-\sqrt {-a}\right )^{3/2}}-\frac {\int \frac {\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 (-a)^{3/4}}-\frac {\int \frac {\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 (-a)^{3/4}}\\ &=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {d \int \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{c+d x} \, dx}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {d \int \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{c+d x} \, dx}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {d \int \frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {d \int \frac {\log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 (-a)^{3/4} \sqrt [4]{b}}\\ &=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{3/4} \sqrt [4]{b}}\\ &=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/4} \sqrt [4]{b}}-\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}+\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \left (-\sqrt {-a}\right )^{3/2} \sqrt [4]{b}}-\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}+\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 359, normalized size = 0.72 \begin {gather*} \frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-i \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+i \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-i \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+i \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/4} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c + d*x]/(a + b*x^4),x]

[Out]

(Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] - I*Log[(d*((-a)^(1/4) - I*b^(1/4)*
x))/(I*b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] + I*Log[(d*((-a)^(1/4) + I*b^(1/4)*x))/((-I)*b^(1/4)*c + (-a)^(
1/4)*d)]*Log[c + d*x] - Log[(d*((-a)^(1/4) + b^(1/4)*x))/(-(b^(1/4)*c) + (-a)^(1/4)*d)]*Log[c + d*x] - PolyLog
[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)] - I*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/
4)*d)] + I*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)] + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1
/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/4)*b^(1/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.47, size = 112, normalized size = 0.23

method	result	size
derivativedivides	\(-\frac {d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 \textit {\_R1}^{2} c -3 \textit {\_R1} \,c^{2}+c^{3}}\right )}{4 b}\)	\(112\)
default	\(-\frac {d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 \textit {\_R1}^{2} c -3 \textit {\_R1} \,c^{2}+c^{3}}\right )}{4 b}\)	\(112\)
risch	\(-\frac {d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 \textit {\_R1}^{2} c -3 \textit {\_R1} \,c^{2}+c^{3}}\right )}{4 b}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

-1/4*d^3/b*sum(1/(-_R1^3+3*_R1^2*c-3*_R1*c^2+c^3)*(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1)),_R1
=RootOf(_Z^4*b-4*_Z^3*b*c+6*_Z^2*b*c^2-4*_Z*b*c^3+a*d^4+b*c^4))

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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(log(d*x+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(log(d*x + c)/(b*x^4 + a), x)

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Fricas [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(log(d*x+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

integral(log(d*x + c)/(b*x^4 + a), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(d*x+c)/(b*x**4+a),x)

[Out]

Timed out

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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(log(d*x+c)/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(log(d*x + c)/(b*x^4 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c + d*x)/(a + b*x^4),x)

[Out]

int(log(c + d*x)/(a + b*x^4), x)

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