3.3.63 \(\int \frac {\log (c (d+e x)^p)}{f+g x^2} \, dx\) [263]

Optimal. Leaf size=229 \[ \frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

[Out]

1/2*ln(c*(e*x+d)^p)*ln(e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(1/2)/g^(1/2)-1/2*ln(c*(e*x+d)^
p)*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^(1/2)-d*g^(1/2)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,-(e*x+d)*g^(1/2)/(
e*(-f)^(1/2)-d*g^(1/2)))/(-f)^(1/2)/g^(1/2)+1/2*p*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/(-f)^(1/
2)/g^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2456, 2441, 2440, 2438} \begin {gather*} -\frac {p \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)^p]/(f + g*x^2),x]

[Out]

(Log[c*(d + e*x)^p]*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d
+ e*x)^p]*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((S
qrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt
[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g])

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 \left (c (d+e x)^p\right )}{f+g x^2} \, dx &=\int \left (\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {-f}}-\frac {\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {-f}}\\ &=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} \sqrt {g}}\\ &=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} \sqrt {g}}\\ &=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 178, normalized size = 0.78 \begin {gather*} \frac {\log \left (c (d+e x)^p\right ) \left (\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )-\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )\right )-p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)^p]/(f + g*x^2),x]

[Out]

(Log[c*(d + e*x)^p]*(Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])] - Log[(e*(Sqrt[-f] + Sqrt[g]*x))
/(e*Sqrt[-f] - d*Sqrt[g])]) - p*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + p*PolyLog[2, (Sq
rt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.56, size = 419, normalized size = 1.83

method	result	size
risch	\(\frac {\left (\ln \left (\left (e x +d \right )^{p}\right )-p \ln \left (e x +d \right )\right ) \arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {f g}}\right )}{\sqrt {f g}}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 \sqrt {-f g}}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 \sqrt {-f g}}+\frac {p \dilog \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 \sqrt {-f g}}-\frac {p \dilog \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 \sqrt {-f g}}+\frac {i \arctan \left (\frac {x g}{\sqrt {f g}}\right ) \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2}}{2 \sqrt {f g}}-\frac {i \arctan \left (\frac {x g}{\sqrt {f g}}\right ) \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2 \sqrt {f g}}-\frac {i \arctan \left (\frac {x g}{\sqrt {f g}}\right ) \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{3}}{2 \sqrt {f g}}+\frac {i \arctan \left (\frac {x g}{\sqrt {f g}}\right ) \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 \sqrt {f g}}+\frac {\arctan \left (\frac {x g}{\sqrt {f g}}\right ) \ln \left (c \right )}{\sqrt {f g}}\)	\(419\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x+d)^p)/(g*x^2+f),x,method=_RETURNVERBOSE)

[Out]

(ln((e*x+d)^p)-p*ln(e*x+d))/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))+1/2*p*ln(e*x+d)/(-f*g)^(
1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2*p*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g
*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*p/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*
g))-1/2*p/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*I/(f*g)^(1/2)*arctan(x*g
/(f*g)^(1/2))*Pi*csgn(I*(e*x+d)^p)*csgn(I*c*(e*x+d)^p)^2-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*(
e*x+d)^p)*csgn(I*c*(e*x+d)^p)*csgn(I*c)-1/2*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x+d)^p)^3+1/2
*I/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*Pi*csgn(I*c*(e*x+d)^p)^2*csgn(I*c)+1/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2)
)*ln(c)

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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 311, normalized size = 1.36 \begin {gather*} \frac {{\left (2 \, \arctan \left (\frac {g x}{\sqrt {f g}}\right ) e^{\left (-1\right )} \log \left (x e + d\right ) + {\left (\arctan \left (\frac {{\left (x e^{2} + d e\right )} \sqrt {f} \sqrt {g}}{d^{2} g + f e^{2}}, \frac {d g x e + d^{2} g}{d^{2} g + f e^{2}}\right ) \log \left (g x^{2} + f\right ) - \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {g x^{2} e^{2} + 2 \, d g x e + d^{2} g}{d^{2} g + f e^{2}}\right ) + i \, {\rm Li}_2\left (-\frac {d g x e + {\left (i \, x e^{2} - i \, d e\right )} \sqrt {f} \sqrt {g} + f e^{2}}{d^{2} g + 2 i \, d \sqrt {f} \sqrt {g} e - f e^{2}}\right ) - i \, {\rm Li}_2\left (-\frac {d g x e - {\left (i \, x e^{2} - i \, d e\right )} \sqrt {f} \sqrt {g} + f e^{2}}{d^{2} g - 2 i \, d \sqrt {f} \sqrt {g} e - f e^{2}}\right )\right )} e^{\left (-1\right )}\right )} p e}{2 \, \sqrt {f g}} - \frac {p \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (x e + d\right )}{\sqrt {f g}} + \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left ({\left (x e + d\right )}^{p} c\right )}{\sqrt {f g}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d)^p)/(g*x^2+f),x, algorithm="maxima")

[Out]

1/2*(2*arctan(g*x/sqrt(f*g))*e^(-1)*log(x*e + d) + (arctan2((x*e^2 + d*e)*sqrt(f)*sqrt(g)/(d^2*g + f*e^2), (d*
g*x*e + d^2*g)/(d^2*g + f*e^2))*log(g*x^2 + f) - arctan(sqrt(g)*x/sqrt(f))*log((g*x^2*e^2 + 2*d*g*x*e + d^2*g)
/(d^2*g + f*e^2)) + I*dilog(-(d*g*x*e + (I*x*e^2 - I*d*e)*sqrt(f)*sqrt(g) + f*e^2)/(d^2*g + 2*I*d*sqrt(f)*sqrt
(g)*e - f*e^2)) - I*dilog(-(d*g*x*e - (I*x*e^2 - I*d*e)*sqrt(f)*sqrt(g) + f*e^2)/(d^2*g - 2*I*d*sqrt(f)*sqrt(g
)*e - f*e^2)))*e^(-1))*p*e/sqrt(f*g) - p*arctan(g*x/sqrt(f*g))*log(x*e + d)/sqrt(f*g) + arctan(g*x/sqrt(f*g))*
log((x*e + d)^p*c)/sqrt(f*g)

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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(c*(e*x+d)^p)/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(log((x*e + d)^p*c)/(g*x^2 + f), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (d + e x\right )^{p} \right )}}{f + g x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(e*x+d)**p)/(g*x**2+f),x)

[Out]

Integral(log(c*(d + e*x)**p)/(f + g*x**2), 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(log(c*(e*x+d)^p)/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(log((x*e + d)^p*c)/(g*x^2 + f), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x\right )}^p\right )}{g\,x^2+f} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x)^p)/(f + g*x^2),x)

[Out]

int(log(c*(d + e*x)^p)/(f + g*x^2), x)

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