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

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

[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 = 0.23, 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 = {2409, 2394, 2393, 2391} \[ -\frac {p \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

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 2391

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 2393

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 2394

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 2409

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 \operatorname {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 \operatorname {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 = 0.13, size = 178, normalized size = 0.78 \[ \frac {\log \left (c (d+e x)^p\right ) \left (\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )-\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )\right )-p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

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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fricas [F]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \]

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((e*x + d)^p*c)/(g*x^2 + f), x)

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

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((e*x + d)^p*c)/(g*x^2 + f), x)

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maple [C]  time = 0.67, size = 419, normalized size = 1.83 \[ -\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )}{2 \sqrt {f g}}+\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2}}{2 \sqrt {f g}}+\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2}}{2 \sqrt {f g}}-\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{3}}{2 \sqrt {f g}}+\frac {p \ln \left (\frac {d g +\sqrt {-f g}\, e -\left (e x +d \right ) g}{d g +\sqrt {-f g}\, e}\right ) \ln \left (e x +d \right )}{2 \sqrt {-f g}}-\frac {p \ln \left (\frac {-d g +\sqrt {-f g}\, e +\left (e x +d \right ) g}{-d g +\sqrt {-f g}\, e}\right ) \ln \left (e x +d \right )}{2 \sqrt {-f g}}+\frac {p \dilog \left (\frac {d g +\sqrt {-f g}\, e -\left (e x +d \right ) g}{d g +\sqrt {-f g}\, e}\right )}{2 \sqrt {-f g}}-\frac {p \dilog \left (\frac {-d g +\sqrt {-f g}\, e +\left (e x +d \right ) g}{-d g +\sqrt {-f g}\, e}\right )}{2 \sqrt {-f g}}+\frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \ln \relax (c )}{\sqrt {f g}}+\frac {\left (-p \ln \left (e x +d \right )+\ln \left (\left (e x +d \right )^{p}\right )\right ) \arctan \left (\frac {-2 d g +2 \left (e x +d \right ) g}{2 \sqrt {f g}\, e}\right )}{\sqrt {f g}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 1.29, size = 309, normalized size = 1.35 \[ \frac {e p {\left (\frac {2 \, \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (e x + d\right )}{e} + \frac {\arctan \left (\frac {{\left (e^{2} x + d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f + d^{2} g}, \frac {d e g x + d^{2} g}{e^{2} f + d^{2} g}\right ) \log \left (g x^{2} + f\right ) - \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {e^{2} g x^{2} + 2 \, d e g x + d^{2} g}{e^{2} f + d^{2} g}\right ) - i \, {\rm Li}_2\left (\frac {d e g x + e^{2} f - {\left (i \, e^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f + 2 i \, d e \sqrt {f} \sqrt {g} - d^{2} g}\right ) + i \, {\rm Li}_2\left (\frac {d e g x + e^{2} f + {\left (i \, e^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f - 2 i \, d e \sqrt {f} \sqrt {g} - d^{2} g}\right )}{e}\right )}}{2 \, \sqrt {f g}} - \frac {p \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (e x + d\right )}{\sqrt {f g}} + \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left ({\left (e x + d\right )}^{p} c\right )}{\sqrt {f g}} \]

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

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

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

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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