3.3.82 \(\int \frac {a+b \log (c x^n)}{\sqrt {d+e x^2}} \, dx\) [282]

Optimal. Leaf size=250 \[ \frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}} \]

[Out]

1/2*b*n*arcsinh(x*e^(1/2)/d^(1/2))^2*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-b*n*arcsinh(x*e^(1/2)/d
^(1/2))*ln(1-(x*e^(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)+arcsin
h(x*e^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)-1/2*b*n*polylog(2,(x*e^
(1/2)/d^(1/2)+(1+e*x^2/d)^(1/2))^2)*d^(1/2)*(1+e*x^2/d)^(1/2)/e^(1/2)/(e*x^2+d)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2364, 2362, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/Sqrt[d + e*x^2],x]

[Out]

(b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*Sqrt[e]*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sq
rt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(Sqrt[e]*Sqrt[d +
e*x^2]) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[e]*Sqrt[d + e*x^
2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[e]*Sqrt[d + e*x
^2])

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2362

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[e, 2]*(x/Sqr
t[d])]*((a + b*Log[c*x^n])/Rt[e, 2]), x] - Dist[b*(n/Rt[e, 2]), Int[ArcSinh[Rt[e, 2]*(x/Sqrt[d])]/x, x], x] /;
FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && PosQ[e]

Rule 2364

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (e/d)*x^2]/Sqr
t[d + e*x^2], Int[(a + b*Log[c*x^n])/Sqrt[1 + (e/d)*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] &&  !GtQ[d, 0
]

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x^2}} \, dx &=\frac {\sqrt {1+\frac {e x^2}{d}} \int \frac {a+b \log \left (c x^n\right )}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{\sqrt {d+e x^2}}\\ &=\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}-\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{\sqrt {e} \sqrt {d+e x^2}}\\ &=\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}-\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\left (2 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {e} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {e} \sqrt {d+e x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 186, normalized size = 0.74 \begin {gather*} \frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{\sqrt {e}}+\frac {b n \sqrt {1+\frac {e x^2}{d}} \left (-\sinh ^{-1}\left (\sqrt {\frac {e}{d}} x\right )^2-2 \sinh ^{-1}\left (\sqrt {\frac {e}{d}} x\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt {\frac {e}{d}} x\right )}\right )+2 \log (x) \log \left (\sqrt {\frac {e}{d}} x+\sqrt {1+\frac {e x^2}{d}}\right )+\text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\sqrt {\frac {e}{d}} x\right )}\right )\right )}{2 \sqrt {\frac {e}{d}} \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/Sqrt[d + e*x^2],x]

[Out]

((a - b*n*Log[x] + b*Log[c*x^n])*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/Sqrt[e] + (b*n*Sqrt[1 + (e*x^2)/d]*(-ArcS
inh[Sqrt[e/d]*x]^2 - 2*ArcSinh[Sqrt[e/d]*x]*Log[1 - E^(-2*ArcSinh[Sqrt[e/d]*x])] + 2*Log[x]*Log[Sqrt[e/d]*x +
Sqrt[1 + (e*x^2)/d]] + PolyLog[2, E^(-2*ArcSinh[Sqrt[e/d]*x])]))/(2*Sqrt[e/d]*Sqrt[d + e*x^2])

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{\sqrt {e \,x^{2}+d}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/(e*x^2+d)^(1/2),x)

[Out]

int((a+b*ln(c*x^n))/(e*x^2+d)^(1/2),x)

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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((a+b*log(c*x^n))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

a*arcsinh(x*e^(1/2)/sqrt(d))*e^(-1/2) + b*integrate((log(c) + log(x^n))/sqrt(x^2*e + d), 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((a+b*log(c*x^n))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((sqrt(x^2*e + d)*b*log(c*x^n) + sqrt(x^2*e + d)*a)/(x^2*e + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/(e*x**2+d)**(1/2),x)

[Out]

Integral((a + b*log(c*x**n))/sqrt(d + e*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((a+b*log(c*x^n))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/sqrt(x^2*e + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*x^2)^(1/2),x)

[Out]

int((a + b*log(c*x^n))/(d + e*x^2)^(1/2), x)

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