3.40 \(\int \log (c (a+\frac {b}{x^2})^p) \, dx\)

Optimal. Leaf size=41 \[ x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}} \]

[Out]

x*ln(c*(a+b/x^2)^p)+2*p*arctan(x*a^(1/2)/b^(1/2))*b^(1/2)/a^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2448, 263, 205} \[ x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x^2)^p],x]

[Out]

(2*Sqrt[b]*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/Sqrt[a] + x*Log[c*(a + b/x^2)^p]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rubi steps

\begin {align*} \int \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx &=x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(2 b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^2} \, dx\\ &=x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(2 b p) \int \frac {1}{b+a x^2} \, dx\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}}+x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 43, normalized size = 1.05 \[ x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x^2)^p],x]

[Out]

(-2*Sqrt[b]*p*ArcTan[Sqrt[b]/(Sqrt[a]*x)])/Sqrt[a] + x*Log[c*(a + b/x^2)^p]

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fricas [A]  time = 0.46, size = 107, normalized size = 2.61 \[ \left [p x \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + p \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) + x \log \relax (c), p x \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, p \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) + x \log \relax (c)\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x^2)^p),x, algorithm="fricas")

[Out]

[p*x*log((a*x^2 + b)/x^2) + p*sqrt(-b/a)*log((a*x^2 + 2*a*x*sqrt(-b/a) - b)/(a*x^2 + b)) + x*log(c), p*x*log((
a*x^2 + b)/x^2) + 2*p*sqrt(b/a)*arctan(a*x*sqrt(b/a)/b) + x*log(c)]

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giac [A]  time = 0.19, size = 42, normalized size = 1.02 \[ p x \log \left (a x^{2} + b\right ) - p x \log \left (x^{2}\right ) + \frac {2 \, b p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}} + x \log \relax (c) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x^2)^p),x, algorithm="giac")

[Out]

p*x*log(a*x^2 + b) - p*x*log(x^2) + 2*b*p*arctan(a*x/sqrt(a*b))/sqrt(a*b) + x*log(c)

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maple [A]  time = 0.05, size = 38, normalized size = 0.93 \[ \frac {2 b p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}}+x \ln \left (c \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{p}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a+b/x^2)^p),x)

[Out]

x*ln(c*((a*x^2+b)/x^2)^p)+2*b*p/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))

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maxima [A]  time = 1.85, size = 33, normalized size = 0.80 \[ \frac {2 \, b p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}} + x \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x^2)^p),x, algorithm="maxima")

[Out]

2*b*p*arctan(a*x/sqrt(a*b))/sqrt(a*b) + x*log((a + b/x^2)^p*c)

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mupad [B]  time = 0.11, size = 33, normalized size = 0.80 \[ x\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )+\frac {2\,\sqrt {b}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x^2)^p),x)

[Out]

x*log(c*(a + b/x^2)^p) + (2*b^(1/2)*p*atan((a^(1/2)*x)/b^(1/2)))/a^(1/2)

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sympy [A]  time = 11.32, size = 109, normalized size = 2.66 \[ \begin {cases} p x \log {\left (a + \frac {b}{x^{2}} \right )} + x \log {\relax (c )} - \frac {i \sqrt {b} p \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{a \sqrt {\frac {1}{a}}} + \frac {i \sqrt {b} p \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{a \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\p x \log {\relax (b )} - 2 p x \log {\relax (x )} + 2 p x + x \log {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a+b/x**2)**p),x)

[Out]

Piecewise((p*x*log(a + b/x**2) + x*log(c) - I*sqrt(b)*p*log(-I*sqrt(b)*sqrt(1/a) + x)/(a*sqrt(1/a)) + I*sqrt(b
)*p*log(I*sqrt(b)*sqrt(1/a) + x)/(a*sqrt(1/a)), Ne(a, 0)), (p*x*log(b) - 2*p*x*log(x) + 2*p*x + x*log(c), True
))

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