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]

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

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Rubi [A]  time = 0.0148343, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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 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]

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*}

Mathematica [A]  time = 0.0075675, 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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Maple [A]  time = 0.063, size = 38, normalized size = 0.9 \begin{align*} x\ln \left ( c \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{p} \right ) +2\,{\frac{bp}{\sqrt{ab}}\arctan \left ({\frac{ax}{\sqrt{ab}}} \right ) } \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.19322, size = 244, normalized size = 5.95 \begin{align*} \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 \left (c\right ), 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 \left (c\right )\right ] \end{align*}

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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Sympy [A]  time = 24.0015, size = 109, normalized size = 2.66 \begin{align*} \begin{cases} p x \log{\left (a + \frac{b}{x^{2}} \right )} + x \log{\left (c \right )} - \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{\left (b \right )} - 2 p x \log{\left (x \right )} + 2 p x + x \log{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

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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Giac [A]  time = 1.24949, size = 57, normalized size = 1.39 \begin{align*} 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 \left (c\right ) \end{align*}

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)