3.1.40 \(\int \log (c (a+\frac {b}{x^2})^p) \, dx\) [40]

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

[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)

________________________________________________________________________________________

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 = {2498, 269, 211} \begin {gather*} \frac {2 \sqrt {b} p \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}}+x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \end {gather*}

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 211

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

Rule 269

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 2498

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*}

________________________________________________________________________________________

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

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]

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 38, normalized size = 0.93

method	result	size
default	\(x \ln \left (c \left (\frac {x^{2} a +b}{x^{2}}\right )^{p}\right )+\frac {2 p b \arctan \left (\frac {a x}{\sqrt {b a}}\right )}{\sqrt {b a}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.56, size = 33, normalized size = 0.80 \begin {gather*} \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 ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 107, normalized size = 2.61 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (39) = 78\).
time = 3.79, size = 95, normalized size = 2.32 \begin {gather*} \begin {cases} x \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\2 p x + x \log {\left (c \left (\frac {b}{x^{2}}\right )^{p} \right )} & \text {for}\: a = 0 \\x \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\x \log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )} + \frac {b p \log {\left (x - \sqrt {- \frac {b}{a}} \right )}}{a \sqrt {- \frac {b}{a}}} - \frac {b p \log {\left (x + \sqrt {- \frac {b}{a}} \right )}}{a \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*log(0**p*c), Eq(a, 0) & Eq(b, 0)), (2*p*x + x*log(c*(b/x**2)**p), Eq(a, 0)), (x*log(a**p*c), Eq(b
, 0)), (x*log(c*(a + b/x**2)**p) + b*p*log(x - sqrt(-b/a))/(a*sqrt(-b/a)) - b*p*log(x + sqrt(-b/a))/(a*sqrt(-b
/a)), True))

________________________________________________________________________________________

Giac [A]
time = 4.15, size = 42, normalized size = 1.02 \begin {gather*} 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 {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.11, size = 33, normalized size = 0.80 \begin {gather*} 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}} \end {gather*}

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)

________________________________________________________________________________________