3.1.42 \(\int \frac {\log (c (a+\frac {b}{x^2})^p)}{x^2} \, dx\) [42]

Optimal. Leaf size=50 \[ \frac {2 p}{x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2505, 269, 331, 211} \begin {gather*} \frac {2 \sqrt {a} p \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+\frac {2 p}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x^2)^p]/x^2,x]

[Out]

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

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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2505

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

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2} \, dx &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}-(2 b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^4} \, dx\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}-(2 b p) \int \frac {1}{x^2 \left (b+a x^2\right )} \, dx\\ &=\frac {2 p}{x}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+(2 a p) \int \frac {1}{b+a x^2} \, dx\\ &=\frac {2 p}{x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 52, normalized size = 1.04 \begin {gather*} \frac {2 p}{x}-\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x^2)^p]/x^2,x]

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.55, size = 49, normalized size = 0.98 \begin {gather*} 2 \, b p {\left (\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {1}{b x}\right )} - \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*p*(a*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b) + 1/(b*x)) - log((a + b/x^2)^p*c)/x

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Fricas [A]
time = 0.36, size = 119, normalized size = 2.38 \begin {gather*} \left [\frac {p x \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - b}{a x^{2} + b}\right ) - p \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, p - \log \left (c\right )}{x}, \frac {2 \, p x \sqrt {\frac {a}{b}} \arctan \left (x \sqrt {\frac {a}{b}}\right ) - p \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, p - \log \left (c\right )}{x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (44) = 88\).
time = 8.70, size = 97, normalized size = 1.94 \begin {gather*} \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 p}{x} - \frac {\log {\left (c \left (\frac {b}{x^{2}}\right )^{p} \right )}}{x} & \text {for}\: a = 0 \\- \frac {\log {\left (a^{p} c \right )}}{x} & \text {for}\: b = 0 \\\frac {p \log {\left (x - \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- \frac {b}{a}}} - \frac {p \log {\left (x + \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- \frac {b}{a}}} + \frac {2 p}{x} - \frac {\log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{x} & \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**2,x)

[Out]

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

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Giac [A]
time = 4.93, size = 54, normalized size = 1.08 \begin {gather*} \frac {2 \, a p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}} - \frac {p \log \left (a x^{2} + b\right )}{x} + \frac {p \log \left (x^{2}\right )}{x} + \frac {2 \, p - \log \left (c\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.24, size = 42, normalized size = 0.84 \begin {gather*} \frac {2\,p}{x}-\frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x}+\frac {2\,\sqrt {a}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x^2)^p)/x^2,x)

[Out]

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

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