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

Optimal. Leaf size=44 \[ \frac{2 \sqrt{b} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.0204981, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2455, 205} \[ \frac{2 \sqrt{b} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]/x^2,x]

[Out]

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

Rule 2455

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]

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

Mathematica [A]  time = 0.0090083, size = 44, normalized size = 1. \[ \frac{2 \sqrt{b} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]/x^2,x]

[Out]

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

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Maple [C]  time = 0.333, size = 183, normalized size = 4.2 \begin{align*} -{\frac{\ln \left ( \left ( b{x}^{2}+a \right ) ^{p} \right ) }{x}}-{\frac{i\pi \,{\it csgn} \left ( i \left ( b{x}^{2}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i \left ( b{x}^{2}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{2}+b{p}^{2} \right ) }{\it \_R}\,\ln \left ( \left ( 3\,{{\it \_R}}^{2}a+2\,b{p}^{2} \right ) x-ap{\it \_R} \right ) x+2\,\ln \left ( c \right ) }{2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x*ln((b*x^2+a)^p)-1/2*(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(
b*x^2+a)^p)*csgn(I*c)-I*Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-2*sum(_R*ln((3*_R^2*
a+2*b*p^2)*x-a*p*_R),_R=RootOf(_Z^2*a+b*p^2))*x+2*ln(c))/x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.05687, size = 228, normalized size = 5.18 \begin{align*} \left [\frac{p x \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - p \log \left (b x^{2} + a\right ) - \log \left (c\right )}{x}, \frac{2 \, p x \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - p \log \left (b x^{2} + a\right ) - \log \left (c\right )}{x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 44.2599, size = 377, normalized size = 8.57 \begin{align*} \begin{cases} - \frac{\log{\left (0^{p} c \right )}}{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{p \log{\left (b \right )}}{x} - \frac{2 p \log{\left (x \right )}}{x} - \frac{2 p}{x} - \frac{\log{\left (c \right )}}{x} & \text{for}\: a = 0 \\- \frac{\log{\left (a^{p} c \right )}}{x} & \text{for}\: b = 0 \\\frac{i a^{\frac{3}{2}} p x \sqrt{\frac{1}{b}} \log{\left (a + b x^{2} \right )}}{\frac{a^{2} x}{b} + a x^{3}} - \frac{2 i a^{\frac{3}{2}} p x \sqrt{\frac{1}{b}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{\frac{a^{2} x}{b} + a x^{3}} + \frac{i a^{\frac{3}{2}} x \sqrt{\frac{1}{b}} \log{\left (c \right )}}{\frac{a^{2} x}{b} + a x^{3}} + \frac{i \sqrt{a} b p x^{3} \sqrt{\frac{1}{b}} \log{\left (a + b x^{2} \right )}}{\frac{a^{2} x}{b} + a x^{3}} - \frac{2 i \sqrt{a} b p x^{3} \sqrt{\frac{1}{b}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{\frac{a^{2} x}{b} + a x^{3}} + \frac{i \sqrt{a} b x^{3} \sqrt{\frac{1}{b}} \log{\left (c \right )}}{\frac{a^{2} x}{b} + a x^{3}} - \frac{a^{2} p \log{\left (a + b x^{2} \right )}}{a^{2} x + a b x^{3}} - \frac{a^{2} \log{\left (c \right )}}{a^{2} x + a b x^{3}} - \frac{a p x^{2} \log{\left (a + b x^{2} \right )}}{\frac{a^{2} x}{b} + a x^{3}} - \frac{a x^{2} \log{\left (c \right )}}{\frac{a^{2} x}{b} + a x^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-log(0**p*c)/x, Eq(a, 0) & Eq(b, 0)), (-p*log(b)/x - 2*p*log(x)/x - 2*p/x - log(c)/x, Eq(a, 0)), (-
log(a**p*c)/x, Eq(b, 0)), (I*a**(3/2)*p*x*sqrt(1/b)*log(a + b*x**2)/(a**2*x/b + a*x**3) - 2*I*a**(3/2)*p*x*sqr
t(1/b)*log(-I*sqrt(a)*sqrt(1/b) + x)/(a**2*x/b + a*x**3) + I*a**(3/2)*x*sqrt(1/b)*log(c)/(a**2*x/b + a*x**3) +
I*sqrt(a)*b*p*x**3*sqrt(1/b)*log(a + b*x**2)/(a**2*x/b + a*x**3) - 2*I*sqrt(a)*b*p*x**3*sqrt(1/b)*log(-I*sqrt
(a)*sqrt(1/b) + x)/(a**2*x/b + a*x**3) + I*sqrt(a)*b*x**3*sqrt(1/b)*log(c)/(a**2*x/b + a*x**3) - a**2*p*log(a
+ b*x**2)/(a**2*x + a*b*x**3) - a**2*log(c)/(a**2*x + a*b*x**3) - a*p*x**2*log(a + b*x**2)/(a**2*x/b + a*x**3)
- a*x**2*log(c)/(a**2*x/b + a*x**3), True))

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Giac [A]  time = 1.16791, size = 54, normalized size = 1.23 \begin{align*} \frac{2 \, b p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b}} - \frac{p \log \left (b x^{2} + a\right )}{x} - \frac{\log \left (c\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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