Integrand size = 16, antiderivative size = 60 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=-\frac {2 b p}{3 a x}-\frac {2 b^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3} \] Output:
-2/3*b*p/a/x-2/3*b^(3/2)*p*arctan(b^(1/2)*x/a^(1/2))/a^(3/2)-1/3*ln(c*(b*x ^2+a)^p)/x^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=-\frac {2 b p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {b x^2}{a}\right )}{3 a x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3} \] Input:
Integrate[Log[c*(a + b*x^2)^p]/x^4,x]
Output:
(-2*b*p*Hypergeometric2F1[-1/2, 1, 1/2, -((b*x^2)/a)])/(3*a*x) - Log[c*(a + b*x^2)^p]/(3*x^3)
Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2905, 264, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {2}{3} b p \int \frac {1}{x^2 \left (b x^2+a\right )}dx-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {2}{3} b p \left (-\frac {b \int \frac {1}{b x^2+a}dx}{a}-\frac {1}{a x}\right )-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2}{3} b p \left (-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right )-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{3 x^3}\) |
Input:
Int[Log[c*(a + b*x^2)^p]/x^4,x]
Output:
(2*b*p*(-(1/(a*x)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)))/3 - L og[c*(a + b*x^2)^p]/(3*x^3)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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] - Simp[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]
Time = 0.83 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{3 x^{3}}+\frac {2 p b \left (-\frac {1}{a x}-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}\right )}{3}\) | \(52\) |
risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{3 x^{3}}-\frac {i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{3} \textit {\_Z}^{2}+b^{3} p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 a^{3} \textit {\_R}^{2}+2 b^{3} p^{2}\right ) x +a^{2} b p \textit {\_R} \right )\right ) a \,x^{3}+4 x^{2} p b +2 \ln \left (c \right ) a}{6 a \,x^{3}}\) | \(211\) |
Input:
int(ln(c*(b*x^2+a)^p)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*ln(c*(b*x^2+a)^p)/x^3+2/3*p*b*(-1/a/x-1/a*b/(a*b)^(1/2)*arctan(b*x/(a *b)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.25 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=\left [\frac {b p x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b p x^{2} - a p \log \left (b x^{2} + a\right ) - a \log \left (c\right )}{3 \, a x^{3}}, -\frac {2 \, b p x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, b p x^{2} + a p \log \left (b x^{2} + a\right ) + a \log \left (c\right )}{3 \, a x^{3}}\right ] \] Input:
integrate(log(c*(b*x^2+a)^p)/x^4,x, algorithm="fricas")
Output:
[1/3*(b*p*x^3*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 2*b*p*x^2 - a*p*log(b*x^2 + a) - a*log(c))/(a*x^3), -1/3*(2*b*p*x^3*sqrt( b/a)*arctan(x*sqrt(b/a)) + 2*b*p*x^2 + a*p*log(b*x^2 + a) + a*log(c))/(a*x ^3)]
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (56) = 112\).
Time = 38.14 (sec) , antiderivative size = 496, normalized size of antiderivative = 8.27 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=\begin {cases} - \frac {\log {\left (0^{p} c \right )}}{3 x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\log {\left (a^{p} c \right )}}{3 x^{3}} & \text {for}\: b = 0 \\- \frac {2 p}{9 x^{3}} - \frac {\log {\left (c \left (b x^{2}\right )^{p} \right )}}{3 x^{3}} & \text {for}\: a = 0 \\- \frac {\log {\left (0^{p} c \right )}}{3 x^{3}} & \text {for}\: a = - b x^{2} \\- \frac {a^{2} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{3 a^{2} x^{3} \sqrt {- \frac {a}{b}} + 3 a b x^{5} \sqrt {- \frac {a}{b}}} - \frac {2 a p x^{3} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} - \frac {2 a p x^{2} \sqrt {- \frac {a}{b}}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} + \frac {a x^{3} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} - \frac {a x^{2} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} - \frac {2 b p x^{5} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} - \frac {2 b p x^{4} \sqrt {- \frac {a}{b}}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} + \frac {b x^{5} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {3 a^{2} x^{3} \sqrt {- \frac {a}{b}}}{b} + 3 a x^{5} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:
integrate(ln(c*(b*x**2+a)**p)/x**4,x)
Output:
Piecewise((-log(0**p*c)/(3*x**3), Eq(a, 0) & Eq(b, 0)), (-log(a**p*c)/(3*x **3), Eq(b, 0)), (-2*p/(9*x**3) - log(c*(b*x**2)**p)/(3*x**3), Eq(a, 0)), (-log(0**p*c)/(3*x**3), Eq(a, -b*x**2)), (-a**2*sqrt(-a/b)*log(c*(a + b*x* *2)**p)/(3*a**2*x**3*sqrt(-a/b) + 3*a*b*x**5*sqrt(-a/b)) - 2*a*p*x**3*log( x - sqrt(-a/b))/(3*a**2*x**3*sqrt(-a/b)/b + 3*a*x**5*sqrt(-a/b)) - 2*a*p*x **2*sqrt(-a/b)/(3*a**2*x**3*sqrt(-a/b)/b + 3*a*x**5*sqrt(-a/b)) + a*x**3*l og(c*(a + b*x**2)**p)/(3*a**2*x**3*sqrt(-a/b)/b + 3*a*x**5*sqrt(-a/b)) - a *x**2*sqrt(-a/b)*log(c*(a + b*x**2)**p)/(3*a**2*x**3*sqrt(-a/b)/b + 3*a*x* *5*sqrt(-a/b)) - 2*b*p*x**5*log(x - sqrt(-a/b))/(3*a**2*x**3*sqrt(-a/b)/b + 3*a*x**5*sqrt(-a/b)) - 2*b*p*x**4*sqrt(-a/b)/(3*a**2*x**3*sqrt(-a/b)/b + 3*a*x**5*sqrt(-a/b)) + b*x**5*log(c*(a + b*x**2)**p)/(3*a**2*x**3*sqrt(-a /b)/b + 3*a*x**5*sqrt(-a/b)), True))
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=-\frac {2}{3} \, b p {\left (\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {1}{a x}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{3 \, x^{3}} \] Input:
integrate(log(c*(b*x^2+a)^p)/x^4,x, algorithm="maxima")
Output:
-2/3*b*p*(b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) + 1/(a*x)) - 1/3*log((b*x^ 2 + a)^p*c)/x^3
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=-\frac {2 \, b^{2} p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a} - \frac {p \log \left (b x^{2} + a\right )}{3 \, x^{3}} - \frac {2 \, b p x^{2} + a \log \left (c\right )}{3 \, a x^{3}} \] Input:
integrate(log(c*(b*x^2+a)^p)/x^4,x, algorithm="giac")
Output:
-2/3*b^2*p*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/3*p*log(b*x^2 + a)/x^3 - 1/3*(2*b*p*x^2 + a*log(c))/(a*x^3)
Time = 26.40 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.77 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{3\,x^3}-\frac {2\,b^{3/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{3\,a^{3/2}}-\frac {2\,b\,p}{3\,a\,x} \] Input:
int(log(c*(a + b*x^2)^p)/x^4,x)
Output:
- log(c*(a + b*x^2)^p)/(3*x^3) - (2*b^(3/2)*p*atan((b^(1/2)*x)/a^(1/2)))/( 3*a^(3/2)) - (2*b*p)/(3*a*x)
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b p \,x^{3}-\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a^{2}-2 a b p \,x^{2}}{3 a^{2} x^{3}} \] Input:
int(log(c*(b*x^2+a)^p)/x^4,x)
Output:
( - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*p*x**3 - log((a + b* x**2)**p*c)*a**2 - 2*a*b*p*x**2)/(3*a**2*x**3)