Optimal. Leaf size=41 \[ -\frac {a+b \tan ^{-1}\left (c x^2\right )}{4 x^4}-\frac {1}{4} b c^2 \tan ^{-1}\left (c x^2\right )-\frac {b c}{4 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5033, 275, 325, 203} \[ -\frac {a+b \tan ^{-1}\left (c x^2\right )}{4 x^4}-\frac {1}{4} b c^2 \tan ^{-1}\left (c x^2\right )-\frac {b c}{4 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 275
Rule 325
Rule 5033
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}\left (c x^2\right )}{x^5} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{2} (b c) \int \frac {1}{x^3 \left (1+c^2 x^4\right )} \, dx\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{4} (b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {b c}{4 x^2}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{4 x^4}-\frac {1}{4} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^2\right )\\ &=-\frac {b c}{4 x^2}-\frac {1}{4} b c^2 \tan ^{-1}\left (c x^2\right )-\frac {a+b \tan ^{-1}\left (c x^2\right )}{4 x^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 48, normalized size = 1.17 \[ -\frac {a}{4 x^4}-\frac {b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^4\right )}{4 x^2}-\frac {b \tan ^{-1}\left (c x^2\right )}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 30, normalized size = 0.73 \[ -\frac {b c x^{2} + {\left (b c^{2} x^{4} + b\right )} \arctan \left (c x^{2}\right ) + a}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.81, size = 74, normalized size = 1.80 \[ \frac {b c^{5} i x^{4} \log \left (c i x^{2} + 1\right ) - b c^{5} i x^{4} \log \left (-c i x^{2} + 1\right ) - 2 \, b c^{4} x^{2} - 2 \, b c^{3} \arctan \left (c x^{2}\right ) - 2 \, a c^{3}}{8 \, c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 39, normalized size = 0.95 \[ -\frac {a}{4 x^{4}}-\frac {b \arctan \left (c \,x^{2}\right )}{4 x^{4}}-\frac {b \,c^{2} \arctan \left (c \,x^{2}\right )}{4}-\frac {b c}{4 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 35, normalized size = 0.85 \[ -\frac {1}{4} \, {\left ({\left (c \arctan \left (c x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {\arctan \left (c x^{2}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.36, size = 41, normalized size = 1.00 \[ -\frac {\frac {b\,c\,x^2}{2}+\frac {a}{2}}{2\,x^4}-\frac {b\,c^2\,\mathrm {atan}\left (c\,x^2\right )}{4}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 24.12, size = 42, normalized size = 1.02 \[ - \frac {a}{4 x^{4}} - \frac {b c^{2} \operatorname {atan}{\left (c x^{2} \right )}}{4} - \frac {b c}{4 x^{2}} - \frac {b \operatorname {atan}{\left (c x^{2} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________