Optimal. Leaf size=116 \[ -\frac {b^2 c^2}{12 x^2}-\frac {b c (a+b \text {ArcTan}(c x))}{6 x^3}+\frac {b c^3 (a+b \text {ArcTan}(c x))}{2 x}+\frac {1}{4} c^4 (a+b \text {ArcTan}(c x))^2-\frac {(a+b \text {ArcTan}(c x))^2}{4 x^4}-\frac {2}{3} b^2 c^4 \log (x)+\frac {1}{3} b^2 c^4 \log \left (1+c^2 x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4946, 5038,
272, 46, 36, 29, 31, 5004} \begin {gather*} \frac {1}{4} c^4 (a+b \text {ArcTan}(c x))^2+\frac {b c^3 (a+b \text {ArcTan}(c x))}{2 x}-\frac {(a+b \text {ArcTan}(c x))^2}{4 x^4}-\frac {b c (a+b \text {ArcTan}(c x))}{6 x^3}-\frac {2}{3} b^2 c^4 \log (x)-\frac {b^2 c^2}{12 x^2}+\frac {1}{3} b^2 c^4 \log \left (c^2 x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 4946
Rule 5004
Rule 5038
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^5} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} (b c) \int \frac {a+b \tan ^{-1}(c x)}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} (b c) \int \frac {a+b \tan ^{-1}(c x)}{x^4} \, dx-\frac {1}{2} \left (b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{6} \left (b^2 c^2\right ) \int \frac {1}{x^3 \left (1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx+\frac {1}{2} \left (b c^5\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}+\frac {b c^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} \left (b^2 c^4\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}+\frac {b c^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{4} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}+\frac {b c^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac {1}{6} b^2 c^4 \log (x)+\frac {1}{12} b^2 c^4 \log \left (1+c^2 x^2\right )-\frac {1}{4} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^6\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}+\frac {b c^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac {2}{3} b^2 c^4 \log (x)+\frac {1}{3} b^2 c^4 \log \left (1+c^2 x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 128, normalized size = 1.10 \begin {gather*} \frac {-3 a^2-2 a b c x-b^2 c^2 x^2+6 a b c^3 x^3+2 b \left (b c x \left (-1+3 c^2 x^2\right )+3 a \left (-1+c^4 x^4\right )\right ) \text {ArcTan}(c x)+3 b^2 \left (-1+c^4 x^4\right ) \text {ArcTan}(c x)^2-8 b^2 c^4 x^4 \log (x)+4 b^2 c^4 x^4 \log \left (1+c^2 x^2\right )}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.25, size = 152, normalized size = 1.31
method | result | size |
derivativedivides | \(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}-\frac {b^{2} \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {b^{2} \arctan \left (c x \right )^{2}}{4}-\frac {b^{2} \arctan \left (c x \right )}{6 c^{3} x^{3}}+\frac {b^{2} \arctan \left (c x \right )}{2 c x}+\frac {b^{2} \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {b^{2}}{12 c^{2} x^{2}}-\frac {2 b^{2} \ln \left (c x \right )}{3}-\frac {a b \arctan \left (c x \right )}{2 c^{4} x^{4}}+\frac {a b \arctan \left (c x \right )}{2}-\frac {a b}{6 c^{3} x^{3}}+\frac {a b}{2 c x}\right )\) | \(152\) |
default | \(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}-\frac {b^{2} \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {b^{2} \arctan \left (c x \right )^{2}}{4}-\frac {b^{2} \arctan \left (c x \right )}{6 c^{3} x^{3}}+\frac {b^{2} \arctan \left (c x \right )}{2 c x}+\frac {b^{2} \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {b^{2}}{12 c^{2} x^{2}}-\frac {2 b^{2} \ln \left (c x \right )}{3}-\frac {a b \arctan \left (c x \right )}{2 c^{4} x^{4}}+\frac {a b \arctan \left (c x \right )}{2}-\frac {a b}{6 c^{3} x^{3}}+\frac {a b}{2 c x}\right )\) | \(152\) |
risch | \(-\frac {b^{2} \left (c^{4} x^{4}-1\right ) \ln \left (i c x +1\right )^{2}}{16 x^{4}}+\frac {i b \left (-3 i b \,c^{4} x^{4} \ln \left (-i c x +1\right )-6 b \,c^{3} x^{3}+2 x b c +6 a +3 i b \ln \left (-i c x +1\right )\right ) \ln \left (i c x +1\right )}{24 x^{4}}-\frac {-12 i \ln \left (\left (4 i b c -a c \right ) x -4 b -i a \right ) a b \,c^{4} x^{4}+12 i \ln \left (\left (-4 i b c -a c \right ) x -4 b +i a \right ) a b \,c^{4} x^{4}+3 c^{4} b^{2} x^{4} \ln \left (-i c x +1\right )^{2}-16 \ln \left (\left (4 i b c -a c \right ) x -4 b -i a \right ) b^{2} c^{4} x^{4}-16 \ln \left (\left (-4 i b c -a c \right ) x -4 b +i a \right ) b^{2} c^{4} x^{4}+32 b^{2} c^{4} \ln \left (-x \right ) x^{4}-12 i b^{2} c^{3} x^{3} \ln \left (-i c x +1\right )-24 a b \,c^{3} x^{3}+4 i b^{2} c x \ln \left (-i c x +1\right )+4 b^{2} c^{2} x^{2}+12 i \ln \left (-i c x +1\right ) a b +8 a b c x -3 b^{2} \ln \left (-i c x +1\right )^{2}+12 a^{2}}{48 x^{4}}\) | \(358\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 152, normalized size = 1.31 \begin {gather*} \frac {1}{6} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} a b + \frac {1}{12} \, {\left (2 \, {\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c \arctan \left (c x\right ) - \frac {{\left (3 \, c^{2} x^{2} \arctan \left (c x\right )^{2} - 4 \, c^{2} x^{2} \log \left (c^{2} x^{2} + 1\right ) + 8 \, c^{2} x^{2} \log \left (x\right ) + 1\right )} c^{2}}{x^{2}}\right )} b^{2} - \frac {b^{2} \arctan \left (c x\right )^{2}}{4 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.15, size = 135, normalized size = 1.16 \begin {gather*} \frac {4 \, b^{2} c^{4} x^{4} \log \left (c^{2} x^{2} + 1\right ) - 8 \, b^{2} c^{4} x^{4} \log \left (x\right ) + 6 \, a b c^{3} x^{3} - b^{2} c^{2} x^{2} - 2 \, a b c x + 3 \, {\left (b^{2} c^{4} x^{4} - b^{2}\right )} \arctan \left (c x\right )^{2} - 3 \, a^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} + 3 \, b^{2} c^{3} x^{3} - b^{2} c x - 3 \, a b\right )} \arctan \left (c x\right )}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.51, size = 170, normalized size = 1.47 \begin {gather*} \begin {cases} - \frac {a^{2}}{4 x^{4}} + \frac {a b c^{4} \operatorname {atan}{\left (c x \right )}}{2} + \frac {a b c^{3}}{2 x} - \frac {a b c}{6 x^{3}} - \frac {a b \operatorname {atan}{\left (c x \right )}}{2 x^{4}} - \frac {2 b^{2} c^{4} \log {\left (x \right )}}{3} + \frac {b^{2} c^{4} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3} + \frac {b^{2} c^{4} \operatorname {atan}^{2}{\left (c x \right )}}{4} + \frac {b^{2} c^{3} \operatorname {atan}{\left (c x \right )}}{2 x} - \frac {b^{2} c^{2}}{12 x^{2}} - \frac {b^{2} c \operatorname {atan}{\left (c x \right )}}{6 x^{3}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{4 x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.26, size = 171, normalized size = 1.47 \begin {gather*} \frac {b^2\,c^4\,{\mathrm {atan}\left (c\,x\right )}^2}{4}-\frac {2\,b^2\,c^4\,\ln \left (x\right )}{3}-\frac {\frac {b^2\,{\mathrm {atan}\left (c\,x\right )}^2}{4}+\frac {a^2}{4}+x\,\left (\frac {c\,\mathrm {atan}\left (c\,x\right )\,b^2}{6}+\frac {a\,c\,b}{6}\right )-x^3\,\left (\frac {b^2\,c^3\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {a\,b\,c^3}{2}\right )+\frac {b^2\,c^2\,x^2}{12}+\frac {a\,b\,\mathrm {atan}\left (c\,x\right )}{2}}{x^4}+\frac {b^2\,c^4\,\ln \left (c\,x+1{}\mathrm {i}\right )}{3}+\frac {b^2\,c^4\,\ln \left (1+c\,x\,1{}\mathrm {i}\right )}{3}+\frac {a\,b\,c^4\,\ln \left (c\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {a\,b\,c^4\,\ln \left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________