Optimal. Leaf size=116 \[ \frac {1}{4} c^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {b c^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}-\frac {2}{3} b^2 c^4 \log (x)-\frac {b^2 c^2}{12 x^2}+\frac {1}{3} b^2 c^4 \log \left (c^2 x^2+1\right ) \]
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Rubi [A] time = 0.22, 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 = {4852, 4918, 266, 44, 36, 29, 31, 4884} \[ \frac {1}{4} c^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {b c^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{6 x^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac {b^2 c^2}{12 x^2}+\frac {1}{3} b^2 c^4 \log \left (c^2 x^2+1\right )-\frac {2}{3} b^2 c^4 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 4852
Rule 4884
Rule 4918
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^6\right ) \operatorname {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*}
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Mathematica [A] time = 0.09, size = 128, normalized size = 1.10 \[ \frac {-3 a^2+6 a b c^3 x^3+2 b \tan ^{-1}(c x) \left (3 a \left (c^4 x^4-1\right )+b c x \left (3 c^2 x^2-1\right )\right )-2 a b c x-8 b^2 c^4 x^4 \log (x)+3 b^2 \left (c^4 x^4-1\right ) \tan ^{-1}(c x)^2-b^2 c^2 x^2+4 b^2 c^4 x^4 \log \left (c^2 x^2+1\right )}{12 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 135, normalized size = 1.16 \[ \frac {4 \, b^{2} c^{4} x^{4} \log \left (c^{2} x^{2} + 1\right ) - 8 \, b^{2} c^{4} x^{4} \log \relax (x) + 6 \, a b c^{3} x^{3} - b^{2} c^{2} x^{2} - 2 \, a b c x + 3 \, {\left (b^{2} c^{4} x^{4} - b^{2}\right )} \arctan \left (c x\right )^{2} - 3 \, a^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} + 3 \, b^{2} c^{3} x^{3} - b^{2} c x - 3 \, a b\right )} \arctan \left (c x\right )}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 147, normalized size = 1.27 \[ -\frac {a^{2}}{4 x^{4}}-\frac {b^{2} \arctan \left (c x \right )^{2}}{4 x^{4}}-\frac {c \,b^{2} \arctan \left (c x \right )}{6 x^{3}}+\frac {c^{3} b^{2} \arctan \left (c x \right )}{2 x}+\frac {c^{4} b^{2} \arctan \left (c x \right )^{2}}{4}-\frac {b^{2} c^{2}}{12 x^{2}}-\frac {2 c^{4} b^{2} \ln \left (c x \right )}{3}+\frac {b^{2} c^{4} \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {a b \arctan \left (c x \right )}{2 x^{4}}-\frac {a b c}{6 x^{3}}+\frac {c^{3} a b}{2 x}+\frac {c^{4} a b \arctan \left (c x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 152, normalized size = 1.31 \[ \frac {1}{6} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} a b + \frac {1}{12} \, {\left (2 \, {\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c \arctan \left (c x\right ) - \frac {{\left (3 \, c^{2} x^{2} \arctan \left (c x\right )^{2} - 4 \, c^{2} x^{2} \log \left (c^{2} x^{2} + 1\right ) + 8 \, c^{2} x^{2} \log \relax (x) + 1\right )} c^{2}}{x^{2}}\right )} b^{2} - \frac {b^{2} \arctan \left (c x\right )^{2}}{4 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 171, normalized size = 1.47 \[ \frac {b^2\,c^4\,{\mathrm {atan}\left (c\,x\right )}^2}{4}-\frac {2\,b^2\,c^4\,\ln \relax (x)}{3}-\frac {\frac {b^2\,{\mathrm {atan}\left (c\,x\right )}^2}{4}+\frac {a^2}{4}+x\,\left (\frac {c\,\mathrm {atan}\left (c\,x\right )\,b^2}{6}+\frac {a\,c\,b}{6}\right )-x^3\,\left (\frac {b^2\,c^3\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {a\,b\,c^3}{2}\right )+\frac {b^2\,c^2\,x^2}{12}+\frac {a\,b\,\mathrm {atan}\left (c\,x\right )}{2}}{x^4}+\frac {b^2\,c^4\,\ln \left (c\,x+1{}\mathrm {i}\right )}{3}+\frac {b^2\,c^4\,\ln \left (1+c\,x\,1{}\mathrm {i}\right )}{3}+\frac {a\,b\,c^4\,\ln \left (c\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {a\,b\,c^4\,\ln \left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.78, size = 170, normalized size = 1.47 \[ \begin {cases} - \frac {a^{2}}{4 x^{4}} + \frac {a b c^{4} \operatorname {atan}{\left (c x \right )}}{2} + \frac {a b c^{3}}{2 x} - \frac {a b c}{6 x^{3}} - \frac {a b \operatorname {atan}{\left (c x \right )}}{2 x^{4}} - \frac {2 b^{2} c^{4} \log {\relax (x )}}{3} + \frac {b^{2} c^{4} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3} + \frac {b^{2} c^{4} \operatorname {atan}^{2}{\left (c x \right )}}{4} + \frac {b^{2} c^{3} \operatorname {atan}{\left (c x \right )}}{2 x} - \frac {b^{2} c^{2}}{12 x^{2}} - \frac {b^{2} c \operatorname {atan}{\left (c x \right )}}{6 x^{3}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{4 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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