Optimal. Leaf size=129 \[ -\frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (a^2+1\right )^3}+\frac {\left (1-3 a^2\right ) b^3 \log \left ((a+b x)^2+1\right )}{6 \left (a^2+1\right )^3}+\frac {a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (a^2+1\right )^3}+\frac {2 a b^2}{3 \left (a^2+1\right )^2 x}-\frac {b}{6 \left (a^2+1\right ) x^2}-\frac {\tan ^{-1}(a+b x)}{3 x^3} \]
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Rubi [A] time = 0.11, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5045, 371, 710, 801, 635, 203, 260} \[ \frac {2 a b^2}{3 \left (a^2+1\right )^2 x}-\frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (a^2+1\right )^3}+\frac {\left (1-3 a^2\right ) b^3 \log \left ((a+b x)^2+1\right )}{6 \left (a^2+1\right )^3}+\frac {a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (a^2+1\right )^3}-\frac {b}{6 \left (a^2+1\right ) x^2}-\frac {\tan ^{-1}(a+b x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 371
Rule 635
Rule 710
Rule 801
Rule 5045
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{x^4} \, dx &=-\frac {\tan ^{-1}(a+b x)}{3 x^3}+\frac {1}{3} b \int \frac {1}{x^3 \left (1+(a+b x)^2\right )} \, dx\\ &=-\frac {\tan ^{-1}(a+b x)}{3 x^3}+\frac {1}{3} b^3 \operatorname {Subst}\left (\int \frac {1}{(-a+x)^3 \left (1+x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {b}{6 \left (1+a^2\right ) x^2}-\frac {\tan ^{-1}(a+b x)}{3 x^3}+\frac {b^3 \operatorname {Subst}\left (\int \frac {-a-x}{(-a+x)^2 \left (1+x^2\right )} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b}{6 \left (1+a^2\right ) x^2}-\frac {\tan ^{-1}(a+b x)}{3 x^3}+\frac {b^3 \operatorname {Subst}\left (\int \left (-\frac {2 a}{\left (1+a^2\right ) (a-x)^2}+\frac {1-3 a^2}{\left (1+a^2\right )^2 (a-x)}+\frac {a \left (3-a^2\right )+\left (1-3 a^2\right ) x}{\left (1+a^2\right )^2 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b}{6 \left (1+a^2\right ) x^2}+\frac {2 a b^2}{3 \left (1+a^2\right )^2 x}-\frac {\tan ^{-1}(a+b x)}{3 x^3}-\frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (1+a^2\right )^3}+\frac {b^3 \operatorname {Subst}\left (\int \frac {a \left (3-a^2\right )+\left (1-3 a^2\right ) x}{1+x^2} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )^3}\\ &=-\frac {b}{6 \left (1+a^2\right ) x^2}+\frac {2 a b^2}{3 \left (1+a^2\right )^2 x}-\frac {\tan ^{-1}(a+b x)}{3 x^3}-\frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (1+a^2\right )^3}+\frac {\left (\left (1-3 a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )^3}+\frac {\left (a \left (3-a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )^3}\\ &=-\frac {b}{6 \left (1+a^2\right ) x^2}+\frac {2 a b^2}{3 \left (1+a^2\right )^2 x}+\frac {a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (1+a^2\right )^3}-\frac {\tan ^{-1}(a+b x)}{3 x^3}-\frac {\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (1+a^2\right )^3}+\frac {\left (1-3 a^2\right ) b^3 \log \left (1+(a+b x)^2\right )}{6 \left (1+a^2\right )^3}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 128, normalized size = 0.99 \[ \frac {2 \left (3 a^2-1\right ) b^3 x^3 \log (x)-(a-i) b x \left ((a+i) \left (a^2-4 a b x+1\right )+i (a-i)^2 b^2 x^2 \log (a+b x+i)\right )-2 \left (a^2+1\right )^3 \tan ^{-1}(a+b x)+i (a+i)^3 b^3 x^3 \log (-a-b x+i)}{6 \left (a^2+1\right )^3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 135, normalized size = 1.05 \[ -\frac {{\left (3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, {\left (3 \, a^{2} - 1\right )} b^{3} x^{3} \log \relax (x) - 4 \, {\left (a^{3} + a\right )} b^{2} x^{2} + {\left (a^{4} + 2 \, a^{2} + 1\right )} b x + 2 \, {\left ({\left (a^{3} - 3 \, a\right )} b^{3} x^{3} + a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} \arctan \left (b x + a\right )}{6 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 162, normalized size = 1.26 \[ -\frac {\arctan \left (b x +a \right )}{3 x^{3}}-\frac {b}{6 \left (a^{2}+1\right ) x^{2}}+\frac {b^{3} \ln \left (b x \right ) a^{2}}{\left (a^{2}+1\right )^{3}}-\frac {b^{3} \ln \left (b x \right )}{3 \left (a^{2}+1\right )^{3}}+\frac {2 a \,b^{2}}{3 \left (a^{2}+1\right )^{2} x}-\frac {b^{3} \ln \left (1+\left (b x +a \right )^{2}\right ) a^{2}}{2 \left (a^{2}+1\right )^{3}}+\frac {b^{3} \ln \left (1+\left (b x +a \right )^{2}\right )}{6 \left (a^{2}+1\right )^{3}}-\frac {b^{3} \arctan \left (b x +a \right ) a^{3}}{3 \left (a^{2}+1\right )^{3}}+\frac {b^{3} \arctan \left (b x +a \right ) a}{\left (a^{2}+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 165, normalized size = 1.28 \[ -\frac {1}{6} \, {\left (\frac {2 \, {\left (a^{3} - 3 \, a\right )} b^{2} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} + \frac {{\left (3 \, a^{2} - 1\right )} b^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {2 \, {\left (3 \, a^{2} - 1\right )} b^{2} \log \relax (x)}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {4 \, a b x - a^{2} - 1}{{\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}}\right )} b - \frac {\arctan \left (b x + a\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 288, normalized size = 2.23 \[ -\frac {\frac {b\,x}{6}+\mathrm {atan}\left (a+b\,x\right )\,\left (\frac {a^2}{3}+\frac {1}{3}\right )+\frac {b^2\,x^2\,\mathrm {atan}\left (a+b\,x\right )}{3}+\frac {x^3\,\left (b^3-7\,a^2\,b^3\right )}{6\,\left (a^4+2\,a^2+1\right )}-\frac {a\,b^2\,x^2}{3\,\left (a^2+1\right )}-\frac {2\,a\,b^4\,x^4}{3\,{\left (a^2+1\right )}^2}+\frac {2\,a\,b\,x\,\mathrm {atan}\left (a+b\,x\right )}{3}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5+x^3}-\frac {\ln \relax (x)\,\left (\frac {b^3}{3}-a^2\,b^3\right )}{a^6+3\,a^4+3\,a^2+1}-\frac {b^3\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )\,\left (3\,a^2-1\right )}{6\,\left (a^6+3\,a^4+3\,a^2+1\right )}-\frac {a\,\mathrm {atan}\left (\frac {2\,x\,b^2+2\,a\,b}{2\,\sqrt {b^2\,\left (a^2+1\right )-a^2\,b^2}}\right )\,\left (a^2-3\right )\,{\left (b^2\right )}^{3/2}}{3\,\left (a^6+3\,a^4+3\,a^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.66, size = 760, normalized size = 5.89 \[ \begin {cases} \frac {i b^{3} \operatorname {atan}{\left (b x - i \right )}}{24} + \frac {i b^{2}}{24 x} - \frac {b}{24 x^{2}} - \frac {\operatorname {atan}{\left (b x - i \right )}}{3 x^{3}} - \frac {i}{18 x^{3}} & \text {for}\: a = - i \\- \frac {i b^{3} \operatorname {atan}{\left (b x + i \right )}}{24} - \frac {i b^{2}}{24 x} - \frac {b}{24 x^{2}} - \frac {\operatorname {atan}{\left (b x + i \right )}}{3 x^{3}} + \frac {i}{18 x^{3}} & \text {for}\: a = i \\- \frac {2 a^{6} \operatorname {atan}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {a^{4} b x}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {6 a^{4} \operatorname {atan}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {2 a^{3} b^{3} x^{3} \operatorname {atan}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {4 a^{3} b^{2} x^{2}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {6 a^{2} b^{3} x^{3} \log {\relax (x )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {3 a^{2} b^{3} x^{3} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {2 a^{2} b x}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {6 a^{2} \operatorname {atan}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {6 a b^{3} x^{3} \operatorname {atan}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {4 a b^{2} x^{2}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {2 b^{3} x^{3} \log {\relax (x )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} + \frac {b^{3} x^{3} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {b x}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} - \frac {2 \operatorname {atan}{\left (a + b x \right )}}{6 a^{6} x^{3} + 18 a^{4} x^{3} + 18 a^{2} x^{3} + 6 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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