Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\tan ^{-1}(a+b x)}{\sqrt [3]{(a+b x)^2+1}},x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\tan ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}(x)}{\sqrt [3]{1+x^2}} \, dx,x,a+b x\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 163, normalized size = 7.09 \[ \frac {\frac {5 \sqrt [3]{2} \sqrt {\pi } \Gamma \left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{(a+b x)^2+1}\right )}{(a+b x)^2+1}+6 \Gamma \left (\frac {11}{6}\right ) \Gamma \left (\frac {7}{3}\right ) \left (\frac {4 (a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{(a+b x)^2+1}\right ) \tan ^{-1}(a+b x)}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right )}{20 b \Gamma \left (\frac {11}{6}\right ) \Gamma \left (\frac {7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] 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 = 1.47, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (b x +a \right )}{\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {atan}\left (a+b\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}{\left (a + b x \right )}}{\sqrt [3]{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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