Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\tan ^{-1}(a+b x)}{\sqrt [3]{(a+b x)^2+1}},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0379871, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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.
[In]
[Out]
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*}
Mathematica [A] time = 0.363002, size = 163, normalized size = 7.41 \[ \frac{\frac{5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{(a+b x)^2+1}\right )}{(a+b x)^2+1}+6 \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (\frac{4 (a+b x) \tan ^{-1}(a+b x) \text{Hypergeometric2F1}\left (1,\frac{4}{3},\frac{11}{6},\frac{1}{(a+b x)^2+1}\right )}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right )}{20 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.044, size = 0, normalized size = 0. \begin{align*} \int{\arctan \left ( bx+a \right ){\frac{1}{\sqrt [3]{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}{\left (a + b x \right )}}{\sqrt [3]{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]