Optimal. Leaf size=55 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 63, 298, 203, 206} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [4]{b+a x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^4}\right )}{a}\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 48, normalized size = 0.87 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.05, size = 55, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 81, normalized size = 1.47 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {\sqrt {a x^{4} + b} + \sqrt {b}}}{b^{\frac {1}{4}}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {\log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}\right )}{4 \, b^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 186, normalized size = 3.38 \begin {gather*} -\frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{4 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{4 \, b} + \frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right )}{8 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right )}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 57, normalized size = 1.04 \begin {gather*} \frac {\arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{2 \, b^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 36, normalized size = 0.65 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )-\mathrm {atanh}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )}{2\,b^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.87, size = 37, normalized size = 0.67 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt [4]{a} x \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________