Optimal. Leaf size=57 \[ \frac{\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac{\tan ^{-1}\left (\sqrt{\frac{-a-b x^4+1}{a+b x^4+1}}\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.115012, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6715, 6313, 1961, 12, 203} \[ \frac{\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac{\tan ^{-1}\left (\sqrt{\frac{-a-b x^4+1}{a+b x^4+1}}\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6715
Rule 6313
Rule 1961
Rule 12
Rule 203
Rubi steps
\begin{align*} \int x^3 \text{sech}^{-1}\left (a+b x^4\right ) \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \text{sech}^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac{\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{\frac{1-a-b x}{1+a+b x}}}{1-a-b x} \, dx,x,x^4\right )\\ &=\frac{\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b}-b \operatorname{Subst}\left (\int \frac{1}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a-b x^4}{1+a+b x^4}}\right )\\ &=\frac{\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a-b x^4}{1+a+b x^4}}\right )}{2 b}\\ &=\frac{\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b}-\frac{\tan ^{-1}\left (\sqrt{\frac{1-a-b x^4}{1+a+b x^4}}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.202167, size = 84, normalized size = 1.47 \[ \frac{\frac{\sqrt{1-\left (a+b x^4\right )^2} \sin ^{-1}\left (a+b x^4\right )}{\sqrt{-\frac{a+b x^4-1}{a+b x^4+1}} \left (a+b x^4+1\right )}+\left (a+b x^4\right ) \text{sech}^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.289, size = 62, normalized size = 1.1 \begin{align*}{\frac{{\rm arcsech} \left (b{x}^{4}+a\right ){x}^{4}}{4}}+{\frac{{\rm arcsech} \left (b{x}^{4}+a\right )a}{4\,b}}-{\frac{1}{4\,b}\arctan \left ( \sqrt{ \left ( b{x}^{4}+a \right ) ^{-1}-1}\sqrt{ \left ( b{x}^{4}+a \right ) ^{-1}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.993601, size = 51, normalized size = 0.89 \begin{align*} \frac{{\left (b x^{4} + a\right )} \operatorname{arsech}\left (b x^{4} + a\right ) - \arctan \left (\sqrt{\frac{1}{{\left (b x^{4} + a\right )}^{2}} - 1}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.47328, size = 613, normalized size = 10.75 \begin{align*} \frac{2 \, b x^{4} \log \left (\frac{{\left (b x^{4} + a\right )} \sqrt{-\frac{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}} + 1}{b x^{4} + a}\right ) + a \log \left (\frac{{\left (b x^{4} + a\right )} \sqrt{-\frac{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}} + 1}{x^{4}}\right ) - a \log \left (\frac{{\left (b x^{4} + a\right )} \sqrt{-\frac{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}} - 1}{x^{4}}\right ) - 2 \, \arctan \left (\frac{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt{-\frac{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arsech}\left (b x^{4} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]