Optimal. Leaf size=54 \[ \frac{\left (a+b x^4\right ) \cosh ^{-1}\left (a+b x^4\right )}{4 b}-\frac{\sqrt{a+b x^4-1} \sqrt{a+b x^4+1}}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0521281, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6715, 5864, 5654, 74} \[ \frac{\left (a+b x^4\right ) \cosh ^{-1}\left (a+b x^4\right )}{4 b}-\frac{\sqrt{a+b x^4-1} \sqrt{a+b x^4+1}}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6715
Rule 5864
Rule 5654
Rule 74
Rubi steps
\begin{align*} \int x^3 \cosh ^{-1}\left (a+b x^4\right ) \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \cosh ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac{\operatorname{Subst}\left (\int \cosh ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac{\left (a+b x^4\right ) \cosh ^{-1}\left (a+b x^4\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,a+b x^4\right )}{4 b}\\ &=-\frac{\sqrt{-1+a+b x^4} \sqrt{1+a+b x^4}}{4 b}+\frac{\left (a+b x^4\right ) \cosh ^{-1}\left (a+b x^4\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0332246, size = 50, normalized size = 0.93 \[ \frac{\left (a+b x^4\right ) \cosh ^{-1}\left (a+b x^4\right )-\sqrt{a+b x^4-1} \sqrt{a+b x^4+1}}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{4\,b} \left ( \left ( b{x}^{4}+a \right ){\rm arccosh} \left (b{x}^{4}+a\right )-\sqrt{b{x}^{4}+a-1}\sqrt{b{x}^{4}+a+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.972995, size = 50, normalized size = 0.93 \begin{align*} \frac{{\left (b x^{4} + a\right )} \operatorname{arcosh}\left (b x^{4} + a\right ) - \sqrt{{\left (b x^{4} + a\right )}^{2} - 1}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.08293, size = 151, normalized size = 2.8 \begin{align*} \frac{{\left (b x^{4} + a\right )} \log \left (b x^{4} + a + \sqrt{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}\right ) - \sqrt{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.21483, size = 61, normalized size = 1.13 \begin{align*} \begin{cases} \frac{a \operatorname{acosh}{\left (a + b x^{4} \right )}}{4 b} + \frac{x^{4} \operatorname{acosh}{\left (a + b x^{4} \right )}}{4} - \frac{\sqrt{a^{2} + 2 a b x^{4} + b^{2} x^{8} - 1}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{acosh}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17469, size = 143, normalized size = 2.65 \begin{align*} \frac{1}{4} \, x^{4} \log \left (b x^{4} + a + \sqrt{{\left (b x^{4} + a\right )}^{2} - 1}\right ) - \frac{1}{4} \, b{\left (\frac{a \log \left ({\left | -a b -{\left (x^{4}{\left | b \right |} - \sqrt{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}\right )}{\left | b \right |} \right |}\right )}{b{\left | b \right |}} + \frac{\sqrt{b^{2} x^{8} + 2 \, a b x^{4} + a^{2} - 1}}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]