Optimal. Leaf size=33 \[ a x-2 b x \sqrt{1-\frac{2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0168854, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5901, 12, 191} \[ a x-2 b x \sqrt{1-\frac{2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5901
Rule 12
Rule 191
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right ) \, dx &=a x+b \int \cosh ^{-1}\left (-1+d x^2\right ) \, dx\\ &=a x+b x \cosh ^{-1}\left (-1+d x^2\right )-b \int \frac{2}{\sqrt{1-\frac{2}{d x^2}}} \, dx\\ &=a x+b x \cosh ^{-1}\left (-1+d x^2\right )-(2 b) \int \frac{1}{\sqrt{1-\frac{2}{d x^2}}} \, dx\\ &=a x-2 b \sqrt{1-\frac{2}{d x^2}} x+b x \cosh ^{-1}\left (-1+d x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0234999, size = 33, normalized size = 1. \[ a x-2 b x \sqrt{1-\frac{2}{d x^2}}+b x \cosh ^{-1}\left (d x^2-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 37, normalized size = 1.1 \begin{align*} ax+b \left ( x{\rm arccosh} \left (d{x}^{2}-1\right )-2\,{\frac{x\sqrt{d{x}^{2}-2}}{\sqrt{d{x}^{2}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.99553, size = 59, normalized size = 1.79 \begin{align*}{\left (x \operatorname{arcosh}\left (d x^{2} - 1\right ) - \frac{2 \,{\left (d^{\frac{3}{2}} x^{2} - 2 \, \sqrt{d}\right )}}{\sqrt{d x^{2} - 2} d}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.1458, size = 132, normalized size = 4. \begin{align*} \frac{b d x^{2} \log \left (d x^{2} + \sqrt{d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) + a d x^{2} - 2 \, \sqrt{d^{2} x^{4} - 2 \, d x^{2}} b}{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acosh}{\left (d x^{2} - 1 \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.11448, size = 95, normalized size = 2.88 \begin{align*}{\left (2 \, d{\left (\frac{\sqrt{2} \sqrt{-d} \mathrm{sgn}\left (x\right )}{d^{2}} - \frac{\sqrt{d^{2} x^{2} - 2 \, d}}{d^{2} \mathrm{sgn}\left (x\right )}\right )} + x \log \left (d x^{2} + \sqrt{{\left (d x^{2} - 1\right )}^{2} - 1} - 1\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]