Optimal. Leaf size=85 \[ -\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{c^2}+\frac{2 a b \sqrt{x}}{c}+x \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2+\frac{b^2 \log \left (1-c^2 x\right )}{c^2}+\frac{2 b^2 \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.0063505, 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 = {} \[ \int \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx &=\int \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2 \, dx\\ \end{align*}
Mathematica [A] time = 0.0614956, size = 115, normalized size = 1.35 \[ \frac{a^2 c^2 x+2 a b c \sqrt{x}+b (a+b) \log \left (1-c \sqrt{x}\right )-a b \log \left (c \sqrt{x}+1\right )+2 b c \sqrt{x} \tanh ^{-1}\left (c \sqrt{x}\right ) \left (a c \sqrt{x}+b\right )+b^2 \left (c^2 x-1\right ) \tanh ^{-1}\left (c \sqrt{x}\right )^2+b^2 \log \left (c \sqrt{x}+1\right )}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.05, size = 272, normalized size = 3.2 \begin{align*}{a}^{2}x+{b}^{2}x \left ({\it Artanh} \left ( c\sqrt{x} \right ) \right ) ^{2}+2\,{\frac{{b}^{2}{\it Artanh} \left ( c\sqrt{x} \right ) \sqrt{x}}{c}}+{\frac{{b}^{2}}{{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) }-{\frac{{b}^{2}}{{c}^{2}}{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{b}^{2}}{4\,{c}^{2}} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+{\frac{{b}^{2}}{{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) }+{\frac{{b}^{2}}{{c}^{2}}\ln \left ( 1+c\sqrt{x} \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{{b}^{2}}{4\,{c}^{2}} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}}+2\,abx{\it Artanh} \left ( c\sqrt{x} \right ) +2\,{\frac{ab\sqrt{x}}{c}}+{\frac{ab}{{c}^{2}}\ln \left ( c\sqrt{x}-1 \right ) }-{\frac{ab}{{c}^{2}}\ln \left ( 1+c\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.01408, size = 236, normalized size = 2.78 \begin{align*}{\left (c{\left (\frac{2 \, \sqrt{x}}{c^{2}} - \frac{\log \left (c \sqrt{x} + 1\right )}{c^{3}} + \frac{\log \left (c \sqrt{x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname{artanh}\left (c \sqrt{x}\right )\right )} a b + \frac{1}{4} \,{\left (4 \, c{\left (\frac{2 \, \sqrt{x}}{c^{2}} - \frac{\log \left (c \sqrt{x} + 1\right )}{c^{3}} + \frac{\log \left (c \sqrt{x} - 1\right )}{c^{3}}\right )} \operatorname{artanh}\left (c \sqrt{x}\right ) + 4 \, x \operatorname{artanh}\left (c \sqrt{x}\right )^{2} - \frac{2 \,{\left (\log \left (c \sqrt{x} - 1\right ) - 2\right )} \log \left (c \sqrt{x} + 1\right ) - \log \left (c \sqrt{x} + 1\right )^{2} - \log \left (c \sqrt{x} - 1\right )^{2} - 4 \, \log \left (c \sqrt{x} - 1\right )}{c^{2}}\right )} b^{2} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.99055, size = 382, normalized size = 4.49 \begin{align*} \frac{4 \, a^{2} c^{2} x + 8 \, a b c \sqrt{x} +{\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )^{2} + 4 \,{\left (a b c^{2} - a b + b^{2}\right )} \log \left (c \sqrt{x} + 1\right ) - 4 \,{\left (a b c^{2} - a b - b^{2}\right )} \log \left (c \sqrt{x} - 1\right ) + 4 \,{\left (a b c^{2} x - a b c^{2} + b^{2} c \sqrt{x}\right )} \log \left (-\frac{c^{2} x + 2 \, c \sqrt{x} + 1}{c^{2} x - 1}\right )}{4 \, c^{2}} \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{atanh}{\left (c \sqrt{x} \right )}\right )^{2}\, dx \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{\left (b \operatorname{artanh}\left (c \sqrt{x}\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]