Optimal. Leaf size=39 \[ a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+\frac {b \sqrt {x}}{c}+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6091, 50, 63, 206} \[ a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+\frac {b \sqrt {x}}{c}+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 206
Rule 6091
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c \sqrt {x}\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {1}{2} (b c) \int \frac {\sqrt {x}}{1-c^2 x} \, dx\\ &=\frac {b \sqrt {x}}{c}+a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx}{2 c}\\ &=\frac {b \sqrt {x}}{c}+a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {b \sqrt {x}}{c}+a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+b x \tanh ^{-1}\left (c \sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 42, normalized size = 1.08 \[ a x-b c \left (\frac {\tanh ^{-1}\left (c \sqrt {x}\right )}{c^3}-\frac {\sqrt {x}}{c^2}\right )+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.21, size = 56, normalized size = 1.44 \[ \frac {2 \, a c^{2} x + 2 \, b c \sqrt {x} + {\left (b c^{2} x - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.20, size = 174, normalized size = 4.46 \[ 2 \, b c {\left (\frac {1}{c^{3} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}} + \frac {{\left (c \sqrt {x} + 1\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{{\left (c \sqrt {x} - 1\right )} c^{3} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{2}}\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 50, normalized size = 1.28 \[ a x +b x \arctanh \left (c \sqrt {x}\right )+\frac {b \sqrt {x}}{c}+\frac {b \ln \left (c \sqrt {x}-1\right )}{2 c^{2}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 53, normalized size = 1.36 \[ \frac {1}{2} \, {\left (c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.90, size = 32, normalized size = 0.82 \[ a\,x+b\,x\,\mathrm {atanh}\left (c\,\sqrt {x}\right )-\frac {b\,\left (\mathrm {atanh}\left (c\,\sqrt {x}\right )-c\,\sqrt {x}\right )}{c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________