Optimal. Leaf size=63 \[ \frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \tanh ^3(x)} \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-2 \coth (x) \sqrt {a \tanh ^3(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3658, 3473, 3476, 329, 212, 206, 203} \[ \frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \tanh ^3(x)} \tan ^{-1}\left (\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}-2 \coth (x) \sqrt {a \tanh ^3(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 329
Rule 3473
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {a \tanh ^3(x)} \, dx &=\frac {\sqrt {a \tanh ^3(x)} \int \tanh ^{\frac {3}{2}}(x) \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\sqrt {a \tanh ^3(x)} \int \frac {1}{\sqrt {\tanh (x)}} \, dx}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}-\frac {\sqrt {a \tanh ^3(x)} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\tanh (x)\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}-\frac {\left (2 \sqrt {a \tanh ^3(x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\sqrt {a \tanh ^3(x)} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \tanh ^3(x)} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh (x)}\right )}{\tanh ^{\frac {3}{2}}(x)}\\ &=-2 \coth (x) \sqrt {a \tanh ^3(x)}+\frac {\tan ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}+\frac {\tanh ^{-1}\left (\sqrt {\tanh (x)}\right ) \sqrt {a \tanh ^3(x)}}{\tanh ^{\frac {3}{2}}(x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 40, normalized size = 0.63 \[ \frac {\sqrt {a \tanh ^3(x)} \left (\tanh ^{-1}\left (\sqrt {\tanh (x)}\right )-2 \sqrt {\tanh (x)}+\tan ^{-1}\left (\sqrt {\tanh (x)}\right )\right )}{\tanh ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.63, size = 376, normalized size = 5.97 \[ \left [-\frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {{\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \sqrt {-a} \sqrt {\frac {a \sinh \relax (x)}{\cosh \relax (x)}}}{a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} - a}\right ) + \frac {1}{4} \, \sqrt {-a} \log \left (-\frac {a \cosh \relax (x)^{4} + 4 \, a \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, a \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, a \cosh \relax (x) \sinh \relax (x)^{3} + a \sinh \relax (x)^{4} + 2 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \sqrt {-a} \sqrt {\frac {a \sinh \relax (x)}{\cosh \relax (x)}} - 2 \, a}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4}}\right ) - 2 \, \sqrt {\frac {a \sinh \relax (x)}{\cosh \relax (x)}}, -\frac {1}{2} \, \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {\frac {a \sinh \relax (x)}{\cosh \relax (x)}}}{a \cosh \relax (x)^{2} + 2 \, a \cosh \relax (x) \sinh \relax (x) + a \sinh \relax (x)^{2} - a}\right ) + \frac {1}{4} \, \sqrt {a} \log \left (2 \, a \cosh \relax (x)^{4} + 8 \, a \cosh \relax (x)^{3} \sinh \relax (x) + 12 \, a \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 8 \, a \cosh \relax (x) \sinh \relax (x)^{3} + 2 \, a \sinh \relax (x)^{4} + 2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )} \sqrt {a} \sqrt {\frac {a \sinh \relax (x)}{\cosh \relax (x)}} - a\right ) - 2 \, \sqrt {\frac {a \sinh \relax (x)}{\cosh \relax (x)}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 115, normalized size = 1.83 \[ \sqrt {a} \arctan \left (-\frac {\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {1}{2} \, \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {4 \, a \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{\sqrt {a} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} - a} + \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 62, normalized size = 0.98 \[ -\frac {\sqrt {a \left (\tanh ^{3}\relax (x )\right )}\, \left (2 \sqrt {a \tanh \relax (x )}-\sqrt {a}\, \arctan \left (\frac {\sqrt {a \tanh \relax (x )}}{\sqrt {a}}\right )-\sqrt {a}\, \arctanh \left (\frac {\sqrt {a \tanh \relax (x )}}{\sqrt {a}}\right )\right )}{\tanh \relax (x ) \sqrt {a \tanh \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \tanh \relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a\,{\mathrm {tanh}\relax (x)}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \tanh ^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________