Optimal. Leaf size=47 \[ 2 a^2 \sqrt {x}-\frac {4 a b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5437, 3773, 3770, 3767, 8} \[ 2 a^2 \sqrt {x}-\frac {4 a b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3773
Rule 5437
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 a^2 \sqrt {x}+(4 a b) \operatorname {Subst}\left (\int \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \operatorname {Subst}\left (\int \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=2 a^2 \sqrt {x}-\frac {4 a b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}-\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int 1 \, dx,x,-i \coth \left (c+d \sqrt {x}\right )\right )}{d}\\ &=2 a^2 \sqrt {x}-\frac {4 a b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 75, normalized size = 1.60 \[ -\frac {-2 a \left (a c+a d \sqrt {x}+2 b \log \left (\tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )\right )+b^2 \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )+b^2 \coth \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 271, normalized size = 5.77 \[ \frac {2 \, {\left (a^{2} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, a^{2} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + a^{2} d \sqrt {x} \sinh \left (d \sqrt {x} + c\right )^{2} - a^{2} d \sqrt {x} - 2 \, b^{2} - 2 \, {\left (a b \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + a b \sinh \left (d \sqrt {x} + c\right )^{2} - a b\right )} \log \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right ) + 1\right ) + 2 \, {\left (a b \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + a b \sinh \left (d \sqrt {x} + c\right )^{2} - a b\right )} \log \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right ) - 1\right )\right )}}{d \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, d \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + d \sinh \left (d \sqrt {x} + c\right )^{2} - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 76, normalized size = 1.62 \[ \frac {2 \, {\left (d \sqrt {x} + c\right )} a^{2}}{d} - \frac {4 \, a b \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right )}{d} + \frac {4 \, a b \log \left ({\left | e^{\left (d \sqrt {x} + c\right )} - 1 \right |}\right )}{d} - \frac {4 \, b^{2}}{d {\left (e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 44, normalized size = 0.94 \[ \frac {2 a^{2} \left (c +d \sqrt {x}\right )-8 a b \arctanh \left ({\mathrm e}^{c +d \sqrt {x}}\right )-2 b^{2} \coth \left (c +d \sqrt {x}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 51, normalized size = 1.09 \[ 2 \, a^{2} \sqrt {x} + \frac {4 \, a b \log \left (\tanh \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )\right )}{d} + \frac {4 \, b^{2}}{d {\left (e^{\left (-2 \, d \sqrt {x} - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 81, normalized size = 1.72 \[ 2\,a^2\,\sqrt {x}-\frac {4\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,\sqrt {x}}-1\right )}-\frac {8\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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