Optimal. Leaf size=68 \[ \frac {2 \sqrt {x}}{a}-\frac {4 b \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}} \]
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Rubi [A] time = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5436, 3783, 2659, 208} \[ \frac {2 \sqrt {x}}{a}-\frac {4 b \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 3783
Rule 5436
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{a+b \text {sech}(c+d x)} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x}}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+\frac {a \cosh (c+d x)}{b}} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {2 \sqrt {x}}{a}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )}{a d}\\ &=\frac {2 \sqrt {x}}{a}-\frac {4 b \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b} d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 69, normalized size = 1.01 \[ \frac {2 \left (\frac {2 b \tan ^{-1}\left (\frac {(b-a) \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )}{d \sqrt {a^2-b^2}}+\frac {c}{d}+\sqrt {x}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 254, normalized size = 3.74 \[ \left [\frac {2 \, {\left ({\left (a^{2} - b^{2}\right )} d \sqrt {x} - \sqrt {-a^{2} + b^{2}} b \log \left (\frac {a b + {\left (b^{2} + \sqrt {-a^{2} + b^{2}} b\right )} \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{2} - b^{2} - \sqrt {-a^{2} + b^{2}} b\right )} \sinh \left (d \sqrt {x} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{a \cosh \left (d \sqrt {x} + c\right ) + b}\right )\right )}}{{\left (a^{3} - a b^{2}\right )} d}, \frac {2 \, {\left ({\left (a^{2} - b^{2}\right )} d \sqrt {x} + 2 \, \sqrt {a^{2} - b^{2}} b \arctan \left (-\frac {\sqrt {a^{2} - b^{2}} a \cosh \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} - b^{2}} a \sinh \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} - b^{2}} b}{a^{2} - b^{2}}\right )\right )}}{{\left (a^{3} - a b^{2}\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 61, normalized size = 0.90 \[ -\frac {4 \, b \arctan \left (\frac {a e^{\left (d \sqrt {x} + c\right )} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a d} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 95, normalized size = 1.40 \[ -\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )-1\right )}{d a}+\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+1\right )}{d a}-\frac {4 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d a \sqrt {\left (a +b \right ) \left (a -b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.78, size = 155, normalized size = 2.28 \[ \frac {2\,\sqrt {x}}{a}+\frac {2\,b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c}{a^2\,\sqrt {x}}-\frac {2\,b\,\left (a+b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\right )}{a^2\,\sqrt {x}\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a\,d\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {2\,b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c}{a^2\,\sqrt {x}}+\frac {2\,b\,\left (a+b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\right )}{a^2\,\sqrt {x}\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a\,d\,\sqrt {a+b}\,\sqrt {b-a}} \]
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 \[ \int \frac {1}{\sqrt {x} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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