Optimal. Leaf size=85 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3776, 3774, 203, 3795} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rule 3776
Rule 3795
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx &=\frac {\int \sqrt {a+a \text {sech}(c+d x)} \, dx}{a}-\int \frac {\text {sech}(c+d x)}{\sqrt {a+a \text {sech}(c+d x)}} \, dx\\ &=\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 118, normalized size = 1.39 \[ \frac {\left (e^{c+d x}+1\right ) \left (\sqrt {2} \sinh ^{-1}\left (e^{c+d x}\right )-2 \tanh ^{-1}\left (\frac {e^{c+d x}-1}{\sqrt {2} \sqrt {e^{2 (c+d x)}+1}}\right )-\sqrt {2} \tanh ^{-1}\left (\sqrt {e^{2 (c+d x)}+1}\right )\right )}{\sqrt {2} d \sqrt {e^{2 (c+d x)}+1} \sqrt {a (\text {sech}(c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 868, normalized size = 10.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +a \,\mathrm {sech}\left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \operatorname {sech}\left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \operatorname {sech}{\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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