Optimal. Leaf size=31 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3970, 65, 213}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 3970
Rubi steps
\begin {align*} \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \text {sech}(c+d x)\right )}{d}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(31)=62\).
time = 0.10, size = 73, normalized size = 2.35 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b+a \cosh (c+d x)}}{\sqrt {a \cosh (c+d x)}}\right ) \sqrt {b+a \cosh (c+d x)}}{d \sqrt {a \cosh (c+d x)} \sqrt {a+b \text {sech}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.62, size = 26, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {2 \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{d \sqrt {a}}\) | \(26\) |
default | \(\frac {2 \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{d \sqrt {a}}\) | \(26\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs.
\(2 (25) = 50\).
time = 0.77, size = 558, normalized size = 18.00 \begin {gather*} \left [\frac {\log \left (-\frac {2 \, a^{2} \cosh \left (d x + c\right )^{4} + 2 \, a^{2} \sinh \left (d x + c\right )^{4} + 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (2 \, a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right ) + {\left (4 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (12 \, a^{2} \cosh \left (d x + c\right )^{2} + 12 \, a b \cosh \left (d x + c\right ) + 4 \, a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, {\left (a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} + b \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right )^{3} + 2 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} + 3 \, b \cosh \left (d x + c\right ) + 2 \, a\right )} \sinh \left (d x + c\right )^{2} + b \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {a} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}} + 2 \, {\left (4 \, a^{2} \cosh \left (d x + c\right )^{3} + 6 \, a b \cosh \left (d x + c\right )^{2} + 2 \, a b + {\left (4 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right )}{2 \, \sqrt {a} d}, -\frac {\sqrt {-a} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + b \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {-a} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )}\right )}{a d}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\tanh {\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.64, size = 27, normalized size = 0.87 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}}{\sqrt {a}}\right )}{\sqrt {a}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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