Optimal. Leaf size=49 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} d} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2738, 211}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2738
Rubi steps
\begin {align*} \int \frac {1}{a+b \cosh (c+d x)} \, dx &=-\frac {(2 i) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 48, normalized size = 0.98 \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {(a-b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 44, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(44\) |
default | \(\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{d \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(44\) |
risch | \(\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d}-\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d}\) | \(123\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 237, normalized size = 4.84 \begin {gather*} \left [\frac {\log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) + b}\right )}{\sqrt {a^{2} - b^{2}} d}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{a^{2} - b^{2}}\right )}{{\left (a^{2} - b^{2}\right )} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs.
\(2 (41) = 82\).
time = 2.47, size = 163, normalized size = 3.33 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x}{\cosh {\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {1}{b d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {for}\: a = - b \\\frac {x}{a + b \cosh {\left (c \right )}} & \text {for}\: d = 0 \\\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{b d} & \text {for}\: a = b \\- \frac {\log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{a d \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b d \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {\log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{a d \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b d \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 39, normalized size = 0.80 \begin {gather*} \frac {2 \, \arctan \left (\frac {b e^{\left (d x + c\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 53, normalized size = 1.08 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {a\,d+b\,d\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{\sqrt {b^2\,d^2-a^2\,d^2}}\right )}{\sqrt {b^2\,d^2-a^2\,d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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