Optimal. Leaf size=38 \[ -\frac {\text {sech}^2(c+b x) \sinh (a-c)}{2 b}+\frac {\cosh (a-c) \tanh (c+b x)}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5746, 2686, 30,
3852, 8} \begin {gather*} \frac {\cosh (a-c) \tanh (b x+c)}{b}-\frac {\sinh (a-c) \text {sech}^2(b x+c)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2686
Rule 3852
Rule 5746
Rubi steps
\begin {align*} \int \cosh (a+b x) \text {sech}^3(c+b x) \, dx &=\cosh (a-c) \int \text {sech}^2(c+b x) \, dx+\sinh (a-c) \int \text {sech}^2(c+b x) \tanh (c+b x) \, dx\\ &=\frac {(i \cosh (a-c)) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+b x))}{b}-\frac {\sinh (a-c) \text {Subst}(\int x \, dx,x,\text {sech}(c+b x))}{b}\\ &=-\frac {\text {sech}^2(c+b x) \sinh (a-c)}{2 b}+\frac {\cosh (a-c) \tanh (c+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 35, normalized size = 0.92 \begin {gather*} -\frac {\text {sech}(c) \text {sech}^2(c+b x) (\sinh (a)-\cosh (a-c) \sinh (c+2 b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.01, size = 56, normalized size = 1.47
method | result | size |
risch | \(-\frac {\left (2 \,{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}+{\mathrm e}^{2 c}\right ) {\mathrm e}^{3 a -c}}{\left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2} b}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (36) = 72\).
time = 0.27, size = 119, normalized size = 3.13 \begin {gather*} \frac {2 \, e^{\left (-2 \, b x + 3 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} + \frac {e^{\left (2 \, a + 3 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} + \frac {e^{\left (5 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} \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. 248 vs.
\(2 (36) = 72\).
time = 0.35, size = 248, normalized size = 6.53 \begin {gather*} -\frac {2 \, {\left (2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right ) - \sinh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{b \cosh \left (b x + c\right )^{3} \cosh \left (-a + c\right )^{2} + 3 \, b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} + {\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{3} + 3 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} - {\left (b \cosh \left (b x + c\right )^{3} + 3 \, b \cosh \left (b x + c\right )\right )} \sinh \left (-a + c\right )^{2} + {\left (3 \, b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} + b \cosh \left (-a + c\right )^{2} - {\left (3 \, b \cosh \left (b x + c\right )^{2} + b\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\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 \cosh {\left (a + b x \right )} \operatorname {sech}^{3}{\left (b x + c \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 49, normalized size = 1.29 \begin {gather*} -\frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{b {\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (c+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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