Optimal. Leaf size=38 \[ \frac {\sinh (a-c) \tanh (b x+c)}{b}-\frac {\cosh (a-c) \text {sech}^2(b x+c)}{2 b} \]
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Rubi [A] time = 0.04, 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 = {5624, 2606, 30, 3767, 8} \[ \frac {\sinh (a-c) \tanh (b x+c)}{b}-\frac {\cosh (a-c) \text {sech}^2(b x+c)}{2 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 3767
Rule 5624
Rubi steps
\begin {align*} \int \text {sech}^3(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int \text {sech}^2(c+b x) \tanh (c+b x) \, dx+\sinh (a-c) \int \text {sech}^2(c+b x) \, dx\\ &=-\frac {\cosh (a-c) \operatorname {Subst}(\int x \, dx,x,\text {sech}(c+b x))}{b}+\frac {(i \sinh (a-c)) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (c+b x))}{b}\\ &=-\frac {\cosh (a-c) \text {sech}^2(c+b x)}{2 b}+\frac {\sinh (a-c) \tanh (c+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 35, normalized size = 0.92 \[ -\frac {\text {sech}(c) \text {sech}^2(b x+c) (\cosh (a)-\sinh (a-c) \sinh (2 b x+c))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 246, normalized size = 6.47 \[ -\frac {2 \, {\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) + \cosh \left (-a + c\right ) \sinh \left (b x + c\right ) - 2 \, \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{b \cosh \left (b x + c\right )^{3} \cosh \left (-a + c\right )^{2} + 3 \, b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} + {\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{3} + 3 \, {\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} - {\left (b \cosh \left (b x + c\right )^{3} + 3 \, b \cosh \left (b x + c\right )\right )} \sinh \left (-a + c\right )^{2} + {\left (3 \, b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} + b \cosh \left (-a + c\right )^{2} - {\left (3 \, b \cosh \left (b x + c\right )^{2} + b\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 51, normalized size = 1.34 \[ -\frac {{\left (2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{b {\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 58, normalized size = 1.53 \[ -\frac {\left (2 \,{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}-{\mathrm e}^{2 c}\right ) {\mathrm e}^{3 a -c}}{\left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 120, normalized size = 3.16 \[ -\frac {2 \, e^{\left (-2 \, b x + 3 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} + \frac {e^{\left (2 \, a + 3 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} - \frac {e^{\left (5 \, c\right )}}{b {\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} + e^{\left (-4 \, b x + a\right )} + e^{\left (a + 4 \, c\right )}\right )}} \]
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 \[ \int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (c+b\,x\right )}^3} \,d x \]
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 \[ \int \sinh {\left (a + b x \right )} \operatorname {sech}^{3}{\left (b x + c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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