Optimal. Leaf size=162 \[ -\frac {e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 x \text {sech}(a c+b c x)}{8 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Rubi [A] time = 0.15, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6720, 2282, 12, 266, 43} \[ -\frac {e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 x \text {sech}(a c+b c x)}{8 \sqrt {\text {sech}^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 2282
Rule 6720
Rubi steps
\begin {align*} \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx &=\frac {\text {sech}(a c+b c x) \int e^{c (a+b x)} \cosh ^3(a c+b c x) \, dx}{\sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{8 x^3} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^3} \, dx,x,e^{c (a+b x)}\right )}{8 b c \sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \operatorname {Subst}\left (\int \frac {(1+x)^3}{x^2} \, dx,x,e^{2 c (a+b x)}\right )}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \operatorname {Subst}\left (\int \left (3+\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,e^{2 c (a+b x)}\right )}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}\\ &=-\frac {e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 x \text {sech}(a c+b c x)}{8 \sqrt {\text {sech}^2(a c+b c x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 0.48 \[ \frac {\left (-e^{-2 c (a+b x)}+3 e^{2 c (a+b x)}+\frac {1}{2} e^{4 c (a+b x)}+6 b c x\right ) \text {sech}^3(c (a+b x))}{16 b c \text {sech}^2(c (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 126, normalized size = 0.78 \[ -\frac {\cosh \left (b c x + a c\right )^{3} + 3 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} - 3 \, \sinh \left (b c x + a c\right )^{3} - 6 \, {\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) + 3 \, {\left (4 \, b c x - 3 \, \cosh \left (b c x + a c\right )^{2} - 2\right )} \sinh \left (b c x + a c\right )}{32 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 82, normalized size = 0.51 \[ \frac {{\left (12 \, b c x e^{\left (-a c\right )} - 2 \, {\left (3 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )} e^{\left (-2 \, b c x - 3 \, a c\right )} + {\left (e^{\left (4 \, b c x + 9 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 7 \, a c\right )}\right )} e^{\left (-6 \, a c\right )}\right )} e^{\left (a c\right )}}{32 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 216, normalized size = 1.33 \[ \frac {3 x \,{\mathrm e}^{c \left (b x +a \right )}}{8 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {{\mathrm e}^{5 c \left (b x +a \right )}}{32 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {3 \,{\mathrm e}^{3 c \left (b x +a \right )}}{16 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}-\frac {{\mathrm e}^{-c \left (b x +a \right )}}{16 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 74, normalized size = 0.46 \[ \frac {3 \, {\left (b c x + a c\right )}}{8 \, b c} + \frac {e^{\left (4 \, b c x + 4 \, a c\right )}}{32 \, b c} + \frac {3 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{16 \, b c} - \frac {e^{\left (-2 \, b c x - 2 \, a c\right )}}{16 \, b c} \]
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 \[ \int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{3/2}} \,d x \]
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 \[ e^{a c} \int \frac {e^{b c x}}{\left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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