Optimal. Leaf size=250 \[ -\frac {e^{-4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {sech}(a c+b c x)}{192 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 x \text {sech}(a c+b c x)}{16 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Rubi [A]
time = 0.14, antiderivative size = 250, 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 = {6852, 2320, 12,
272, 45} \begin {gather*} -\frac {e^{-4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {sech}(a c+b c x)}{192 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 x \text {sech}(a c+b c x)}{16 \sqrt {\text {sech}^2(a c+b c x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 2320
Rule 6852
Rubi steps
\begin {align*} \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx &=\frac {\text {sech}(a c+b c x) \int e^{c (a+b x)} \cosh ^5(a c+b c x) \, dx}{\sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{32 x^5} \, 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) \text {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^5} \, dx,x,e^{c (a+b x)}\right )}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \frac {(1+x)^5}{x^3} \, dx,x,e^{2 c (a+b x)}\right )}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}\\ &=\frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \left (10+\frac {1}{x^3}+\frac {5}{x^2}+\frac {10}{x}+5 x+x^2\right ) \, dx,x,e^{2 c (a+b x)}\right )}{64 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)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {sech}(a c+b c x)}{192 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 x \text {sech}(a c+b c x)}{16 \sqrt {\text {sech}^2(a c+b c x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 106, normalized size = 0.42 \begin {gather*} \frac {\left (-\frac {1}{2} e^{-4 c (a+b x)}-5 e^{-2 c (a+b x)}+10 e^{2 c (a+b x)}+\frac {5}{2} e^{4 c (a+b x)}+\frac {1}{3} e^{6 c (a+b x)}+20 b c x\right ) \text {sech}^5(c (a+b x))}{64 b c \text {sech}^2(c (a+b x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.54, size = 326, normalized size = 1.30
method | result | size |
risch | \(\frac {5 x \,{\mathrm e}^{c \left (b x +a \right )}}{16 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {{\mathrm e}^{7 c \left (b x +a \right )}}{192 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {5 \,{\mathrm e}^{5 c \left (b x +a \right )}}{128 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {5 \,{\mathrm e}^{3 c \left (b x +a \right )}}{32 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}-\frac {5 \,{\mathrm e}^{-c \left (b x +a \right )}}{64 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}-\frac {{\mathrm e}^{-3 c \left (b x +a \right )}}{128 c b \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}\) | \(326\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 112, normalized size = 0.45 \begin {gather*} \frac {5 \, {\left (b c x + a c\right )}}{16 \, b c} + \frac {e^{\left (6 \, b c x + 6 \, a c\right )}}{192 \, b c} + \frac {5 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{128 \, b c} + \frac {5 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{32 \, b c} - \frac {5 \, e^{\left (-2 \, b c x - 2 \, a c\right )}}{64 \, b c} - \frac {e^{\left (-4 \, b c x - 4 \, a c\right )}}{128 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 218, normalized size = 0.87 \begin {gather*} -\frac {\cosh \left (b c x + a c\right )^{5} + 5 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} - 5 \, \sinh \left (b c x + a c\right )^{5} - 5 \, {\left (10 \, \cosh \left (b c x + a c\right )^{2} + 9\right )} \sinh \left (b c x + a c\right )^{3} + 15 \, \cosh \left (b c x + a c\right )^{3} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{3} + 9 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 60 \, {\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) - 5 \, {\left (5 \, \cosh \left (b c x + a c\right )^{4} - 24 \, b c x + 27 \, \cosh \left (b c x + a c\right )^{2} + 12\right )} \sinh \left (b c x + a c\right )}{384 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\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. 532 vs.
\(2 (238) = 476\).
time = 129.76, size = 532, normalized size = 2.13 \begin {gather*} \begin {cases} x & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\\frac {x e^{a c}}{\left (\operatorname {sech}^{2}{\left (a c \right )}\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {5 x e^{a c} e^{b c x} \tanh ^{5}{\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x} \tanh ^{4}{\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x} \tanh ^{3}{\left (a c + b c x \right )}}{8 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {5 x e^{a c} e^{b c x} \tanh ^{2}{\left (a c + b c x \right )}}{8 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {5 x e^{a c} e^{b c x} \tanh {\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {8 e^{a c} e^{b c x} \tanh ^{5}{\left (a c + b c x \right )}}{15 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {53 e^{a c} e^{b c x} \tanh ^{4}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {331 e^{a c} e^{b c x} \tanh ^{3}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {131 e^{a c} e^{b c x} \tanh ^{2}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {253 e^{a c} e^{b c x} \tanh {\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {11 e^{a c} e^{b c x}}{30 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 110, normalized size = 0.44 \begin {gather*} \frac {{\left (120 \, b c x e^{\left (-a c\right )} - 3 \, {\left (30 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 10 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )} e^{\left (-4 \, b c x - 5 \, a c\right )} + {\left (2 \, e^{\left (6 \, b c x + 20 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 18 \, a c\right )} + 60 \, e^{\left (2 \, b c x + 16 \, a c\right )}\right )} e^{\left (-15 \, a c\right )}\right )} e^{\left (a c\right )}}{384 \, b c} \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 = -0.00 \begin {gather*} \int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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