Optimal. Leaf size=141 \[ -\frac {4 \cosh (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\cosh ^2(a c+b c x)}}+\frac {32 \cosh (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\cosh ^2(a c+b c x)}}-\frac {8 \cosh (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\cosh ^2(a c+b c x)}} \]
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Rubi [A]
time = 0.14, antiderivative size = 141, 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 {8 \cosh (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt {\cosh ^2(a c+b c x)}}+\frac {32 \cosh (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3 \sqrt {\cosh ^2(a c+b c x)}}-\frac {4 \cosh (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4 \sqrt {\cosh ^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)}}{\cosh ^2(a c+b c x)^{5/2}} \, dx &=\frac {\cosh (a c+b c x) \int e^{c (a+b x)} \text {sech}^5(a c+b c x) \, dx}{\sqrt {\cosh ^2(a c+b c x)}}\\ &=\frac {\cosh (a c+b c x) \text {Subst}\left (\int \frac {32 x^5}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\cosh ^2(a c+b c x)}}\\ &=\frac {(32 \cosh (a c+b c x)) \text {Subst}\left (\int \frac {x^5}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\cosh ^2(a c+b c x)}}\\ &=\frac {(16 \cosh (a c+b c x)) \text {Subst}\left (\int \frac {x^2}{(1+x)^5} \, dx,x,e^{2 c (a+b x)}\right )}{b c \sqrt {\cosh ^2(a c+b c x)}}\\ &=\frac {(16 \cosh (a c+b c x)) \text {Subst}\left (\int \left (\frac {1}{(1+x)^5}-\frac {2}{(1+x)^4}+\frac {1}{(1+x)^3}\right ) \, dx,x,e^{2 c (a+b x)}\right )}{b c \sqrt {\cosh ^2(a c+b c x)}}\\ &=-\frac {4 \cosh (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\cosh ^2(a c+b c x)}}+\frac {32 \cosh (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\cosh ^2(a c+b c x)}}-\frac {8 \cosh (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\cosh ^2(a c+b c x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 72, normalized size = 0.51 \begin {gather*} -\frac {4 \left (1+4 e^{2 c (a+b x)}+6 e^{4 c (a+b x)}\right ) \cosh (c (a+b x))}{3 b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\cosh ^2(c (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.89, size = 80, normalized size = 0.57
method | result | size |
risch | \(-\frac {4 \left (6 \,{\mathrm e}^{4 c \left (b x +a \right )}+4 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) {\mathrm e}^{-c \left (b x +a \right )}}{3 c b \sqrt {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 209, normalized size = 1.48 \begin {gather*} -\frac {8 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {4}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\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. 315 vs.
\(2 (130) = 260\).
time = 0.36, size = 315, normalized size = 2.23 \begin {gather*} -\frac {4 \, {\left (7 \, \cosh \left (b c x + a c\right )^{2} + 10 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 7 \, \sinh \left (b c x + a c\right )^{2} + 4\right )}}{3 \, {\left (b c \cosh \left (b c x + a c\right )^{6} + 6 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{5} + b c \sinh \left (b c x + a c\right )^{6} + 4 \, b c \cosh \left (b c x + a c\right )^{4} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \, b c\right )} \sinh \left (b c x + a c\right )^{4} + 7 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (5 \, b c \cosh \left (b c x + a c\right )^{3} + 4 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{3} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{4} + 24 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )^{2} + 4 \, b c + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{5} + 8 \, b c \cosh \left (b c x + a c\right )^{3} + 5 \, b c \cosh \left (b c x + a c\right )\right )} \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 51, normalized size = 0.36 \begin {gather*} -\frac {4 \, {\left (6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{3 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 89, normalized size = 0.63 \begin {gather*} -\frac {8\,{\mathrm {e}}^{a\,c+b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}\,\left (4\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+6\,{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{3\,b\,c\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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