Optimal. Leaf size=121 \[ \frac {x}{32 a^5}-\frac {1}{10 d (a+a \tanh (c+d x))^5}-\frac {1}{16 a d (a+a \tanh (c+d x))^4}-\frac {1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac {1}{32 a d \left (a^2+a^2 \tanh (c+d x)\right )^2}-\frac {1}{32 d \left (a^5+a^5 \tanh (c+d x)\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3560, 8}
\begin {gather*} -\frac {1}{32 d \left (a^5 \tanh (c+d x)+a^5\right )}+\frac {x}{32 a^5}-\frac {1}{32 a d \left (a^2 \tanh (c+d x)+a^2\right )^2}-\frac {1}{24 a^2 d (a \tanh (c+d x)+a)^3}-\frac {1}{16 a d (a \tanh (c+d x)+a)^4}-\frac {1}{10 d (a \tanh (c+d x)+a)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3560
Rubi steps
\begin {align*} \int \frac {1}{(a+a \tanh (c+d x))^5} \, dx &=-\frac {1}{10 d (a+a \tanh (c+d x))^5}+\frac {\int \frac {1}{(a+a \tanh (c+d x))^4} \, dx}{2 a}\\ &=-\frac {1}{10 d (a+a \tanh (c+d x))^5}-\frac {1}{16 a d (a+a \tanh (c+d x))^4}+\frac {\int \frac {1}{(a+a \tanh (c+d x))^3} \, dx}{4 a^2}\\ &=-\frac {1}{10 d (a+a \tanh (c+d x))^5}-\frac {1}{16 a d (a+a \tanh (c+d x))^4}-\frac {1}{24 a^2 d (a+a \tanh (c+d x))^3}+\frac {\int \frac {1}{(a+a \tanh (c+d x))^2} \, dx}{8 a^3}\\ &=-\frac {1}{10 d (a+a \tanh (c+d x))^5}-\frac {1}{16 a d (a+a \tanh (c+d x))^4}-\frac {1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac {1}{32 a^3 d (a+a \tanh (c+d x))^2}+\frac {\int \frac {1}{a+a \tanh (c+d x)} \, dx}{16 a^4}\\ &=-\frac {1}{10 d (a+a \tanh (c+d x))^5}-\frac {1}{16 a d (a+a \tanh (c+d x))^4}-\frac {1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac {1}{32 a^3 d (a+a \tanh (c+d x))^2}-\frac {1}{32 d \left (a^5+a^5 \tanh (c+d x)\right )}+\frac {\int 1 \, dx}{32 a^5}\\ &=\frac {x}{32 a^5}-\frac {1}{10 d (a+a \tanh (c+d x))^5}-\frac {1}{16 a d (a+a \tanh (c+d x))^4}-\frac {1}{24 a^2 d (a+a \tanh (c+d x))^3}-\frac {1}{32 a^3 d (a+a \tanh (c+d x))^2}-\frac {1}{32 d \left (a^5+a^5 \tanh (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 109, normalized size = 0.90 \begin {gather*} \frac {\text {sech}^5(c+d x) (-500 \cosh (c+d x)-375 \cosh (3 (c+d x))-12 \cosh (5 (c+d x))+120 d x \cosh (5 (c+d x))-100 \sinh (c+d x)-225 \sinh (3 (c+d x))+12 \sinh (5 (c+d x))+120 d x \sinh (5 (c+d x)))}{3840 a^5 d (1+\tanh (c+d x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 91, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{64}-\frac {1}{10 \left (\tanh \left (d x +c \right )+1\right )^{5}}-\frac {1}{16 \left (\tanh \left (d x +c \right )+1\right )^{4}}-\frac {1}{24 \left (\tanh \left (d x +c \right )+1\right )^{3}}-\frac {1}{32 \left (\tanh \left (d x +c \right )+1\right )^{2}}-\frac {1}{32 \left (\tanh \left (d x +c \right )+1\right )}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{64}}{d \,a^{5}}\) | \(91\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{64}-\frac {1}{10 \left (\tanh \left (d x +c \right )+1\right )^{5}}-\frac {1}{16 \left (\tanh \left (d x +c \right )+1\right )^{4}}-\frac {1}{24 \left (\tanh \left (d x +c \right )+1\right )^{3}}-\frac {1}{32 \left (\tanh \left (d x +c \right )+1\right )^{2}}-\frac {1}{32 \left (\tanh \left (d x +c \right )+1\right )}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{64}}{d \,a^{5}}\) | \(91\) |
risch | \(\frac {x}{32 a^{5}}-\frac {5 \,{\mathrm e}^{-2 d x -2 c}}{64 a^{5} d}-\frac {5 \,{\mathrm e}^{-4 d x -4 c}}{64 a^{5} d}-\frac {5 \,{\mathrm e}^{-6 d x -6 c}}{96 a^{5} d}-\frac {5 \,{\mathrm e}^{-8 d x -8 c}}{256 a^{5} d}-\frac {{\mathrm e}^{-10 d x -10 c}}{320 a^{5} d}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 78, normalized size = 0.64 \begin {gather*} \frac {d x + c}{32 \, a^{5} d} - \frac {300 \, e^{\left (-2 \, d x - 2 \, c\right )} + 300 \, e^{\left (-4 \, d x - 4 \, c\right )} + 200 \, e^{\left (-6 \, d x - 6 \, c\right )} + 75 \, e^{\left (-8 \, d x - 8 \, c\right )} + 12 \, e^{\left (-10 \, d x - 10 \, c\right )}}{3840 \, a^{5} d} \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. 287 vs.
\(2 (109) = 218\).
time = 0.40, size = 287, normalized size = 2.37 \begin {gather*} \frac {12 \, {\left (10 \, d x - 1\right )} \cosh \left (d x + c\right )^{5} + 60 \, {\left (10 \, d x - 1\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 12 \, {\left (10 \, d x + 1\right )} \sinh \left (d x + c\right )^{5} + 15 \, {\left (8 \, {\left (10 \, d x + 1\right )} \cosh \left (d x + c\right )^{2} - 15\right )} \sinh \left (d x + c\right )^{3} - 375 \, \cosh \left (d x + c\right )^{3} + 15 \, {\left (8 \, {\left (10 \, d x - 1\right )} \cosh \left (d x + c\right )^{3} - 75 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 5 \, {\left (12 \, {\left (10 \, d x + 1\right )} \cosh \left (d x + c\right )^{4} - 135 \, \cosh \left (d x + c\right )^{2} - 20\right )} \sinh \left (d x + c\right ) - 500 \, \cosh \left (d x + c\right )}{3840 \, {\left (a^{5} d \cosh \left (d x + c\right )^{5} + 5 \, a^{5} d \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, a^{5} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, a^{5} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, a^{5} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a^{5} d \sinh \left (d x + c\right )^{5}\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. 1018 vs.
\(2 (102) = 204\).
time = 1.56, size = 1018, normalized size = 8.41 \begin {gather*} \begin {cases} \frac {15 d x \tanh ^{5}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} + \frac {75 d x \tanh ^{4}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} + \frac {150 d x \tanh ^{3}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} + \frac {150 d x \tanh ^{2}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} + \frac {75 d x \tanh {\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} + \frac {15 d x}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} - \frac {15 \tanh ^{4}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} - \frac {75 \tanh ^{3}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} - \frac {155 \tanh ^{2}{\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} - \frac {175 \tanh {\left (c + d x \right )}}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} - \frac {128}{480 a^{5} d \tanh ^{5}{\left (c + d x \right )} + 2400 a^{5} d \tanh ^{4}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{3}{\left (c + d x \right )} + 4800 a^{5} d \tanh ^{2}{\left (c + d x \right )} + 2400 a^{5} d \tanh {\left (c + d x \right )} + 480 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \tanh {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 75, normalized size = 0.62 \begin {gather*} -\frac {\frac {{\left (300 \, e^{\left (8 \, d x + 8 \, c\right )} + 300 \, e^{\left (6 \, d x + 6 \, c\right )} + 200 \, e^{\left (4 \, d x + 4 \, c\right )} + 75 \, e^{\left (2 \, d x + 2 \, c\right )} + 12\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{a^{5}} - \frac {120 \, {\left (d x + c\right )}}{a^{5}}}{3840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 92, normalized size = 0.76 \begin {gather*} \frac {x}{32\,a^5}-\frac {5\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{64\,a^5\,d}-\frac {5\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,a^5\,d}-\frac {5\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{96\,a^5\,d}-\frac {5\,{\mathrm {e}}^{-8\,c-8\,d\,x}}{256\,a^5\,d}-\frac {{\mathrm {e}}^{-10\,c-10\,d\,x}}{320\,a^5\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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