Optimal. Leaf size=123 \[ -\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}+\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Rubi [A]
time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3399, 4270,
4269, 3556} \begin {gather*} \frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3399
Rule 3556
Rule 4269
Rule 4270
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+a \cosh (e+f x))^2} \, dx &=\frac {\int (c+d x) \csc ^4\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}\\ &=\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {d \int \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}+\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 114, normalized size = 0.93 \begin {gather*} \frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (-2 d \cosh \left (\frac {3}{2} (e+f x)\right ) \log \left (\cosh \left (\frac {1}{2} (e+f x)\right )\right )+\cosh \left (\frac {1}{2} (e+f x)\right ) \left (2 d-6 d \log \left (\cosh \left (\frac {1}{2} (e+f x)\right )\right )\right )+f (c+d x) \left (3 \sinh \left (\frac {1}{2} (e+f x)\right )+\sinh \left (\frac {3}{2} (e+f x)\right )\right )\right )}{3 a^2 f^2 (1+\cosh (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.57, size = 108, normalized size = 0.88
method | result | size |
risch | \(\frac {2 d x}{3 a^{2} f}+\frac {2 d e}{3 a^{2} f^{2}}-\frac {2 \left (3 d f x \,{\mathrm e}^{f x +e}+3 c f \,{\mathrm e}^{f x +e}+d x f -d \,{\mathrm e}^{2 f x +2 e}+c f -d \,{\mathrm e}^{f x +e}\right )}{3 f^{2} a^{2} \left ({\mathrm e}^{f x +e}+1\right )^{3}}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}+1\right )}{3 a^{2} f^{2}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs.
\(2 (100) = 200\).
time = 0.29, size = 255, normalized size = 2.07 \begin {gather*} \frac {2}{3} \, d {\left (\frac {f x e^{\left (3 \, f x + 3 \, e\right )} + {\left (3 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac {2}{3} \, c {\left (\frac {3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f} + \frac {1}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f}\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. 472 vs.
\(2 (100) = 200\).
time = 0.38, size = 472, normalized size = 3.84 \begin {gather*} \frac {2 \, {\left (d f x \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + d f x \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + {\left (3 \, d f x + d\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + {\left (3 \, d f x \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 3 \, d f x + d\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} - c f - {\left (3 \, c f - d\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + d \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 3 \, d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 3 \, {\left (d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + d\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 3 \, d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 3 \, {\left (d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + d\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + d\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right ) + {\left (3 \, d f x \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} - 3 \, c f + 2 \, {\left (3 \, d f x + d\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + d\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}}{3 \, {\left (a^{2} f^{2} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + a^{2} f^{2} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 3 \, a^{2} f^{2} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 3 \, a^{2} f^{2} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a^{2} f^{2} + 3 \, {\left (a^{2} f^{2} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a^{2} f^{2}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 3 \, {\left (a^{2} f^{2} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 2 \, a^{2} f^{2} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a^{2} f^{2}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 156, normalized size = 1.27 \begin {gather*} \begin {cases} - \frac {c \tanh ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f} + \frac {c \tanh {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{2 a^{2} f} - \frac {d x \tanh ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f} + \frac {d x \tanh {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{2 a^{2} f} - \frac {d x}{3 a^{2} f} + \frac {2 d \log {\left (\tanh {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )}}{3 a^{2} f^{2}} - \frac {d \tanh ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{6 a^{2} f^{2}} & \text {for}\: f \neq 0 \\\frac {c x + \frac {d x^{2}}{2}}{\left (a \cosh {\left (e \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs.
\(2 (95) = 190\).
time = 0.40, size = 192, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (d f x e^{\left (3 \, f x + 3 \, e\right )} + 3 \, d f x e^{\left (2 \, f x + 2 \, e\right )} - 3 \, c f e^{\left (f x + e\right )} - d e^{\left (3 \, f x + 3 \, e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - 3 \, d e^{\left (2 \, f x + 2 \, e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - 3 \, d e^{\left (f x + e\right )} \log \left (e^{\left (f x + e\right )} + 1\right ) - c f + d e^{\left (2 \, f x + 2 \, e\right )} + d e^{\left (f x + e\right )} - d \log \left (e^{\left (f x + e\right )} + 1\right )\right )}}{3 \, {\left (a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.89, size = 138, normalized size = 1.12 \begin {gather*} \frac {2\,d}{3\,a^2\,f^2\,\left ({\mathrm {e}}^{e+f\,x}+1\right )}-\frac {2\,\left (d+c\,f+d\,f\,x\right )}{3\,a^2\,f^2\,\left (2\,{\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}+\frac {2\,d\,x}{3\,a^2\,f}-\frac {2\,d\,\ln \left ({\mathrm {e}}^{f\,x}\,{\mathrm {e}}^e+1\right )}{3\,a^2\,f^2}-\frac {4\,{\mathrm {e}}^{e+f\,x}\,\left (c+d\,x\right )}{3\,a^2\,f\,\left (3\,{\mathrm {e}}^{e+f\,x}+3\,{\mathrm {e}}^{2\,e+2\,f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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