Optimal. Leaf size=102 \[ \frac {(c+d x) \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i d \text {PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4270, 4265,
2317, 2438} \begin {gather*} \frac {(c+d x) \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i d \text {Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {Li}_2\left (i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4265
Rule 4270
Rubi steps
\begin {align*} \int (c+d x) \text {sech}^3(a+b x) \, dx &=\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {1}{2} \int (c+d x) \text {sech}(a+b x) \, dx\\ &=\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b}-\frac {(i d) \int \log \left (1-i e^{a+b x}\right ) \, dx}{2 b}+\frac {(i d) \int \log \left (1+i e^{a+b x}\right ) \, dx}{2 b}\\ &=\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b}-\frac {(i d) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}+\frac {(i d) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {i d \text {Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {Li}_2\left (i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 2.11, size = 178, normalized size = 1.75 \begin {gather*} \frac {b c \text {ArcTan}(\sinh (a+b x))+\frac {1}{2} d \left (-\left ((-2 i a+\pi -2 i b x) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )\right )+(-2 i a+\pi ) \log \left (\cot \left (\frac {1}{4} (2 i a+\pi +2 i b x)\right )\right )-2 i \left (\text {PolyLog}\left (2,-i e^{a+b x}\right )-\text {PolyLog}\left (2,i e^{a+b x}\right )\right )\right )+b d x \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)+d \text {sech}(a+b x) (1+b x \tanh (a))+b c \text {sech}(a+b x) \tanh (a+b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 215 vs. \(2 (87 ) = 174\).
time = 0.99, size = 216, normalized size = 2.12
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a} \left (b d x \,{\mathrm e}^{2 b x +2 a}+b c \,{\mathrm e}^{2 b x +2 a}-b d x +{\mathrm e}^{2 b x +2 a} d -b c +d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}+\frac {c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {i d \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {i d \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1267 vs. \(2 (81) = 162\).
time = 0.39, size = 1267, normalized size = 12.42 \begin {gather*} \frac {2 \, {\left (b d x + b c + d\right )} \cosh \left (b x + a\right )^{3} + 6 \, {\left (b d x + b c + d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, {\left (b d x + b c + d\right )} \sinh \left (b x + a\right )^{3} - 2 \, {\left (b d x + b c - d\right )} \cosh \left (b x + a\right ) + {\left (i \, d \cosh \left (b x + a\right )^{4} + 4 i \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + i \, d \sinh \left (b x + a\right )^{4} + 2 i \, d \cosh \left (b x + a\right )^{2} - 2 \, {\left (-3 i \, d \cosh \left (b x + a\right )^{2} - i \, d\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (-i \, d \cosh \left (b x + a\right )^{3} - i \, d \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + i \, d\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + {\left (-i \, d \cosh \left (b x + a\right )^{4} - 4 i \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - i \, d \sinh \left (b x + a\right )^{4} - 2 i \, d \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 i \, d \cosh \left (b x + a\right )^{2} + i \, d\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (i \, d \cosh \left (b x + a\right )^{3} + i \, d \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - i \, d\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left ({\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (i \, b c - i \, a d\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{2} - i \, b c + i \, a d\right )} \sinh \left (b x + a\right )^{2} + i \, b c - i \, a d - 4 \, {\left ({\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + {\left ({\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (-i \, b c + i \, a d\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{2} + i \, b c - i \, a d\right )} \sinh \left (b x + a\right )^{2} - i \, b c + i \, a d - 4 \, {\left ({\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left ({\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (-i \, b d x - i \, a d\right )} \sinh \left (b x + a\right )^{4} - i \, b d x - 2 \, {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (i \, b d x + 3 \, {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{2} + i \, a d\right )} \sinh \left (b x + a\right )^{2} - i \, a d - 4 \, {\left ({\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + {\left ({\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (i \, b d x + i \, a d\right )} \sinh \left (b x + a\right )^{4} + i \, b d x - 2 \, {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (-i \, b d x + 3 \, {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{2} - i \, a d\right )} \sinh \left (b x + a\right )^{2} + i \, a d - 4 \, {\left ({\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (b d x - 3 \, {\left (b d x + b c + d\right )} \cosh \left (b x + a\right )^{2} + b c - d\right )} \sinh \left (b x + a\right )}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} + 2 \, b^{2} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} + b^{2}\right )} \sinh \left (b x + a\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + b^{2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}
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 \begin {gather*} \int \left (c + d x\right ) \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {c+d\,x}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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