Optimal. Leaf size=70 \[ \frac {(4 a+3 b) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {(4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4131, 3853,
3855} \begin {gather*} \frac {(4 a+3 b) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {(4 a+3 b) \tanh (c+d x) \text {sech}(c+d x)}{8 d}+\frac {b \tanh (c+d x) \text {sech}^3(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3853
Rule 3855
Rule 4131
Rubi steps
\begin {align*} \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {b \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {1}{4} (4 a+3 b) \int \text {sech}^3(c+d x) \, dx\\ &=\frac {(4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {1}{8} (4 a+3 b) \int \text {sech}(c+d x) \, dx\\ &=\frac {(4 a+3 b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac {(4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 60, normalized size = 0.86 \begin {gather*} \frac {(4 a+3 b) \text {ArcTan}(\sinh (c+d x))+(4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)+2 b \text {sech}^3(c+d x) \tanh (c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.36, size = 173, normalized size = 2.47
method | result | size |
risch | \(\frac {{\mathrm e}^{d x +c} \left (4 a \,{\mathrm e}^{6 d x +6 c}+3 b \,{\mathrm e}^{6 d x +6 c}+4 a \,{\mathrm e}^{4 d x +4 c}+11 b \,{\mathrm e}^{4 d x +4 c}-4 a \,{\mathrm e}^{2 d x +2 c}-11 b \,{\mathrm e}^{2 d x +2 c}-4 a -3 b \right )}{4 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d}\) | \(173\) |
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. 184 vs.
\(2 (64) = 128\).
time = 0.50, size = 184, normalized size = 2.63 \begin {gather*} -\frac {1}{4} \, b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\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. 1112 vs.
\(2 (64) = 128\).
time = 0.38, size = 1112, normalized size = 15.89 \begin {gather*} \frac {{\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + {\left (4 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{7} + {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )^{5} + {\left (21 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a + 11 \, b\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} + {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )^{3} + {\left (35 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )^{2} - 4 \, a - 11 \, b\right )} \sinh \left (d x + c\right )^{3} + {\left (21 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{8} + 8 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + {\left (4 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{8} + 4 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + 30 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 12 \, a + 9 \, b\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{6} + 15 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left ({\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{7} + 3 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{5} + 3 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} + {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, a + 3 \, b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (7 \, {\left (4 \, a + 3 \, b\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )^{4} - 3 \, {\left (4 \, a + 11 \, b\right )} \cosh \left (d x + c\right )^{2} - 4 \, a - 3 \, b\right )} \sinh \left (d x + c\right )}{4 \, {\left (d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} + 4 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} + 30 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} + 15 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} + 3 \, d \cosh \left (d x + c\right )^{5} + 3 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\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 (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \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. 156 vs.
\(2 (64) = 128\).
time = 0.40, size = 156, normalized size = 2.23 \begin {gather*} \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (4 \, a + 3 \, b\right )} + \frac {4 \, {\left (4 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 16 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 20 \, b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 283, normalized size = 4.04 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (4\,a\,\sqrt {d^2}+3\,b\,\sqrt {d^2}\right )}{d\,\sqrt {16\,a^2+24\,a\,b+9\,b^2}}\right )\,\sqrt {16\,a^2+24\,a\,b+9\,b^2}}{4\,\sqrt {d^2}}-\frac {\frac {a\,{\mathrm {e}}^{5\,c+5\,d\,x}}{d}+\frac {2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+2\,b\right )}{d}+\frac {a\,{\mathrm {e}}^{c+d\,x}}{d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (2\,a-b\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a+3\,b\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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