Optimal. Leaf size=42 \[ \frac {(a+b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {(a-b) \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 42, 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 = {3269, 393, 209}
\begin {gather*} \frac {(a+b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {(a-b) \tanh (c+d x) \text {sech}(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 393
Rule 3269
Rubi steps
\begin {align*} \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b) \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac {(a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {(a-b) \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 1.69 \begin {gather*} \frac {a \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {b \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {a \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 2.00, size = 109, normalized size = 2.60
method | result | size |
risch | \(\frac {{\mathrm e}^{d x +c} \left (a -b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{2 d}\) | \(109\) |
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. 136 vs.
\(2 (38) = 76\).
time = 0.47, size = 136, normalized size = 3.24 \begin {gather*} -b {\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 )} - 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. 324 vs.
\(2 (38) = 76\).
time = 0.38, size = 324, normalized size = 7.71 \begin {gather*} \frac {{\left (a - b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a - b\right )} \sinh \left (d x + c\right )^{3} + {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} - a + b\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \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 \sinh ^{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. 105 vs.
\(2 (38) = 76\).
time = 0.42, size = 105, normalized size = 2.50 \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 (a + b\right )} + \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 127, normalized size = 3.02 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {d^2}+b\,\sqrt {d^2}\right )}{d\,\sqrt {a^2+2\,a\,b+b^2}}\right )\,\sqrt {a^2+2\,a\,b+b^2}}{\sqrt {d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a-b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a-b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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