Optimal. Leaf size=28 \[ \frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3756}
\begin {gather*} \frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3756
Rubi steps
\begin {align*} \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \left (a+b x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs.
\(2(26)=52\).
time = 1.54, size = 60, normalized size = 2.14
method | result | size |
risch | \(-\frac {2 \left (3 a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}+6 a \,{\mathrm e}^{2 d x +2 c}+3 a +b \right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 34, normalized size = 1.21 \begin {gather*} \frac {b \tanh \left (d x + c\right )^{3}}{3 \, d} + \frac {2 \, a}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\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. 159 vs.
\(2 (26) = 52\).
time = 0.35, size = 159, normalized size = 5.68 \begin {gather*} -\frac {4 \, {\left ({\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (3 \, a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 3 \, a\right )}}{3 \, {\left (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} + 4 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + 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 ) + 3 \, 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 \tanh ^{2}{\left (c + d x \right )}\right ) \operatorname {sech}^{2}{\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. 59 vs.
\(2 (26) = 52\).
time = 0.44, size = 59, normalized size = 2.11 \begin {gather*} -\frac {2 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a + b\right )}}{3 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 59, normalized size = 2.11 \begin {gather*} -\frac {2\,\left (3\,a+b+6\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}+3\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}\right )}{3\,d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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