Optimal. Leaf size=32 \[ -\frac {a \tanh (c+d x)}{d}+a x+\frac {b \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4141, 1802, 206} \[ -\frac {a \tanh (c+d x)}{d}+a x+\frac {b \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \tanh ^2(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b \left (1-x^2\right )\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a+b x^2+\frac {a}{1-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}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a x-\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 = 0.02, size = 41, normalized size = 1.28 \[ \frac {a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 155, normalized size = 4.84 \[ \frac {{\left (3 \, a d x + 3 \, a - b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a d x + 3 \, a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a - b\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a d x + 3 \, a - b\right )} \cosh \left (d x + c\right ) - 3 \, {\left ({\left (3 \, a - b\right )} \cosh \left (d x + c\right )^{2} + a + b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 69, normalized size = 2.16 \[ \frac {3 \, a d x + \frac {2 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a - b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 60, normalized size = 1.88 \[ \frac {a \left (d x +c -\tanh \left (d x +c \right )\right )+b \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 42, normalized size = 1.31 \[ \frac {b \tanh \left (d x + c\right )^{3}}{3 \, d} + a {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 163, normalized size = 5.09 \[ \frac {\frac {2\,\left (a+b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a-b\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+a\,x+\frac {\frac {2\,\left (a-b\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a-b\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {2\,\left (a-b\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \tanh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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