Optimal. Leaf size=49 \[ \frac {(a-b) \text {sech}^2(c+d x)}{2 d}+\frac {a \log (\cosh (c+d x))}{d}+\frac {b \text {sech}^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.06, antiderivative size = 49, 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 = {4138, 446, 76} \[ \frac {(a-b) \text {sech}^2(c+d x)}{2 d}+\frac {a \log (\cosh (c+d x))}{d}+\frac {b \text {sech}^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \tanh ^3(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (b+a x^2\right )}{x^5} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(1-x) (b+a x)}{x^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {b}{x^3}+\frac {a-b}{x^2}-\frac {a}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {a \log (\cosh (c+d x))}{d}+\frac {(a-b) \text {sech}^2(c+d x)}{2 d}+\frac {b \text {sech}^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.92 \[ -\frac {a \tanh ^2(c+d x)}{2 d}+\frac {a \log (\cosh (c+d x))}{d}+\frac {b \tanh ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 1072, normalized size = 21.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 116, normalized size = 2.37 \[ -\frac {12 \, a d x - 12 \, a \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {25 \, a e^{\left (8 \, d x + 8 \, c\right )} + 76 \, a e^{\left (6 \, d x + 6 \, c\right )} + 24 \, b e^{\left (6 \, d x + 6 \, c\right )} + 102 \, a e^{\left (4 \, d x + 4 \, c\right )} + 76 \, a e^{\left (2 \, d x + 2 \, c\right )} + 24 \, b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 64, normalized size = 1.31 \[ \frac {a \ln \left (\cosh \left (d x +c \right )\right )}{d}-\frac {\left (\tanh ^{2}\left (d x +c \right )\right ) a}{2 d}-\frac {b \left (\sinh ^{2}\left (d x +c \right )\right )}{2 d \cosh \left (d x +c \right )^{4}}-\frac {b}{4 d \cosh \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 78, normalized size = 1.59 \[ \frac {b \tanh \left (d x + c\right )^{4}}{4 \, d} + a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 173, normalized size = 3.53 \[ \frac {2\,\left (a-b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a\,x-\frac {2\,\left (a-3\,b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {8\,b}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {4\,b}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {a\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.66, size = 80, normalized size = 1.63 \[ \begin {cases} a x - \frac {a \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac {b \tanh ^{2}{\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} - \frac {b \operatorname {sech}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\relax (c )}\right ) \tanh ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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