Optimal. Leaf size=48 \[ \frac {a^2 \log (\cosh (c+d x))}{d}-\frac {a b \text {sech}^2(c+d x)}{d}-\frac {b^2 \text {sech}^4(c+d x)}{4 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 48, 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 = {4223, 272, 45}
\begin {gather*} \frac {a^2 \log (\cosh (c+d x))}{d}-\frac {a b \text {sech}^2(c+d x)}{d}-\frac {b^2 \text {sech}^4(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4223
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^2 \tanh (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (b+a x^2\right )^2}{x^5} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(b+a x)^2}{x^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^2}{x^3}+\frac {2 a b}{x^2}+\frac {a^2}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {a^2 \log (\cosh (c+d x))}{d}-\frac {a b \text {sech}^2(c+d x)}{d}-\frac {b^2 \text {sech}^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 81, normalized size = 1.69 \begin {gather*} \frac {\left (b+a \cosh ^2(c+d x)\right )^2 \left (-b^2-4 a b \cosh ^2(c+d x)+4 a^2 \cosh ^4(c+d x) \log (\cosh (c+d x))\right ) \text {sech}^4(c+d x)}{d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 42, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {\frac {\mathrm {sech}\left (d x +c \right )^{4} b^{2}}{4}+\mathrm {sech}\left (d x +c \right )^{2} a b +a^{2} \ln \left (\mathrm {sech}\left (d x +c \right )\right )}{d}\) | \(42\) |
default | \(-\frac {\frac {\mathrm {sech}\left (d x +c \right )^{4} b^{2}}{4}+\mathrm {sech}\left (d x +c \right )^{2} a b +a^{2} \ln \left (\mathrm {sech}\left (d x +c \right )\right )}{d}\) | \(42\) |
risch | \(-a^{2} x -\frac {2 a^{2} c}{d}-\frac {4 b \,{\mathrm e}^{2 d x +2 c} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{2 d x +2 c}+a \right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {a^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 55, normalized size = 1.15 \begin {gather*} \frac {a b \tanh \left (d x + c\right )^{2}}{d} + \frac {a^{2} \log \left (\cosh \left (d x + c\right )\right )}{d} - \frac {4 \, b^{2}}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4}} \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. 1180 vs.
\(2 (46) = 92\).
time = 0.37, size = 1180, normalized size = 24.58 \begin {gather*} -\frac {a^{2} d x \cosh \left (d x + c\right )^{8} + 8 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + a^{2} d x \sinh \left (d x + c\right )^{8} + 4 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{2} + a^{2} d x + a b\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, a^{2} d x \cosh \left (d x + c\right )^{4} + 3 \, a^{2} d x + 30 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{2} d x + 8 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{5} + 10 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{6} + 15 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{4} + a^{2} d x + 3 \, {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - {\left (a^{2} \cosh \left (d x + c\right )^{8} + 8 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + a^{2} \sinh \left (d x + c\right )^{8} + 4 \, a^{2} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2}\right )} \sinh \left (d x + c\right )^{6} + 6 \, a^{2} \cosh \left (d x + c\right )^{4} + 8 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (35 \, a^{2} \cosh \left (d x + c\right )^{4} + 30 \, a^{2} \cosh \left (d x + c\right )^{2} + 3 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right )^{2} + 8 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{5} + 10 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{6} + 15 \, a^{2} \cosh \left (d x + c\right )^{4} + 9 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 8 \, {\left (a^{2} \cosh \left (d x + c\right )^{7} + 3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \cosh \left (d x + c\right )^{3} + a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (a^{2} d x \cosh \left (d x + c\right )^{7} + 3 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{5} + {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 63, normalized size = 1.31 \begin {gather*} \begin {cases} a^{2} x - \frac {a^{2} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a b \operatorname {sech}^{2}{\left (c + d x \right )}}{d} - \frac {b^{2} \operatorname {sech}^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\left (c \right )}\right )^{2} \tanh {\left (c \right )} & \text {otherwise} \end {cases} \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. 162 vs.
\(2 (46) = 92\).
time = 0.41, size = 162, normalized size = 3.38 \begin {gather*} -\frac {12 \, {\left (d x + c\right )} a^{2} - 12 \, a^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {25 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 100 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 150 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 100 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.69, size = 182, normalized size = 3.79 \begin {gather*} \frac {4\,\left (a\,b-b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-a^2\,x+\frac {8\,b^2}{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^2}{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^2\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}-\frac {4\,a\,b}{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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