Optimal. Leaf size=53 \[ -\frac {b (2 a+b) \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\sinh (c+d x))}{d}+\frac {b^2 \text {sech}^2(c+d x)}{2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 53, 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, 457, 90}
\begin {gather*} \frac {(a+b)^2 \log (\sinh (c+d x))}{d}-\frac {b (2 a+b) \log (\cosh (c+d x))}{d}+\frac {b^2 \text {sech}^2(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (b+a x^2\right )^2}{x^3 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {(b+a x)^2}{(1-x) x^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^2}{-1+x}+\frac {b^2}{x^2}+\frac {b (2 a+b)}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b (2 a+b) \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\sinh (c+d x))}{d}+\frac {b^2 \text {sech}^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 84, normalized size = 1.58 \begin {gather*} \frac {2 \left (b^2+2 \cosh ^2(c+d x) \left (-b (2 a+b) \log (\cosh (c+d x))+(a+b)^2 \log (\sinh (c+d x))\right )\right ) (a \cosh (c+d x)+b \text {sech}(c+d x))^2}{d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.58, size = 50, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )+2 a b \ln \left (\tanh \left (d x +c \right )\right )+b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(50\) |
default | \(\frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )+2 a b \ln \left (\tanh \left (d x +c \right )\right )+b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(50\) |
risch | \(-a^{2} x -\frac {2 a^{2} c}{d}+\frac {2 b^{2} {\mathrm e}^{2 d x +2 c}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {2 b \ln \left (1+{\mathrm e}^{2 d x +2 c}\right ) a}{d}-\frac {b^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a b}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{d}\) | \(143\) |
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. 161 vs.
\(2 (51) = 102\).
time = 0.47, size = 161, normalized size = 3.04 \begin {gather*} b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{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 )} + 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \frac {a^{2} \log \left (\sinh \left (d x + c\right )\right )}{d} \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. 665 vs.
\(2 (51) = 102\).
time = 0.41, size = 665, normalized size = 12.55 \begin {gather*} -\frac {a^{2} d x \cosh \left (d x + c\right )^{4} + 4 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} d x \sinh \left (d x + c\right )^{4} + a^{2} d x + 2 \, {\left (a^{2} d x - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} d x \cosh \left (d x + c\right )^{2} + a^{2} d x - b^{2}\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{2} + 2 \, a b + b^{2} + 4 \, {\left ({\left (2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (2 \, a b + b^{2}\right )} \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 ) - {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (a^{2} d x \cosh \left (d x + c\right )^{3} + {\left (a^{2} d x - b^{2}\right )} \cosh \left (d x + c\right )\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 \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \coth {\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. 150 vs.
\(2 (51) = 102\).
time = 0.42, size = 150, normalized size = 2.83 \begin {gather*} -\frac {{\left (2 \, a b + b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) - {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {2 \, a b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 4 \, a b + 6 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 308, normalized size = 5.81 \begin {gather*} \frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^2\,x+\frac {a^2\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d}-\frac {2\,b^2}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^4\,\sqrt {-d^2}+4\,b^4\,\sqrt {-d^2}+16\,a\,b^3\,\sqrt {-d^2}+8\,a^3\,b\,\sqrt {-d^2}+20\,a^2\,b^2\,\sqrt {-d^2}\right )}{a^2\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}+2\,b^2\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}+4\,a\,b\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}}\right )\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}}{\sqrt {-d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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