Optimal. Leaf size=53 \[ \frac {(a+b)^2 \log (\sinh (c+d x))}{d}-\frac {b (2 a+b) \log (\cosh (c+d x))}{d}+\frac {b^2 \text {sech}^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.08, 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 = {4138, 446, 88} \[ \frac {(a+b)^2 \log (\sinh (c+d x))}{d}-\frac {b (2 a+b) \log (\cosh (c+d x))}{d}+\frac {b^2 \text {sech}^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=-\frac {\operatorname {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 {\operatorname {Subst}\left (\int \frac {(b+a x)^2}{(1-x) x^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {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 = 0.25, size = 84, normalized size = 1.58 \[ \frac {2 (a \cosh (c+d x)+b \text {sech}(c+d x))^2 \left (2 \cosh ^2(c+d x) \left ((a+b)^2 \log (\sinh (c+d x))-b (2 a+b) \log (\cosh (c+d x))\right )+b^2\right )}{d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 665, normalized size = 12.55 \[ -\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} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 171, normalized size = 3.23 \[ -\frac {2 \, a^{2} d x + 2 \, {\left (2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 2 \, {\left (a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) - \frac {6 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 10 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b + 3 \, b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 60, normalized size = 1.13 \[ \frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )}{d}+\frac {2 a b \ln \left (\tanh \left (d x +c \right )\right )}{d}+\frac {b^{2}}{2 d \cosh \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (\tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 161, normalized size = 3.04 \[ b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \frac {a^{2} \log \left (\sinh \left (d x + c\right )\right )}{d} \]
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 \[ \frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^2\,x+\frac {a^2\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d}-\frac {2\,b^2}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^4\,\sqrt {-d^2}+4\,b^4\,\sqrt {-d^2}+16\,a\,b^3\,\sqrt {-d^2}+8\,a^3\,b\,\sqrt {-d^2}+20\,a^2\,b^2\,\sqrt {-d^2}\right )}{a^2\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}+2\,b^2\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}+4\,a\,b\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}}\right )\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}}{\sqrt {-d^2}} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \coth {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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