Optimal. Leaf size=55 \[ \frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4138, 446, 88} \[ \frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \coth ^3(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 \left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^2}{(1-x)^2 x} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {(a+b)^2}{(-1+x)^2}+\frac {a^2-b^2}{-1+x}+\frac {b^2}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {(a+b)^2 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 \log (\cosh (c+d x))}{d}+\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 82, normalized size = 1.49 \[ -\frac {2 \left (a \cosh ^2(c+d x)+b\right )^2 \left ((a+b)^2 \text {csch}^2(c+d x)-2 \left (\left (a^2-b^2\right ) \log (\sinh (c+d x))+b^2 \log (\cosh (c+d x))\right )\right )}{d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 637, normalized size = 11.58 \[ -\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 - a^{2} - 2 \, a b - 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 + a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{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 ) - {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - 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 - a^{2} - 2 \, a b - 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 = 0.25, size = 161, normalized size = 2.93 \[ -\frac {2 \, a^{2} d x - 2 \, b^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 2 \, {\left (a^{2} e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 10 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} - 3 \, b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 78, normalized size = 1.42 \[ \frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )}{d}-\frac {a^{2} \left (\coth ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a b}{d \sinh \left (d x +c \right )^{2}}-\frac {b^{2}}{2 d \sinh \left (d x +c \right )^{2}}-\frac {b^{2} \ln \left (\tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 206, normalized size = 3.75 \[ a^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {4 \, a b}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 240, normalized size = 4.36 \[ \frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (d\,\left (a^2-b^2\right )+b^2\,d\right )}{2\,d^2}-a^2\,x-\frac {2\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^4\,\sqrt {-d^2}+4\,b^4\,\sqrt {-d^2}-4\,a^2\,b^2\,\sqrt {-d^2}\right )}{a^2\,d\,\sqrt {a^4-4\,a^2\,b^2+4\,b^4}-2\,b^2\,d\,\sqrt {a^4-4\,a^2\,b^2+4\,b^4}}\right )\,\sqrt {a^4-4\,a^2\,b^2+4\,b^4}}{\sqrt {-d^2}}-\frac {2\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \coth ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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