Optimal. Leaf size=52 \[ \frac {a^2 \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {a (a+b) \text {csch}^2(c+d x)}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 52, 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, 444, 43} \[ \frac {a^2 \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {a (a+b) \text {csch}^2(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 4138
Rubi steps
\begin {align*} \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x \left (b+a x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^2}{(1-x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {(a+b)^2}{(-1+x)^3}-\frac {2 a (a+b)}{(-1+x)^2}-\frac {a^2}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {a (a+b) \text {csch}^2(c+d x)}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}+\frac {a^2 \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 77, normalized size = 1.48 \[ -\frac {\left (a \cosh ^2(c+d x)+b\right )^2 \left (-4 a^2 \log (\sinh (c+d x))+(a+b)^2 \text {csch}^4(c+d x)+4 a (a+b) \text {csch}^2(c+d x)\right )}{d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 1252, normalized size = 24.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 147, normalized size = 2.83 \[ -\frac {12 \, a^{2} d x - 12 \, a^{2} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {25 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 52 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 102 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 52 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 102, normalized size = 1.96 \[ \frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )}{d}-\frac {a^{2} \left (\coth ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (\coth ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a b \left (\cosh ^{2}\left (d x +c \right )\right )}{d \sinh \left (d x +c \right )^{4}}+\frac {a b}{2 d \sinh \left (d x +c \right )^{4}}-\frac {b^{2}}{4 d \sinh \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 282, normalized size = 5.42 \[ a^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 4 \, a b {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} + \frac {e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - \frac {4 \, b^{2}}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 207, normalized size = 3.98 \[ \frac {a^2\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-\frac {4\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {4\,\left (2\,a^2+3\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^2\,x-\frac {4\,\left (a^2+b\,a\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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