Optimal. Leaf size=31 \[ \frac {a \log (\sinh (c+d x))}{d}-\frac {(a+b) \text {csch}^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.06, antiderivative size = 31, 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, 444, 43} \[ \frac {a \log (\sinh (c+d x))}{d}-\frac {(a+b) \text {csch}^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 4138
Rubi steps
\begin {align*} \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x \left (b+a x^2\right )}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {b+a x}{(1-x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a+b}{(-1+x)^2}+\frac {a}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {(a+b) \text {csch}^2(c+d x)}{2 d}+\frac {a \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 52, normalized size = 1.68 \[ -\frac {a \left (\coth ^2(c+d x)-2 \log (\tanh (c+d x))-2 \log (\cosh (c+d x))\right )}{2 d}-\frac {b \text {csch}^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 378, normalized size = 12.19 \[ -\frac {a d x \cosh \left (d x + c\right )^{4} + 4 \, a d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a d x \sinh \left (d x + c\right )^{4} + a d x - 2 \, {\left (a d x - a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a d x \cosh \left (d x + c\right )^{2} - a d x + a + b\right )} \sinh \left (d x + c\right )^{2} - {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (a d x \cosh \left (d x + c\right )^{3} - {\left (a d x - a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 81, normalized size = 2.61 \[ -\frac {2 \, a d x - 2 \, a \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 42, normalized size = 1.35 \[ \frac {a \ln \left (\sinh \left (d x +c \right )\right )}{d}-\frac {a \left (\coth ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b}{2 d \sinh \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 108, normalized size = 3.48 \[ a {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {2 \, b}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 76, normalized size = 2.45 \[ \frac {a\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-a\,x-\frac {2\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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