Integrand size = 12, antiderivative size = 73 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^3 x+\frac {b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))}{2 d}+\frac {5 a b^2 \tanh (c+d x)}{2 d}+\frac {b^2 (a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3867, 3855, 3852, 8} \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^3 x+\frac {b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))}{2 d}+\frac {5 a b^2 \tanh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x) (a+b \text {sech}(c+d x))}{2 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3867
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 (a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a^3+b \left (6 a^2+b^2\right ) \text {sech}(c+d x)+5 a b^2 \text {sech}^2(c+d x)\right ) \, dx \\ & = a^3 x+\frac {b^2 (a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d}+\frac {1}{2} \left (5 a b^2\right ) \int \text {sech}^2(c+d x) \, dx+\frac {1}{2} \left (b \left (6 a^2+b^2\right )\right ) \int \text {sech}(c+d x) \, dx \\ & = a^3 x+\frac {b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))}{2 d}+\frac {b^2 (a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d}+\frac {\left (5 i a b^2\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 d} \\ & = a^3 x+\frac {b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))}{2 d}+\frac {5 a b^2 \tanh (c+d x)}{2 d}+\frac {b^2 (a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {2 a^3 d x+b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))+b^2 (6 a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d} \]
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Time = 0.91 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {a^{3} \left (d x +c \right )+6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \tanh \left (d x +c \right )+b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(66\) |
default | \(\frac {a^{3} \left (d x +c \right )+6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \tanh \left (d x +c \right )+b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(66\) |
parts | \(a^{3} x +\frac {b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}+\frac {3 a \,b^{2} \tanh \left (d x +c \right )}{d}+\frac {3 a^{2} b \arctan \left (\sinh \left (d x +c \right )\right )}{d}\) | \(67\) |
parallelrisch | \(\frac {-3 i \left (1+\cosh \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {b^{2}}{6}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+3 i \left (1+\cosh \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {b^{2}}{6}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+a^{3} d x \cosh \left (2 d x +2 c \right )+a^{3} d x +3 a \,b^{2} \sinh \left (2 d x +2 c \right )+b^{3} \sinh \left (d x +c \right )}{d \left (1+\cosh \left (2 d x +2 c \right )\right )}\) | \(139\) |
risch | \(a^{3} x -\frac {b^{2} \left (-{\mathrm e}^{3 d x +3 c} b +6 \,{\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{d x +c} b +6 a \right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}+\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}-\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(142\) |
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Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (67) = 134\).
Time = 0.27 (sec) , antiderivative size = 521, normalized size of antiderivative = 7.14 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {a^{3} d x \cosh \left (d x + c\right )^{4} + a^{3} d x \sinh \left (d x + c\right )^{4} + b^{3} \cosh \left (d x + c\right )^{3} + a^{3} d x - b^{3} \cosh \left (d x + c\right ) + {\left (4 \, a^{3} d x \cosh \left (d x + c\right ) + b^{3}\right )} \sinh \left (d x + c\right )^{3} - 6 \, a b^{2} + 2 \, {\left (a^{3} d x - 3 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (6 \, a^{3} d x \cosh \left (d x + c\right )^{2} + 2 \, a^{3} d x + 3 \, b^{3} \cosh \left (d x + c\right ) - 6 \, a b^{2}\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{4} + 6 \, a^{2} b + b^{3} + 2 \, {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{2} b + b^{3} + 3 \, {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + {\left (4 \, a^{3} d x \cosh \left (d x + c\right )^{3} + 3 \, b^{3} \cosh \left (d x + c\right )^{2} - b^{3} + 4 \, {\left (a^{3} d x - 3 \, a 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} \]
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\[ \int (a+b \text {sech}(c+d x))^3 \, dx=\int \left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{3}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^{3} x - b^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {3 \, a^{2} b \arctan \left (\sinh \left (d x + c\right )\right )}{d} + \frac {6 \, a b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {{\left (d x + c\right )} a^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + \frac {b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 6 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (d x + c\right )} - 6 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \]
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Time = 2.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.26 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^3\,x-\frac {\frac {6\,a\,b^2}{d}-\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+6\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^4\,b^2+12\,a^2\,b^4+b^6}}\right )\,\sqrt {36\,a^4\,b^2+12\,a^2\,b^4+b^6}}{\sqrt {d^2}}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
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