Optimal. Leaf size=73 \[ a^3 x+\frac {b \left (6 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {5 a b^2 \tanh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x) (a+b \text {sech}(c+d x))}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3782, 3770, 3767, 8} \[ \frac {b \left (6 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d}+a^3 x+\frac {5 a b^2 \tanh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x) (a+b \text {sech}(c+d x))}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3770
Rule 3782
Rubi steps
\begin {align*} \int (a+b \text {sech}(c+d x))^3 \, dx &=\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 ) \tan ^{-1}(\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 ) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 d}\\ &=a^3 x+\frac {b \left (6 a^2+b^2\right ) \tan ^{-1}(\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 55, normalized size = 0.75 \[ \frac {2 a^3 d x+b \left (6 a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))+b^2 \tanh (c+d x) (6 a+b \text {sech}(c+d x))}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 521, normalized size = 7.14 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 92, normalized size = 1.26 \[ \frac {{\left (d x + c\right )} a^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + \frac {b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 6 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (d x + c\right )} - 6 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 80, normalized size = 1.10 \[ a^{3} x +\frac {a^{3} c}{d}+\frac {6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {3 a \,b^{2} \tanh \left (d x +c \right )}{d}+\frac {b^{3} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}+\frac {b^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 114, normalized size = 1.56 \[ a^{3} x - b^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {3 \, a^{2} b \arctan \left (\sinh \left (d x + c\right )\right )}{d} + \frac {6 \, a b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.40, size = 165, normalized size = 2.26 \[ a^3\,x-\frac {\frac {6\,a\,b^2}{d}-\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+6\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^4\,b^2+12\,a^2\,b^4+b^6}}\right )\,\sqrt {36\,a^4\,b^2+12\,a^2\,b^4+b^6}}{\sqrt {d^2}}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________