Optimal. Leaf size=72 \[ \frac {a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} a^2 x (a+6 b)+\frac {b^2 (3 a+b) \tanh (c+d x)}{d}-\frac {b^3 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4146, 390, 385, 206} \[ \frac {1}{2} a^2 x (a+6 b)+\frac {a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {b^2 (3 a+b) \tanh (c+d x)}{d}-\frac {b^3 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 390
Rule 4146
Rubi steps
\begin {align*} \int \cosh ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-b x^2\right )^3}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^2 (3 a+b)-b^3 x^2+\frac {a^2 (a+3 b)-3 a^2 b x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 (3 a+b) \tanh (c+d x)}{d}-\frac {b^3 \tanh ^3(c+d x)}{3 d}+\frac {\operatorname {Subst}\left (\int \frac {a^2 (a+3 b)-3 a^2 b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 (3 a+b) \tanh (c+d x)}{d}-\frac {b^3 \tanh ^3(c+d x)}{3 d}+\frac {\left (a^2 (a+6 b)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {1}{2} a^2 (a+6 b) x+\frac {a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 (3 a+b) \tanh (c+d x)}{d}-\frac {b^3 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 64, normalized size = 0.89 \[ \frac {3 a^3 \sinh (2 (c+d x))+6 a^2 (a+6 b) (c+d x)+4 b^2 \tanh (c+d x) \left (9 a+b \text {sech}^2(c+d x)+2 b\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 270, normalized size = 3.75 \[ \frac {3 \, a^{3} \sinh \left (d x + c\right )^{5} - 4 \, {\left (18 \, a b^{2} + 4 \, b^{3} - 3 \, {\left (a^{3} + 6 \, a^{2} b\right )} d x\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (18 \, a b^{2} + 4 \, b^{3} - 3 \, {\left (a^{3} + 6 \, a^{2} b\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (30 \, a^{3} \cosh \left (d x + c\right )^{2} + 9 \, a^{3} + 72 \, a b^{2} + 16 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} - 12 \, {\left (18 \, a b^{2} + 4 \, b^{3} - 3 \, {\left (a^{3} + 6 \, a^{2} b\right )} d x\right )} \cosh \left (d x + c\right ) + 3 \, {\left (5 \, a^{3} \cosh \left (d x + c\right )^{4} + 2 \, a^{3} + 24 \, a b^{2} + 16 \, b^{3} + {\left (9 \, a^{3} + 72 \, a b^{2} + 16 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 152, normalized size = 2.11 \[ \frac {3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, {\left (a^{3} + 6 \, a^{2} b\right )} {\left (d x + c\right )} - 3 \, {\left (2 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + a^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 2 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 77, normalized size = 1.07 \[ \frac {a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \tanh \left (d x +c \right )+b^{3} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 160, normalized size = 2.22 \[ \frac {1}{8} \, a^{3} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 3 \, a^{2} b x + \frac {4}{3} \, b^{3} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {6 \, a b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 221, normalized size = 3.07 \[ \frac {a^2\,x\,\left (a+6\,b\right )}{2}-\frac {\frac {2\,a\,b^2}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,b^3+3\,a\,b^2\right )}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {2\,\left (2\,b^3+3\,a\,b^2\right )}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {a^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {2\,a\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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