Optimal. Leaf size=82 \[ \frac {1}{2} b^2 x (6 a-5 b)-\frac {(a-b)^3 \tanh ^3(c+d x)}{3 d}+\frac {(a-b)^2 (a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \sinh (c+d x) \cosh (c+d x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 82, 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 = {3191, 390, 385, 206} \[ \frac {1}{2} b^2 x (6 a-5 b)-\frac {(a-b)^3 \tanh ^3(c+d x)}{3 d}+\frac {(a-b)^2 (a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \sinh (c+d x) \cosh (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 390
Rule 3191
Rubi steps
\begin {align*} \int \text {sech}^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-(a-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 ((a-b)^2 (a+2 b)-(a-b)^3 x^2+\frac {(3 a-2 b) b^2-3 (a-b) b^2 x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a-b)^2 (a+2 b) \tanh (c+d x)}{d}-\frac {(a-b)^3 \tanh ^3(c+d x)}{3 d}+\frac {\operatorname {Subst}\left (\int \frac {(3 a-2 b) b^2-3 (a-b) b^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {(a-b)^2 (a+2 b) \tanh (c+d x)}{d}-\frac {(a-b)^3 \tanh ^3(c+d x)}{3 d}+\frac {\left ((6 a-5 b) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {1}{2} (6 a-5 b) b^2 x+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {(a-b)^2 (a+2 b) \tanh (c+d x)}{d}-\frac {(a-b)^3 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 84, normalized size = 1.02 \[ \frac {6 b^2 (6 a-5 b) (c+d x)+2 (a-b)^2 \tanh (c+d x) \text {sech}^2(c+d x) ((2 a+7 b) \cosh (2 (c+d x))+4 a+5 b)+3 b^3 \sinh (2 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 321, normalized size = 3.91 \[ \frac {3 \, b^{3} \sinh \left (d x + c\right )^{5} - 4 \, {\left (4 \, a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 14 \, b^{3} - 3 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (4 \, a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 14 \, b^{3} - 3 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (30 \, b^{3} \cosh \left (d x + c\right )^{2} + 16 \, a^{3} + 24 \, a^{2} b - 96 \, a b^{2} + 65 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} - 12 \, {\left (4 \, a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 14 \, b^{3} - 3 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 16 \, a^{3} - 24 \, a^{2} b + 10 \, b^{3} + {\left (16 \, a^{3} + 24 \, a^{2} b - 96 \, a b^{2} + 65 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 208, normalized size = 2.54 \[ \frac {3 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} {\left (d x + c\right )} - 3 \, {\left (12 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (9 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} + 3 \, a^{2} b - 12 \, a b^{2} + 7 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 148, normalized size = 1.80 \[ \frac {a^{3} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )+3 a \,b^{2} \left (d x +c -\tanh \left (d x +c \right )-\frac {\left (\tanh ^{3}\left (d x +c \right )\right )}{3}\right )+b^{3} \left (\frac {\sinh ^{5}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \left (\tanh ^{3}\left (d x +c \right )\right )}{6}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 382, normalized size = 4.66 \[ a b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} - \frac {1}{24} \, b^{3} {\left (\frac {60 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {121 \, e^{\left (-2 \, d x - 2 \, c\right )} + 201 \, e^{\left (-4 \, d x - 4 \, c\right )} + 147 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} + \frac {4}{3} \, a^{3} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + 2 \, a^{2} b {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 273, normalized size = 3.33 \[ \frac {b^2\,x\,\left (6\,a-5\,b\right )}{2}-\frac {\frac {2\,\left (a^2\,b-2\,a\,b^2+b^3\right )}{d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2\,b-2\,a\,b^2+b^3\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^3-3\,a^2\,b+b^3\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {2\,\left (2\,a^3-3\,a^2\,b+b^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b-2\,a\,b^2+b^3\right )}{d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {2\,\left (a^2\,b-2\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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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