Optimal. Leaf size=61 \[ a^3 x-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.10, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 207} \[ a^3 x-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \coth ^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 \left (1-x^2\right )\right )^3}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b^2 (3 a+2 b)+\frac {(a+b)^3}{x^2}+b^3 x^2-\frac {a^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 1.80, size = 126, normalized size = 2.07 \[ \frac {8 (a \cosh (c+d x)+b \text {sech}(c+d x))^3 \left (3 a^3 d x \cosh ^3(c+d x)+\sinh (d x) \cosh ^2(c+d x) \left (3 (a+b)^3 \text {csch}(c) \coth (c+d x)-b^2 (9 a+5 b) \text {sech}(c)\right )-b^3 \tanh (c) \cosh (c+d x)+b^3 (-\text {sech}(c)) \sinh (d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 359, normalized size = 5.89 \[ -\frac {{\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 4 \, {\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 9 \, a^{3} + 27 \, a^{2} b + 18 \, a b^{2} + 4 \, {\left (3 \, a^{3} + 9 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{3} + 18 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3} + 3 \, {\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 4 \, {\left ({\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{12 \, {\left (d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 132, normalized size = 2.16 \[ \frac {3 \, a^{3} d x - \frac {6 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {2 \, {\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 5 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 111, normalized size = 1.82 \[ \frac {a^{3} \left (d x +c -\coth \left (d x +c \right )\right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )+b^{3} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}-4 \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 172, normalized size = 2.82 \[ a^{3} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} - \frac {16}{3} \, b^{3} {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}} + \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac {12 \, a b^{2}}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 234, normalized size = 3.84 \[ \frac {\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+a^3\,x+\frac {\frac {2\,\left (b^3+a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \coth ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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