Optimal. Leaf size=75 \[ \frac {b (2 a+3 b) \tanh (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}+\frac {(a+b) (a+3 b) \coth (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4132, 448} \[ \frac {b (2 a+3 b) \tanh (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}+\frac {(a+b) (a+3 b) \coth (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 448
Rule 4132
Rubi steps
\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (a+b-b x^2\right )^2}{x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b (2 a+3 b)+\frac {(a+b)^2}{x^4}+\frac {(-a-3 b) (a+b)}{x^2}-b^2 x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b) (a+3 b) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}+\frac {b (2 a+3 b) \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 1.32, size = 151, normalized size = 2.01 \[ -\frac {\text {csch}(2 c) \text {csch}^3(2 (c+d x)) \left (-3 a^2 \sinh (2 (c+d x))+a^2 \sinh (6 (c+d x))+3 a^2 \sinh (4 c+2 d x)+a^2 \sinh (4 c+6 d x)-6 a b \sinh (2 (c+d x))+2 a b \sinh (6 (c+d x))+8 a b \sinh (4 c+6 d x)+8 a (a+2 b) \sinh (2 c)-6 (a+2 b)^2 \sinh (2 d x)+8 b^2 \sinh (4 c+6 d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 408, normalized size = 5.44 \[ -\frac {8 \, {\left ({\left (a^{2} - 4 \, a b - 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 8 \, {\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} - 4 \, a b - 4 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - 4 \, a b - 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} + 4 \, a b\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} + 12 \, a b + 12 \, b^{2} + 8 \, {\left ({\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{8} + 56 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{6} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} - 4 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} - 6 \, d \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} - d \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 115, normalized size = 1.53 \[ -\frac {4 \, {\left (3 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} - 8 \, a b - 8 \, b^{2}\right )}}{3 \, d {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 138, normalized size = 1.84 \[ \frac {a^{2} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+2 a b \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )}+\frac {4}{3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}+\frac {8 \tanh \left (d x +c \right )}{3}\right )+b^{2} \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )^{3}}+\frac {2}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}+8 \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 285, normalized size = 3.80 \[ \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {32}{3} \, a b {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} - \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + \frac {32}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 115, normalized size = 1.53 \[ -\frac {4\,\left (6\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}-a^2-8\,b^2-8\,a\,b+8\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}+3\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}+24\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}+24\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}+16\,a\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}\right )}{3\,d\,{\left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}^3} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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