Optimal. Leaf size=113 \[ \frac {a \left (2 a^2-5 a b-2 b^2\right ) \coth (c+d x)}{2 d}-\frac {a^2 (2 a+3 b) \coth ^3(c+d x)}{6 d}+\frac {1}{2} b^2 x (6 a-b)+\frac {b \cosh ^2(c+d x) \coth ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 113, 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 = {3187, 468, 570, 207} \[ \frac {a \left (2 a^2-5 a b-2 b^2\right ) \coth (c+d x)}{2 d}-\frac {a^2 (2 a+3 b) \coth ^3(c+d x)}{6 d}+\frac {1}{2} b^2 x (6 a-b)+\frac {b \cosh ^2(c+d x) \coth ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 468
Rule 570
Rule 3187
Rubi steps
\begin {align*} \int \text {csch}^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}{x^4 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \cosh ^2(c+d x) \coth ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{2 d}+\frac {\operatorname {Subst}\left (\int \frac {\left (a (2 a+3 b)-(a-b) (2 a-b) x^2\right ) \left (a+(-a+b) x^2\right )}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {b \cosh ^2(c+d x) \coth ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{2 d}+\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 (2 a+3 b)}{x^4}-\frac {a \left (2 a^2-5 a b-2 b^2\right )}{x^2}+\frac {b^2 (-6 a+b)}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {a \left (2 a^2-5 a b-2 b^2\right ) \coth (c+d x)}{2 d}-\frac {a^2 (2 a+3 b) \coth ^3(c+d x)}{6 d}+\frac {b \cosh ^2(c+d x) \coth ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{2 d}-\frac {\left ((6 a-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-b) b^2 x+\frac {a \left (2 a^2-5 a b-2 b^2\right ) \coth (c+d x)}{2 d}-\frac {a^2 (2 a+3 b) \coth ^3(c+d x)}{6 d}+\frac {b \cosh ^2(c+d x) \coth ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 2.45, size = 107, normalized size = 0.95 \[ \frac {2 \sinh ^6(c+d x) \left (a \text {csch}^2(c+d x)+b\right )^3 \left (3 b^2 (2 (6 a-b) (c+d x)+b \sinh (2 (c+d x)))-4 a^2 \coth (c+d x) \left (a \text {csch}^2(c+d x)-2 a+9 b\right )\right )}{3 d (2 a+b \cosh (2 (c+d x))-b)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 281, normalized size = 2.49 \[ \frac {3 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (16 \, a^{3} - 72 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 4 \, {\left (4 \, a^{3} - 18 \, a^{2} b - 3 \, {\left (6 \, a b^{2} - b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (16 \, a^{3} - 72 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 6 \, {\left (8 \, a^{3} - 12 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right ) + 12 \, {\left (4 \, a^{3} - 18 \, a^{2} b - 3 \, {\left (6 \, a b^{2} - b^{3}\right )} d x - {\left (4 \, a^{3} - 18 \, a^{2} b - 3 \, {\left (6 \, a b^{2} - b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 154, normalized size = 1.36 \[ \frac {3 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, {\left (6 \, a b^{2} - b^{3}\right )} {\left (d x + c\right )} - 3 \, {\left (12 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (9 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 18 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{3} + 9 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 77, normalized size = 0.68 \[ \frac {a^{3} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (d x +c \right )+b^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 161, normalized size = 1.42 \[ -\frac {1}{8} \, b^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 3 \, a b^{2} x + \frac {4}{3} \, a^{3} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 222, normalized size = 1.96 \[ \frac {\frac {2\,\left (3\,a^2\,b-2\,a^3\right )}{3\,d}-\frac {2\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,a^2\,b}{d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{3\,d}+\frac {2\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {b^2\,x\,\left (6\,a-b\right )}{2}-\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {2\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
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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