Optimal. Leaf size=109 \[ a^4 x-\frac {b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac {2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3782, 4048, 3770, 3767, 8} \[ -\frac {b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac {2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}+a^4 x-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 4048
Rubi steps
\begin {align*} \int (a+b \text {csch}(c+d x))^4 \, dx &=-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}+\frac {1}{3} \int (a+b \text {csch}(c+d x)) \left (3 a^3+b \left (9 a^2-2 b^2\right ) \text {csch}(c+d x)+8 a b^2 \text {csch}^2(c+d x)\right ) \, dx\\ &=-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}+\frac {1}{6} \int \left (6 a^4+12 a b \left (2 a^2-b^2\right ) \text {csch}(c+d x)+2 b^2 \left (17 a^2-2 b^2\right ) \text {csch}^2(c+d x)\right ) \, dx\\ &=a^4 x-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}+\frac {1}{3} \left (b^2 \left (17 a^2-2 b^2\right )\right ) \int \text {csch}^2(c+d x) \, dx+\left (2 a b \left (2 a^2-b^2\right )\right ) \int \text {csch}(c+d x) \, dx\\ &=a^4 x-\frac {2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}-\frac {\left (i b^2 \left (17 a^2-2 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{3 d}\\ &=a^4 x-\frac {2 a b \left (2 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}\\ \end {align*}
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Mathematica [B] time = 6.24, size = 508, normalized size = 4.66 \[ \frac {a^4 (c+d x) \sinh ^4(c+d x) (a+b \text {csch}(c+d x))^4}{d (a \sinh (c+d x)+b)^4}+\frac {2 a b \left (2 a^2-b^2\right ) \sinh ^4(c+d x) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \text {csch}(c+d x))^4}{d (a \sinh (c+d x)+b)^4}+\frac {\sinh ^4(c+d x) \text {csch}\left (\frac {1}{2} (c+d x)\right ) \left (b^4 \cosh \left (\frac {1}{2} (c+d x)\right )-9 a^2 b^2 \cosh \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \text {csch}(c+d x))^4}{3 d (a \sinh (c+d x)+b)^4}+\frac {\sinh ^4(c+d x) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (b^4 \sinh \left (\frac {1}{2} (c+d x)\right )-9 a^2 b^2 \sinh \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \text {csch}(c+d x))^4}{3 d (a \sinh (c+d x)+b)^4}-\frac {b^4 \sinh ^4(c+d x) \coth \left (\frac {1}{2} (c+d x)\right ) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4}{24 d (a \sinh (c+d x)+b)^4}+\frac {b^4 \sinh ^4(c+d x) \tanh \left (\frac {1}{2} (c+d x)\right ) \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4}{24 d (a \sinh (c+d x)+b)^4}-\frac {a b^3 \sinh ^4(c+d x) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4}{2 d (a \sinh (c+d x)+b)^4}-\frac {a b^3 \sinh ^4(c+d x) \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4}{2 d (a \sinh (c+d x)+b)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 1440, normalized size = 13.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 169, normalized size = 1.55 \[ \frac {3 \, {\left (d x + c\right )} a^{4} - 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {4 \, {\left (3 \, a b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b^{3} e^{\left (d x + c\right )} + 9 \, a^{2} b^{2} - b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 92, normalized size = 0.84 \[ \frac {a^{4} \left (d x +c \right )-8 a^{3} b \arctanh \left ({\mathrm e}^{d x +c}\right )-6 a^{2} b^{2} \coth \left (d x +c \right )+4 a \,b^{3} \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+b^{4} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 234, normalized size = 2.15 \[ a^{4} x + 2 \, a b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac {4}{3} \, b^{4} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a^{3} b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {12 \, a^{2} b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 239, normalized size = 2.19 \[ a^4\,x-\frac {\frac {12\,a^2\,b^2}{d}+\frac {4\,a\,b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\frac {4\,b^4}{d}+\frac {8\,a\,b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {8\,b^4}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}+\frac {4\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,b^3\,\sqrt {-d^2}-2\,a^3\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {4\,a^6\,b^2-4\,a^4\,b^4+a^2\,b^6}}\right )\,\sqrt {4\,a^6\,b^2-4\,a^4\,b^4+a^2\,b^6}}{\sqrt {-d^2}} \]
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 \[ \int \left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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