Optimal. Leaf size=47 \[ \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3215, 1157, 388, 206} \[ \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 388
Rule 1157
Rule 3215
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b-2 b x^2+b x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {-a-2 b+2 b x^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 1.74 \[ -\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {b \sinh (c) \sinh (d x)}{d}+\frac {b \cosh (c) \cosh (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.58, size = 690, normalized size = 14.68 \[ \frac {b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, b \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, b \cosh \left (d x + c\right )^{4} - 6 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + {\left (a \cosh \left (d x + c\right )^{5} + 5 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{5} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{3} - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + a \cosh \left (d x + c\right ) + {\left (5 \, a \cosh \left (d x + c\right )^{4} - 6 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (a \cosh \left (d x + c\right )^{5} + 5 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{5} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{3} - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + a \cosh \left (d x + c\right ) + {\left (5 \, a \cosh \left (d x + c\right )^{4} - 6 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{5} - 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{2 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5} - 2 \, d \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{4} - 6 \, 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 [B] time = 0.19, size = 107, normalized size = 2.28 \[ \frac {2 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + a \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - a \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 38, normalized size = 0.81 \[ \frac {a \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+b \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 115, normalized size = 2.45 \[ \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 126, normalized size = 2.68 \[ \frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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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