Optimal. Leaf size=42 \[ -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \cosh ^3(c+d x)}{3 d}-\frac {b \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3215, 1153, 206} \[ -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \cosh ^3(c+d x)}{3 d}-\frac {b \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1153
Rule 3215
Rubi steps
\begin {align*} \int \text {csch}(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}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (b-b x^2+\frac {a}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{3 d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 70, normalized size = 1.67 \[ \frac {a \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {3 b \cosh (c+d x)}{4 d}+\frac {b \cosh (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.14, size = 395, normalized size = 9.40 \[ \frac {b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 9 \, b \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{2} - 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{4} - 18 \, b \cosh \left (d x + c\right )^{2} - 3 \, b\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 78, normalized size = 1.86 \[ \frac {b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b e^{\left (d x + c\right )} - {\left (9 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 24 \, a \log \left (e^{\left (d x + c\right )} + 1\right ) + 24 \, a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 36, normalized size = 0.86 \[ \frac {-2 a \arctanh \left ({\mathrm e}^{d x +c}\right )+b \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 71, normalized size = 1.69 \[ \frac {1}{24} \, b {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {a \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 96, normalized size = 2.29 \[ \frac {b\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b\,{\mathrm {e}}^{-c-d\,x}}{8\,d}+\frac {b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {3\,b\,{\mathrm {e}}^{c+d\,x}}{8\,d}-\frac {2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}} \]
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 \[ \int \left (a + b \sinh ^{4}{\left (c + d x \right )}\right ) \operatorname {csch}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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