Optimal. Leaf size=31 \[ -\frac {a \coth ^3(c+d x)}{3 d}+\frac {a \coth (c+d x)}{d}+b x \]
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Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3217, 1261, 207} \[ -\frac {a \coth ^3(c+d x)}{3 d}+\frac {a \coth (c+d x)}{d}+b x \]
Antiderivative was successfully verified.
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Rule 207
Rule 1261
Rule 3217
Rubi steps
\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-2 a x^2+(a+b) x^4}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x^4}-\frac {a}{x^2}-\frac {b}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=b x+\frac {a \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.29 \[ \frac {2 a \coth (c+d x)}{3 d}-\frac {a \coth (c+d x) \text {csch}^2(c+d x)}{3 d}+b x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 129, normalized size = 4.16 \[ \frac {2 \, a \cosh \left (d x + c\right )^{3} + 6 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (3 \, b d x - 2 \, a\right )} \sinh \left (d x + c\right )^{3} - 6 \, a \cosh \left (d x + c\right ) - 3 \, {\left (3 \, b d x - {\left (3 \, b d x - 2 \, a\right )} \cosh \left (d x + c\right )^{2} - 2 \, a\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 45, normalized size = 1.45 \[ \frac {3 \, {\left (d x + c\right )} b - \frac {4 \, {\left (3 \, a e^{\left (2 \, d x + 2 \, c\right )} - a\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 33, normalized size = 1.06 \[ \frac {a \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+\left (d x +c \right ) b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 97, normalized size = 3.13 \[ b x + \frac {4}{3} \, a {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 81, normalized size = 2.61 \[ \frac {4\,a-12\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,b\,d\,x+9\,b\,d\,x\,{\mathrm {e}}^{2\,c+2\,d\,x}-9\,b\,d\,x\,{\mathrm {e}}^{4\,c+4\,d\,x}+3\,b\,d\,x\,{\mathrm {e}}^{6\,c+6\,d\,x}}{3\,d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}^3} \]
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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