Optimal. Leaf size=53 \[ \frac {3 \coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}+\frac {3 \tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2700, 276}
\begin {gather*} -\frac {\tanh ^3(a+b x)}{3 b}+\frac {3 \tanh (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}+\frac {3 \coth (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2700
Rubi steps
\begin {align*} \int \text {csch}^4(a+b x) \text {sech}^4(a+b x) \, dx &=-\frac {i \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac {i \text {Subst}\left (\int \left (3+\frac {1}{x^4}+\frac {3}{x^2}+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {3 \coth (a+b x)}{b}-\frac {\coth ^3(a+b x)}{3 b}+\frac {3 \tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.81 \begin {gather*} 16 \left (\frac {\coth (2 (a+b x))}{3 b}-\frac {\coth (2 (a+b x)) \text {csch}^2(2 (a+b x))}{6 b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.75, size = 45, normalized size = 0.85
method | result | size |
risch | \(-\frac {32 \left (3 \,{\mathrm e}^{4 b x +4 a}-1\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{3} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{3}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 90, normalized size = 1.70 \begin {gather*} \frac {32 \, e^{\left (-4 \, b x - 4 \, a\right )}}{b {\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} - \frac {32}{3 \, b {\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs.
\(2 (49) = 98\).
time = 0.35, size = 330, normalized size = 6.23 \begin {gather*} -\frac {64 \, {\left (\cosh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{10} + 120 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{7} + 45 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{8} + 10 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + b \sinh \left (b x + a\right )^{10} - 3 \, b \cosh \left (b x + a\right )^{6} + 3 \, {\left (70 \, b \cosh \left (b x + a\right )^{4} - b\right )} \sinh \left (b x + a\right )^{6} + 18 \, {\left (14 \, b \cosh \left (b x + a\right )^{5} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 15 \, {\left (14 \, b \cosh \left (b x + a\right )^{6} - 3 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{4} + 60 \, {\left (2 \, b \cosh \left (b x + a\right )^{7} - b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )^{2} + {\left (45 \, b \cosh \left (b x + a\right )^{8} - 45 \, b \cosh \left (b x + a\right )^{4} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (5 \, b \cosh \left (b x + a\right )^{9} - 9 \, b \cosh \left (b x + a\right )^{5} + 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}
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 \begin {gather*} \int \operatorname {csch}^{4}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 31, normalized size = 0.58 \begin {gather*} -\frac {32 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}}{3 \, b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 31, normalized size = 0.58 \begin {gather*} -\frac {32\,\left (3\,{\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}{3\,b\,{\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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