Optimal. Leaf size=64 \[ -\frac {(3 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {3 a \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a \coth (c+d x) \text {csch}^3(c+d x)}{4 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, 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 = {3294, 1171,
393, 212} \begin {gather*} -\frac {(3 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {a \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {3 a \coth (c+d x) \text {csch}(c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 1171
Rule 3294
Rubi steps
\begin {align*} \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=-\frac {\text {Subst}\left (\int \frac {a+b-2 b x^2+b x^4}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {\text {Subst}\left (\int \frac {-3 a-4 b+4 b x^2}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 d}\\ &=\frac {3 a \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {(3 a+8 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=-\frac {(3 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {3 a \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a \coth (c+d x) \text {csch}^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(139\) vs. \(2(64)=128\).
time = 0.03, size = 139, normalized size = 2.17 \begin {gather*} \frac {3 a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {3 a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs.
\(2(58)=116\).
time = 1.28, size = 121, normalized size = 1.89
method | result | size |
risch | \(\frac {a \,{\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c}-11 \,{\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c}+3\right )}{4 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs.
\(2 (58) = 116\).
time = 0.28, size = 174, normalized size = 2.72 \begin {gather*} -\frac {1}{8} \, a {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - b {\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}\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. 1476 vs.
\(2 (58) = 116\).
time = 0.47, size = 1476, normalized size = 23.06 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (58) = 116\).
time = 0.43, size = 124, normalized size = 1.94 \begin {gather*} -\frac {{\left (3 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - {\left (3 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (3 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 20 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{2}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.74, size = 242, normalized size = 3.78 \begin {gather*} \frac {3\,a\,{\mathrm {e}}^{c+d\,x}}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a\,\sqrt {-d^2}+8\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {9\,a^2+48\,a\,b+64\,b^2}}\right )\,\sqrt {9\,a^2+48\,a\,b+64\,b^2}}{4\,\sqrt {-d^2}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {6\,a\,{\mathrm {e}}^{c+d\,x}}{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\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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