Optimal. Leaf size=148 \[ \frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3294, 1171,
1824, 212} \begin {gather*} \frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1171
Rule 1824
Rule 3294
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\text {Subst}\left (\int \frac {-a^3-6 a^2 b-6 a b^2-2 b^3+2 b \left (3 a^2+9 a b+5 b^2\right ) x^2-2 b^2 (9 a+10 b) x^4+2 b^2 (3 a+10 b) x^6-10 b^3 x^8+2 b^3 x^{10}}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\text {Subst}\left (\int \left (-2 b \left (3 a^2+3 a b+b^2\right )+4 b^2 (3 a+2 b) x^2-6 b^2 (a+2 b) x^4+8 b^3 x^6-2 b^3 x^8-\frac {a^3}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}-\frac {2 b^2 (3 a+2 b) \cosh ^3(c+d x)}{3 d}+\frac {3 b^2 (a+2 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^3 \cosh ^7(c+d x)}{7 d}+\frac {b^3 \cosh ^9(c+d x)}{9 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 155, normalized size = 1.05 \begin {gather*} \frac {1890 b \left (128 a^2+80 a b+21 b^2\right ) \cosh (c+d x)-1260 b^2 (20 a+7 b) \cosh (3 (c+d x))+3024 a b^2 \cosh (5 (c+d x))+2268 b^3 \cosh (5 (c+d x))-405 b^3 \cosh (7 (c+d x))+35 b^3 \cosh (9 (c+d x))-10080 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-40320 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-10080 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{80640 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. \(378\) vs.
\(2(136)=272\).
time = 1.48, size = 379, normalized size = 2.56
method | result | size |
risch | \(\frac {b^{3} {\mathrm e}^{9 d x +9 c}}{4608 d}-\frac {9 b^{3} {\mathrm e}^{7 d x +7 c}}{3584 d}+\frac {3 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {9 \,{\mathrm e}^{5 d x +5 c} b^{3}}{640 d}-\frac {5 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{32 d}-\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{3}}{128 d}+\frac {3 b \,{\mathrm e}^{d x +c} a^{2}}{2 d}+\frac {15 a \,{\mathrm e}^{d x +c} b^{2}}{16 d}+\frac {63 b^{3} {\mathrm e}^{d x +c}}{256 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}+\frac {15 a \,{\mathrm e}^{-d x -c} b^{2}}{16 d}+\frac {63 b^{3} {\mathrm e}^{-d x -c}}{256 d}-\frac {5 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{32 d}-\frac {7 b^{3} {\mathrm e}^{-3 d x -3 c}}{128 d}+\frac {3 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{160 d}+\frac {9 \,{\mathrm e}^{-5 d x -5 c} b^{3}}{640 d}-\frac {9 b^{3} {\mathrm e}^{-7 d x -7 c}}{3584 d}+\frac {b^{3} {\mathrm e}^{-9 d x -9 c}}{4608 d}-\frac {a^{3} {\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}\) | \(379\) |
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. 334 vs.
\(2 (136) = 272\).
time = 0.29, size = 334, normalized size = 2.26 \begin {gather*} -\frac {1}{161280} \, b^{3} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} + \frac {1}{160} \, a b^{2} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a^{3} {\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 )} \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. 4895 vs.
\(2 (136) = 272\).
time = 0.42, size = 4895, normalized size = 33.07 \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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 300 vs.
\(2 (136) = 272\).
time = 0.58, size = 300, normalized size = 2.03 \begin {gather*} \frac {35 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 720 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 3024 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 6048 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 40320 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 26880 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 241920 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 241920 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 80640 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 40320 \, a^{3} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 40320 \, a^{3} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {161280 \, a^{3} {\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}}{161280 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 326, normalized size = 2.20 \begin {gather*} \frac {\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}-\frac {9\,b^3\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{3584\,d}-\frac {9\,b^3\,{\mathrm {e}}^{7\,c+7\,d\,x}}{3584\,d}+\frac {b^3\,{\mathrm {e}}^{-9\,c-9\,d\,x}}{4608\,d}+\frac {b^3\,{\mathrm {e}}^{9\,c+9\,d\,x}}{4608\,d}+\frac {3\,b\,{\mathrm {e}}^{-c-d\,x}\,\left (128\,a^2+80\,a\,b+21\,b^2\right )}{256\,d}+\frac {3\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}\,\left (4\,a+3\,b\right )}{640\,d}+\frac {3\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (4\,a+3\,b\right )}{640\,d}-\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (20\,a+7\,b\right )}{128\,d}-\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (20\,a+7\,b\right )}{128\,d}+\frac {3\,b\,{\mathrm {e}}^{c+d\,x}\,\left (128\,a^2+80\,a\,b+21\,b^2\right )}{256\,d}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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