Optimal. Leaf size=156 \[ \frac {a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}+\frac {b^2 (3 a+b) \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3294, 1171,
1828, 1824, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}+\frac {a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1171
Rule 1824
Rule 1828
Rule 3294
Rubi steps
\begin {align*} \int \text {csch}^7(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 )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-5 a^3-18 a^2 b-18 a b^2-6 b^3+6 b \left (3 a^2+9 a b+5 b^2\right ) x^2-6 b^2 (9 a+10 b) x^4+6 b^2 (3 a+10 b) x^6-30 b^3 x^8+6 b^3 x^{10}}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{6 d}\\ &=\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {3 \left (5 a^3+24 a^2 b+24 a b^2+8 b^3\right )-48 b^2 (3 a+2 b) x^2+72 b^2 (a+2 b) x^4-96 b^3 x^6+24 b^3 x^8}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{24 d}\\ &=-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-3 \left (5 a^3+24 a^2 b+48 a b^2+16 b^3\right )+144 b^2 (a+b) x^2-144 b^3 x^4+48 b^3 x^6}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {\text {Subst}\left (\int \left (-48 b^2 (3 a+b)+96 b^3 x^2-48 b^3 x^4-\frac {3 \left (5 a^3+24 a^2 b\right )}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=\frac {b^2 (3 a+b) \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {\left (a^2 (5 a+24 b)\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{16 d}\\ &=\frac {a^2 (5 a+24 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}+\frac {b^2 (3 a+b) \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^2 (5 a+24 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 223, normalized size = 1.43 \begin {gather*} -\frac {-240 b^2 (24 a+5 b) \cosh (c+d x)+200 b^3 \cosh (3 (c+d x))-24 b^3 \cosh (5 (c+d x))+150 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+720 a^2 b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-30 a^3 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+5 a^3 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )+600 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+2880 a^2 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+150 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+720 a^2 b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+30 a^3 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+5 a^3 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{1920 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. \(357\) vs.
\(2(144)=288\).
time = 1.52, size = 358, normalized size = 2.29
method | result | size |
risch | \(\frac {{\mathrm e}^{5 d x +5 c} b^{3}}{160 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} b^{3}}{96 d}+\frac {3 a \,{\mathrm e}^{d x +c} b^{2}}{2 d}+\frac {5 b^{3} {\mathrm e}^{d x +c}}{16 d}+\frac {3 a \,{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {5 b^{3} {\mathrm e}^{-d x -c}}{16 d}-\frac {5 b^{3} {\mathrm e}^{-3 d x -3 c}}{96 d}+\frac {{\mathrm e}^{-5 d x -5 c} b^{3}}{160 d}-\frac {a^{2} {\mathrm e}^{d x +c} \left (15 a \,{\mathrm e}^{10 d x +10 c}+72 b \,{\mathrm e}^{10 d x +10 c}-85 a \,{\mathrm e}^{8 d x +8 c}-216 b \,{\mathrm e}^{8 d x +8 c}+198 a \,{\mathrm e}^{6 d x +6 c}+144 b \,{\mathrm e}^{6 d x +6 c}+198 a \,{\mathrm e}^{4 d x +4 c}+144 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{2 d x +2 c}-216 b \,{\mathrm e}^{2 d x +2 c}+15 a +72 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}\) | \(358\) |
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. 390 vs.
\(2 (144) = 288\).
time = 0.29, size = 390, normalized size = 2.50 \begin {gather*} \frac {1}{480} \, b^{3} {\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 b^{2} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{48} \, a^{3} {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + \frac {3}{2} \, a^{2} 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} + \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. 8547 vs.
\(2 (144) = 288\).
time = 0.49, size = 8547, normalized size = 54.79 \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. 321 vs.
\(2 (144) = 288\).
time = 0.59, size = 321, normalized size = 2.06 \begin {gather*} \frac {3 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 40 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 240 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, {\left (5 \, a^{3} + 24 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (5 \, a^{3} + 24 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {20 \, {\left (15 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 72 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 160 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 576 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 1152 \, a^{2} b {\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 )}^{3}}}{480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 633, normalized size = 4.06 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^3\,\sqrt {-d^2}+24\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^6+240\,a^5\,b+576\,a^4\,b^2}}\right )\,\sqrt {25\,a^6+240\,a^5\,b+576\,a^4\,b^2}}{8\,\sqrt {-d^2}}-\frac {\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (8\,a^3+9\,b\,a^2\right )}{3\,d}-\frac {8\,a^2\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d}-\frac {8\,a^2\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{d}+\frac {2\,a^2\,b\,{\mathrm {e}}^{9\,c+9\,d\,x}}{d}+\frac {2\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{d}}{15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {5\,b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{96\,d}-\frac {5\,b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{96\,d}+\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+24\,b\,a^2\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 {b^2\,{\mathrm {e}}^{c+d\,x}\,\left (24\,a+5\,b\right )}{16\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (48\,a^2\,b-5\,a^3\right )}{12\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {22\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 )}-\frac {16\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}+\frac {b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (24\,a+5\,b\right )}{16\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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