Optimal. Leaf size=133 \[ \frac {5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]
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Rubi [A]
time = 0.12, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3299, 3855,
3852, 3853} \begin {gather*} \frac {5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}-\frac {2 a b \coth ^3(c+d x)}{3 d}+\frac {2 a b \coth (c+d x)}{d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3299
Rule 3852
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\left (i \int \left (i b^2 \text {csch}(c+d x)+2 i a b \text {csch}^4(c+d x)+i a^2 \text {csch}^7(c+d x)\right ) \, dx\right )\\ &=a^2 \int \text {csch}^7(c+d x) \, dx+(2 a b) \int \text {csch}^4(c+d x) \, dx+b^2 \int \text {csch}(c+d x) \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {1}{6} \left (5 a^2\right ) \int \text {csch}^5(c+d x) \, dx+\frac {(2 i a b) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {1}{8} \left (5 a^2\right ) \int \text {csch}^3(c+d x) \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}-\frac {1}{16} \left (5 a^2\right ) \int \text {csch}(c+d x) \, dx\\ &=\frac {5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {2 a b \coth (c+d x)}{d}-\frac {2 a b \coth ^3(c+d x)}{3 d}-\frac {5 a^2 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 235, normalized size = 1.77 \begin {gather*} \frac {4 a b \coth (c+d x)}{3 d}-\frac {5 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {2 a b \coth (c+d x) \text {csch}^2(c+d x)}{3 d}-\frac {b^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {5 a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.27, size = 209, normalized size = 1.57
method | result | size |
risch | \(-\frac {a \left (15 a \,{\mathrm e}^{11 d x +11 c}-85 a \,{\mathrm e}^{9 d x +9 c}+192 b \,{\mathrm e}^{8 d x +8 c}+198 a \,{\mathrm e}^{7 d x +7 c}-640 b \,{\mathrm e}^{6 d x +6 c}+198 a \,{\mathrm e}^{5 d x +5 c}+768 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{3 d x +3 c}-384 b \,{\mathrm e}^{2 d x +2 c}+15 a \,{\mathrm e}^{d x +c}+64 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}\) | \(209\) |
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. 316 vs.
\(2 (123) = 246\).
time = 0.28, size = 316, normalized size = 2.38 \begin {gather*} \frac {1}{48} \, a^{2} {\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 )} - b^{2} {\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 )} + \frac {8}{3} \, a b {\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 )} \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. 3607 vs.
\(2 (123) = 246\).
time = 0.47, size = 3607, normalized size = 27.12 \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 [A]
time = 0.50, size = 204, normalized size = 1.53 \begin {gather*} \frac {3 \, {\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, {\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{2} e^{\left (11 \, d x + 11 \, c\right )} - 85 \, a^{2} e^{\left (9 \, d x + 9 \, c\right )} + 192 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 198 \, a^{2} e^{\left (7 \, d x + 7 \, c\right )} - 640 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 198 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} + 768 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 85 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 384 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} e^{\left (d x + c\right )} + 64 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{48 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 434, normalized size = 3.26 \begin {gather*} \frac {\frac {5\,a^2\,{\mathrm {e}}^{c+d\,x}}{12\,d}-\frac {8\,a\,b}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d}+\frac {16\,a\,b}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^2\,\sqrt {-d^2}-16\,b^2\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^4-160\,a^2\,b^2+256\,b^4}}\right )\,\sqrt {25\,a^4-160\,a^2\,b^2+256\,b^4}}{8\,\sqrt {-d^2}}-\frac {18\,a^2\,{\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 )}-\frac {80\,a^2\,{\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 {32\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (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\right )}-\frac {5\,a^2\,{\mathrm {e}}^{c+d\,x}}{8\,d\,\left ({\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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