Optimal. Leaf size=141 \[ -\frac {a^3 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^6 d}+\frac {a^2 \left (a^2+b^2\right ) \sinh (c+d x)}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^3(c+d x)}{3 b^3 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {\sinh ^5(c+d x)}{5 b d} \]
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Rubi [A]
time = 0.15, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 908}
\begin {gather*} \frac {a^2 \left (a^2+b^2\right ) \sinh (c+d x)}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^3(c+d x)}{3 b^3 d}-\frac {a^3 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^6 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {\sinh ^5(c+d x)}{5 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 908
Rule 2916
Rubi steps
\begin {align*} \int \frac {\cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^3 \left (-b^2-x^2\right )}{b^3 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^3 d}\\ &=-\frac {\text {Subst}\left (\int \frac {x^3 \left (-b^2-x^2\right )}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^6 d}\\ &=-\frac {\text {Subst}\left (\int \left (-a^2 \left (a^2+b^2\right )+a \left (a^2+b^2\right ) x-\left (a^2+b^2\right ) x^2+a x^3-x^4+\frac {a^3 \left (a^2+b^2\right )}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^6 d}\\ &=-\frac {a^3 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^6 d}+\frac {a^2 \left (a^2+b^2\right ) \sinh (c+d x)}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^3(c+d x)}{3 b^3 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {\sinh ^5(c+d x)}{5 b d}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 123, normalized size = 0.87 \begin {gather*} -\frac {\frac {60 a^3 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^6}-\frac {60 a^2 \left (a^2+b^2\right ) \sinh (c+d x)}{b^5}+\frac {30 a \left (a^2+b^2\right ) \sinh ^2(c+d x)}{b^4}-\frac {20 \left (a^2+b^2\right ) \sinh ^3(c+d x)}{b^3}+\frac {15 a \sinh ^4(c+d x)}{b^2}-\frac {12 \sinh ^5(c+d x)}{b}}{60 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. \(420\) vs.
\(2(133)=266\).
time = 1.16, size = 421, normalized size = 2.99
method | result | size |
derivativedivides | \(\frac {-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {2 b +a}{4 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +7 b^{2}}{12 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +5 a \,b^{2}+3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a \left (8 a^{3}+4 a^{2} b +8 a \,b^{2}+3 b^{3}\right )}{8 b^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{3} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{6}}-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {-2 b +a}{4 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4 a^{2}-6 a b +7 b^{2}}{12 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{3}-4 a^{2} b +5 a \,b^{2}-3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a \left (8 a^{3}-4 a^{2} b +8 a \,b^{2}-3 b^{3}\right )}{8 b^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a^{3} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{6}}-\frac {2 a^{3} \left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{6}}}{d}\) | \(421\) |
default | \(\frac {-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {2 b +a}{4 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +7 b^{2}}{12 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +5 a \,b^{2}+3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a \left (8 a^{3}+4 a^{2} b +8 a \,b^{2}+3 b^{3}\right )}{8 b^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{3} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{6}}-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {-2 b +a}{4 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4 a^{2}-6 a b +7 b^{2}}{12 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{3}-4 a^{2} b +5 a \,b^{2}-3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a \left (8 a^{3}-4 a^{2} b +8 a \,b^{2}-3 b^{3}\right )}{8 b^{5} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a^{3} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{6}}-\frac {2 a^{3} \left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{6}}}{d}\) | \(421\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{96 b d}-\frac {a \,{\mathrm e}^{-2 d x -2 c}}{16 b^{2} d}-\frac {a \,{\mathrm e}^{4 d x +4 c}}{64 b^{2} d}+\frac {a^{5} x}{b^{6}}-\frac {{\mathrm e}^{-5 d x -5 c}}{160 b d}+\frac {{\mathrm e}^{5 d x +5 c}}{160 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{16 b^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{96 b d}-\frac {{\mathrm e}^{d x +c}}{16 b d}+\frac {a^{3} x}{b^{4}}+\frac {{\mathrm e}^{-d x -c}}{16 b d}-\frac {a \,{\mathrm e}^{-4 d x -4 c}}{64 b^{2} d}+\frac {2 a^{5} c}{b^{6} d}-\frac {a^{5} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{6} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{4} d}+\frac {2 a^{3} c}{b^{4} d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 b^{3} d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 b^{3} d}+\frac {{\mathrm e}^{d x +c} a^{4}}{2 b^{5} d}-\frac {{\mathrm e}^{-d x -c} a^{4}}{2 b^{5} d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 b^{3} d}-\frac {a^{3} {\mathrm e}^{2 d x +2 c}}{8 b^{4} d}-\frac {a^{3} {\mathrm e}^{-2 d x -2 c}}{8 b^{4} d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 b^{3} d}\) | \(437\) |
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. 300 vs.
\(2 (133) = 266\).
time = 0.28, size = 300, normalized size = 2.13 \begin {gather*} -\frac {{\left (15 \, a b^{3} e^{\left (-d x - c\right )} - 6 \, b^{4} - 10 \, {\left (4 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 60 \, {\left (2 \, a^{3} b + a b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 60 \, {\left (8 \, a^{4} + 6 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{960 \, b^{5} d} - \frac {{\left (a^{5} + a^{3} b^{2}\right )} {\left (d x + c\right )}}{b^{6} d} - \frac {15 \, a b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + 6 \, b^{4} e^{\left (-5 \, d x - 5 \, c\right )} + 60 \, {\left (8 \, a^{4} + 6 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-d x - c\right )} + 60 \, {\left (2 \, a^{3} b + a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, {\left (4 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{960 \, b^{5} d} - \frac {{\left (a^{5} + a^{3} b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{6} d} \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. 1660 vs.
\(2 (133) = 266\).
time = 0.43, size = 1660, normalized size = 11.77 \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 = 0.45, size = 258, normalized size = 1.83 \begin {gather*} \frac {\frac {6 \, b^{4} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 15 \, a b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4} + 40 \, a^{2} b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 40 \, b^{4} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 120 \, a^{3} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 120 \, a b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 480 \, a^{4} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 480 \, a^{2} b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{5}} - \frac {960 \, {\left (a^{5} + a^{3} b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{6}}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 307, normalized size = 2.18 \begin {gather*} \frac {{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,b\,d}-\frac {{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,b\,d}-\frac {a\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b^2\,d}-\frac {a\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b^2\,d}-\frac {\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^5+a^3\,b^2\right )}{b^6\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a^4+6\,a^2\,b^2-b^4\right )}{16\,b^5\,d}+\frac {a^3\,x\,\left (a^2+b^2\right )}{b^6}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (2\,a^3+a\,b^2\right )}{16\,b^4\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^3+a\,b^2\right )}{16\,b^4\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a^4+6\,a^2\,b^2-b^4\right )}{16\,b^5\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (4\,a^2+b^2\right )}{96\,b^3\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (4\,a^2+b^2\right )}{96\,b^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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