Optimal. Leaf size=113 \[ \frac {a^2 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh (c+d x)}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^3 d}-\frac {a \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^4(c+d x)}{4 b d} \]
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Rubi [A]
time = 0.11, antiderivative size = 113, 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 ) \log (a+b \sinh (c+d x))}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh (c+d x)}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^3 d}-\frac {a \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^4(c+d x)}{4 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 ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^2 \left (-b^2-x^2\right )}{b^2 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^3 d}\\ &=-\frac {\text {Subst}\left (\int \frac {x^2 \left (-b^2-x^2\right )}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^5 d}\\ &=-\frac {\text {Subst}\left (\int \left (a \left (a^2+b^2\right )-\left (a^2+b^2\right ) x+a x^2-x^3-\frac {a^2 \left (a^2+b^2\right )}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^5 d}\\ &=\frac {a^2 \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^5 d}-\frac {a \left (a^2+b^2\right ) \sinh (c+d x)}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh ^2(c+d x)}{2 b^3 d}-\frac {a \sinh ^3(c+d x)}{3 b^2 d}+\frac {\sinh ^4(c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 104, normalized size = 0.92 \begin {gather*} \frac {12 \left (2 a^2 b^2+b^4\right ) \cosh (2 (c+d x))+3 b^4 \cosh (4 (c+d x))+8 a \left (12 a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))-3 b \left (4 a^2+3 b^2\right ) \sinh (c+d x)-b^3 \sinh (3 (c+d x))\right )}{96 b^5 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. \(342\) vs.
\(2(107)=214\).
time = 1.12, size = 343, normalized size = 3.04
method | result | size |
risch | \(-\frac {a^{4} x}{b^{5}}-\frac {x \,a^{2}}{b^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}-\frac {a \,{\mathrm e}^{3 d x +3 c}}{24 b^{2} d}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 b^{3} d}+\frac {{\mathrm e}^{2 d x +2 c}}{16 b d}-\frac {a^{3} {\mathrm e}^{d x +c}}{2 b^{4} d}-\frac {3 a \,{\mathrm e}^{d x +c}}{8 b^{2} d}+\frac {a^{3} {\mathrm e}^{-d x -c}}{2 b^{4} d}+\frac {3 a \,{\mathrm e}^{-d x -c}}{8 b^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 b^{3} d}+\frac {{\mathrm e}^{-2 d x -2 c}}{16 b d}+\frac {a \,{\mathrm e}^{-3 d x -3 c}}{24 b^{2} d}+\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}-\frac {2 a^{4} c}{b^{5} d}-\frac {2 a^{2} c}{b^{3} d}+\frac {a^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{5} d}+\frac {a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{3} d}\) | \(326\) |
derivativedivides | \(\frac {\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a^{2}+4 a b -5 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-8 a^{3}+4 a^{2} b -8 a \,b^{2}+3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a^{2} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -5 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-8 a^{3}-4 a^{2} b -8 a \,b^{2}-3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a^{2} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}+\frac {2 a^{2} \left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{5}}}{d}\) | \(343\) |
default | \(\frac {\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a^{2}+4 a b -5 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-8 a^{3}+4 a^{2} b -8 a \,b^{2}+3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a^{2} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 b -2 a}{6 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b -5 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-8 a^{3}-4 a^{2} b -8 a \,b^{2}-3 b^{3}}{8 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a^{2} \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}+\frac {2 a^{2} \left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{5}}}{d}\) | \(343\) |
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. 234 vs.
\(2 (107) = 214\).
time = 0.27, size = 234, normalized size = 2.07 \begin {gather*} -\frac {{\left (8 \, a b^{2} e^{\left (-d x - c\right )} - 3 \, b^{3} - 12 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 24 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (a^{4} + a^{2} b^{2}\right )} {\left (d x + c\right )}}{b^{5} d} + \frac {8 \, a b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} + 24 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-d x - c\right )} + 12 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{192 \, b^{4} d} + \frac {{\left (a^{4} + a^{2} 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^{5} 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. 1069 vs.
\(2 (107) = 214\).
time = 0.38, size = 1069, normalized size = 9.46 \begin {gather*} \frac {3 \, b^{4} \cosh \left (d x + c\right )^{8} + 3 \, b^{4} \sinh \left (d x + c\right )^{8} - 8 \, a b^{3} \cosh \left (d x + c\right )^{7} + 8 \, {\left (3 \, b^{4} \cosh \left (d x + c\right ) - a b^{3}\right )} \sinh \left (d x + c\right )^{7} - 192 \, {\left (a^{4} + a^{2} b^{2}\right )} d x \cosh \left (d x + c\right )^{4} + 12 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (21 \, b^{4} \cosh \left (d x + c\right )^{2} - 14 \, a b^{3} \cosh \left (d x + c\right ) + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sinh \left (d x + c\right )^{6} - 24 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{5} + 24 \, {\left (7 \, b^{4} \cosh \left (d x + c\right )^{3} - 7 \, a b^{3} \cosh \left (d x + c\right )^{2} - 4 \, a^{3} b - 3 \, a b^{3} + 3 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 8 \, a b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (105 \, b^{4} \cosh \left (d x + c\right )^{4} - 140 \, a b^{3} \cosh \left (d x + c\right )^{3} - 96 \, {\left (a^{4} + a^{2} b^{2}\right )} d x + 90 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{2} - 60 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 3 \, b^{4} + 24 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{3} + 8 \, {\left (21 \, b^{4} \cosh \left (d x + c\right )^{5} - 35 \, a b^{3} \cosh \left (d x + c\right )^{4} + 12 \, a^{3} b + 9 \, a b^{3} - 96 \, {\left (a^{4} + a^{2} b^{2}\right )} d x \cosh \left (d x + c\right ) + 30 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{3} - 30 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 12 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{2} + 12 \, {\left (7 \, b^{4} \cosh \left (d x + c\right )^{6} - 14 \, a b^{3} \cosh \left (d x + c\right )^{5} - 96 \, {\left (a^{4} + a^{2} b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 15 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{4} + 2 \, a^{2} b^{2} + b^{4} - 20 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 192 \, {\left ({\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (d x + c\right )^{4}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (3 \, b^{4} \cosh \left (d x + c\right )^{7} - 7 \, a b^{3} \cosh \left (d x + c\right )^{6} - 96 \, {\left (a^{4} + a^{2} b^{2}\right )} d x \cosh \left (d x + c\right )^{3} + 9 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{5} - 15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{4} + a b^{3} + 9 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{192 \, {\left (b^{5} d \cosh \left (d x + c\right )^{4} + 4 \, b^{5} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b^{5} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b^{5} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{5} d \sinh \left (d x + c\right )^{4}\right )}} \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 = 0.47, size = 202, normalized size = 1.79 \begin {gather*} \frac {\frac {3 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4} - 8 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 24 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 96 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 96 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{4}} + \frac {192 \, {\left (a^{4} + a^{2} 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^{5}}}{192 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 238, normalized size = 2.11 \begin {gather*} \frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}-\frac {x\,\left (a^4+a^2\,b^2\right )}{b^5}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}+\frac {a\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b^2\,d}-\frac {a\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,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^4+a^2\,b^2\right )}{b^5\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^3+3\,a\,b^2\right )}{8\,b^4\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^3+3\,a\,b^2\right )}{8\,b^4\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (2\,a^2+b^2\right )}{16\,b^3\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+b^2\right )}{16\,b^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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