Optimal. Leaf size=141 \[ -\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}-\frac {2 a^2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d} \]
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Rubi [A]
time = 0.35, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3129,
3128, 3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {2 a^2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a x \left (2 a^2+b^2\right )}{2 b^4}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}+\frac {\sinh ^2(c+d x) \cosh (c+d x)}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2968
Rule 3102
Rule 3128
Rule 3129
Rubi steps
\begin {align*} \int \frac {\cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\int \frac {\sinh ^2(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx\\ &=\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}+\frac {\int \frac {\sinh (c+d x) \left (-2 a+b \sinh (c+d x)-3 a \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{3 b}\\ &=-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}+\frac {\int \frac {3 a^2-a b \sinh (c+d x)+2 \left (3 a^2+b^2\right ) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{6 b^2}\\ &=\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}+\frac {i \int \frac {-3 i a^2 b+3 i a \left (2 a^2+b^2\right ) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{6 b^3}\\ &=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}+\frac {\left (a^2 \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{b^4}\\ &=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}-\frac {\left (2 i a^2 \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b^4 d}\\ &=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}+\frac {\left (4 i a^2 \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{b^4 d}\\ &=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}-\frac {2 a^2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {\left (3 a^2+b^2\right ) \cosh (c+d x)}{3 b^3 d}-\frac {a \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}+\frac {\cosh (c+d x) \sinh ^2(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 123, normalized size = 0.87 \begin {gather*} \frac {3 b \left (4 a^2+b^2\right ) \cosh (c+d x)+b^3 \cosh (3 (c+d x))-3 a \left (2 \left (2 a^2+b^2\right ) (c+d x)+8 a \sqrt {-a^2-b^2} \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+b^2 \sinh (2 (c+d x))\right )}{12 b^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.16, size = 247, normalized size = 1.75
method | result | size |
risch | \(-\frac {a^{3} x}{b^{4}}-\frac {a x}{2 b^{2}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}+\frac {{\mathrm e}^{d x +c}}{8 b d}+\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}+\frac {{\mathrm e}^{-d x -c}}{8 b d}+\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\frac {\sqrt {a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {\sqrt {a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}\) | \(244\) |
derivativedivides | \(\frac {-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b -a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}}{d}\) | \(247\) |
default | \(\frac {-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}+\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {b -a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b -b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}\, \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{4}}}{d}\) | \(247\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 209, normalized size = 1.48 \begin {gather*} \frac {\sqrt {a^{2} + b^{2}} a^{2} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{b^{4} d} - \frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac {{\left (2 \, a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{2 \, b^{4} d} + \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} + b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} 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. 745 vs.
\(2 (130) = 260\).
time = 0.38, size = 745, normalized size = 5.28 \begin {gather*} \frac {b^{3} \cosh \left (d x + c\right )^{6} + b^{3} \sinh \left (d x + c\right )^{6} - 3 \, a b^{2} \cosh \left (d x + c\right )^{5} - 12 \, {\left (2 \, a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right ) - a b^{2}\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right ) + 4 \, a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{4} + 3 \, a b^{2} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} - 15 \, a b^{2} \cosh \left (d x + c\right )^{2} - 6 \, {\left (2 \, a^{3} + a b^{2}\right )} d x + 6 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + b^{3} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} - 10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 12 \, {\left (2 \, a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right ) + 4 \, a^{2} b + b^{3} + 6 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right )^{5} - 5 \, a b^{2} \cosh \left (d x + c\right )^{4} - 12 \, {\left (2 \, a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 4 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{3} + a b^{2} + 2 \, {\left (4 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (b^{4} d \cosh \left (d x + c\right )^{3} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{3}\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 = 0.43, size = 211, normalized size = 1.50 \begin {gather*} -\frac {\frac {12 \, {\left (2 \, a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{b^{4}} - \frac {b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a^{2} e^{\left (d x + c\right )} + 3 \, b^{2} e^{\left (d x + c\right )}}{b^{3}} - \frac {{\left (3 \, a b^{2} e^{\left (d x + c\right )} + b^{3} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{b^{4}} - \frac {24 \, {\left (a^{4} + a^{2} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 278, normalized size = 1.97 \begin {gather*} \frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}-\frac {x\,\left (2\,a^3+a\,b^2\right )}{2\,b^4}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}+\frac {a\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^2\,d}-\frac {a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^2\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^2+b^2\right )}{8\,b^3\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^2+b^2\right )}{8\,b^3\,d}-\frac {a^2\,\ln \left (-\frac {2\,a^2\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^5}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^5}\right )\,\sqrt {a^2+b^2}}{b^4\,d}+\frac {a^2\,\ln \left (\frac {2\,a^2\,\sqrt {a^2+b^2}\,\left (b-a\,{\mathrm {e}}^{c+d\,x}\right )}{b^5}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2+b^2\right )}{b^5}\right )\,\sqrt {a^2+b^2}}{b^4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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