Optimal. Leaf size=153 \[ -\frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \cosh (x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {\log (\cosh (x))}{a}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \cosh (x)\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cosh (x)+b^{2/3} \cosh ^2(x)\right )}{6 a^{5/3}}-\frac {\log \left (a+b \cosh ^3(x)\right )}{3 a}+\frac {\text {sech}^2(x)}{2 a} \]
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Rubi [A]
time = 0.15, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3309, 1848,
1885, 206, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \cosh (x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cosh (x)+b^{2/3} \cosh ^2(x)\right )}{6 a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \cosh (x)\right )}{3 a^{5/3}}-\frac {\log \left (a+b \cosh ^3(x)\right )}{3 a}+\frac {\text {sech}^2(x)}{2 a}+\frac {\log (\cosh (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1848
Rule 1885
Rule 3309
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{a+b \cosh ^3(x)} \, dx &=-\text {Subst}\left (\int \frac {1-x^2}{x^3 \left (a+b x^3\right )} \, dx,x,\cosh (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {1}{a x}+\frac {b \left (-1+x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=\frac {\log (\cosh (x))}{a}+\frac {\text {sech}^2(x)}{2 a}-\frac {b \text {Subst}\left (\int \frac {-1+x^2}{a+b x^3} \, dx,x,\cosh (x)\right )}{a}\\ &=\frac {\log (\cosh (x))}{a}+\frac {\text {sech}^2(x)}{2 a}+\frac {b \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\cosh (x)\right )}{a}-\frac {b \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\cosh (x)\right )}{a}\\ &=\frac {\log (\cosh (x))}{a}-\frac {\log \left (a+b \cosh ^3(x)\right )}{3 a}+\frac {\text {sech}^2(x)}{2 a}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\cosh (x)\right )}{3 a^{5/3}}+\frac {b \text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\cosh (x)\right )}{3 a^{5/3}}\\ &=\frac {\log (\cosh (x))}{a}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \cosh (x)\right )}{3 a^{5/3}}-\frac {\log \left (a+b \cosh ^3(x)\right )}{3 a}+\frac {\text {sech}^2(x)}{2 a}-\frac {b^{2/3} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\cosh (x)\right )}{6 a^{5/3}}+\frac {b \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\cosh (x)\right )}{2 a^{4/3}}\\ &=\frac {\log (\cosh (x))}{a}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \cosh (x)\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cosh (x)+b^{2/3} \cosh ^2(x)\right )}{6 a^{5/3}}-\frac {\log \left (a+b \cosh ^3(x)\right )}{3 a}+\frac {\text {sech}^2(x)}{2 a}+\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \cosh (x)}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=-\frac {b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \cosh (x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3}}+\frac {\log (\cosh (x))}{a}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \cosh (x)\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cosh (x)+b^{2/3} \cosh ^2(x)\right )}{6 a^{5/3}}-\frac {\log \left (a+b \cosh ^3(x)\right )}{3 a}+\frac {\text {sech}^2(x)}{2 a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.92, size = 145, normalized size = 0.95 \begin {gather*} \frac {-6 x+6 \log (\cosh (x))-2 \text {RootSum}\left [b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3+3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-b x+b \log \left (e^x-\text {$\#$1}\right )-4 a x \text {$\#$1}^3+4 a \log \left (e^x-\text {$\#$1}\right ) \text {$\#$1}^3-3 b x \text {$\#$1}^4+3 b \log \left (e^x-\text {$\#$1}\right ) \text {$\#$1}^4}{b+2 b \text {$\#$1}^2+4 a \text {$\#$1}^3+b \text {$\#$1}^4}\&\right ]+3 \text {sech}^2(x)}{6 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.00, size = 145, normalized size = 0.95
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2} a}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{a}+\left (\munderset {\textit {\_R} =\RootOf \left (27 a^{5} \textit {\_Z}^{3}+27 a^{4} \textit {\_Z}^{2}+9 \textit {\_Z} \,a^{3}+a^{2}-b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (\frac {6 a^{2} \textit {\_R}}{b}+\frac {2 a}{b}\right ) {\mathrm e}^{x}+1\right )\right )\) | \(93\) |
default | \(\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )-\frac {2}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+\frac {2}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}}{a}-\frac {\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (3 a -3 b \right ) \textit {\_Z} -a -b \right )}{\sum }\frac {\left (\textit {\_R}^{2} a -\textit {\_R}^{2} b -2 \textit {\_R} a -4 \textit {\_R} b +a +b \right ) \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{2} a -\textit {\_R}^{2} b -2 \textit {\_R} a -2 \textit {\_R} b +a -b}}{3 a}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.20, size = 1138, normalized size = 7.44 \begin {gather*} -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sqrt {\frac {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{2} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a + 4}{a^{2}}} \arctan \left (-\frac {{\left (2 \, \sqrt {\frac {1}{3}} \sqrt {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{4} e^{\left (2 \, x\right )} + b^{2} e^{\left (4 \, x\right )} - 2 \, a b e^{\left (3 \, x\right )} - 2 \, a b e^{x} + {\left (a^{2} b e^{\left (3 \, x\right )} - 4 \, a^{3} e^{\left (2 \, x\right )} + a^{2} b e^{x}\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} + b^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} e^{\left (2 \, x\right )}} {\left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a^{3} - 2 \, a^{2}\right )} \sqrt {\frac {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{2} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a + 4}{a^{2}}} - \sqrt {\frac {1}{3}} {\left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{5} e^{x} - 4 \, a^{2} b e^{\left (2 \, x\right )} + 4 \, a^{3} e^{x} - 4 \, a^{2} b + 2 \, {\left (a^{3} b e^{\left (2 \, x\right )} - 2 \, a^{4} e^{x} + a^{3} b\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}\right )} \sqrt {\frac {{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{2} - 4 \, {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a + 4}{a^{2}}}\right )} e^{\left (-x\right )}}{8 \, b^{2}}\right ) + 2 \, {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} \log \left (-{\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} a^{2} e^{x} + b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right ) - {\left ({\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} - 6 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (2 \, x\right )} - 6\right )} \log \left ({\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )}^{2} a^{4} e^{\left (2 \, x\right )} + b^{2} e^{\left (4 \, x\right )} - 2 \, a b e^{\left (3 \, x\right )} - 2 \, a b e^{x} + {\left (a^{2} b e^{\left (3 \, x\right )} - 4 \, a^{3} e^{\left (2 \, x\right )} + a^{2} b e^{x}\right )} {\left (\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} {\left (\frac {1}{a^{3}} + \frac {b^{2}}{a^{5}} - \frac {a^{2} - b^{2}}{a^{5}}\right )}^{\frac {1}{3}} + \frac {2}{a}\right )} + b^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} e^{\left (2 \, x\right )}\right ) - 12 \, {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - 24 \, e^{\left (2 \, x\right )}}{12 \, {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{3}{\left (x \right )}}{a + b \cosh ^{3}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 191, normalized size = 1.25 \begin {gather*} -\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x} \right |}\right )}{3 \, a^{2}} + \frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {\log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 8 \, a \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} {\left (e^{\left (-x\right )} + e^{x}\right )} + 4 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} - \frac {3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4}{2 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 1173, normalized size = 7.67 \begin {gather*} \frac {2}{a+a\,{\mathrm {e}}^{2\,x}}-\frac {2}{a+2\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{4\,x}}+\left (\sum _{k=1}^3\ln \left (-\frac {50331648\,a^6\,{\mathrm {e}}^{2\,x}-786432\,b^6\,{\mathrm {e}}^{2\,x}+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^7\,452984832+50331648\,a^6-786432\,b^6+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^2\,a^8\,1358954496+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^9\,1358954496+50593792\,a^2\,b^4-102498304\,a^4\,b^2+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^2\,a^8\,{\mathrm {e}}^{2\,x}\,1358954496+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^9\,{\mathrm {e}}^{2\,x}\,1358954496+50593792\,a^2\,b^4\,{\mathrm {e}}^{2\,x}-102498304\,a^4\,b^2\,{\mathrm {e}}^{2\,x}+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^3\,b^4\,7602176-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^5\,b^2\,465305600+524288\,a\,b^5\,{\mathrm {e}}^x+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^2\,a^4\,b^4\,24379392-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^2\,a^6\,b^2\,1383333888+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^5\,b^4\,18874368-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^7\,b^2\,1370750976+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^7\,{\mathrm {e}}^{2\,x}\,452984832-5242880\,a^3\,b^3\,{\mathrm {e}}^x-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^2\,b^5\,{\mathrm {e}}^x\,524288-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^4\,b^3\,{\mathrm {e}}^x\,8912896+\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^3\,b^4\,{\mathrm {e}}^{2\,x}\,7602176-\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\,a^5\,b^2\,{\mathrm {e}}^{2\,x}\,465305600+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^6\,b^3\,{\mathrm {e}}^x\,14155776+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^2\,a^4\,b^4\,{\mathrm {e}}^{2\,x}\,24379392-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^2\,a^6\,b^2\,{\mathrm {e}}^{2\,x}\,1383333888+{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^5\,b^4\,{\mathrm {e}}^{2\,x}\,18874368-{\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )}^3\,a^7\,b^2\,{\mathrm {e}}^{2\,x}\,1370750976}{a^6\,b^6\,3}\right )\,\mathrm {root}\left (27\,a^5\,z^3+27\,a^4\,z^2+9\,a^3\,z+a^2-b^2,z,k\right )\right )+\frac {\ln \left (3221225472\,a^6\,{\mathrm {e}}^{2\,x}-786432\,b^6\,{\mathrm {e}}^{2\,x}+3221225472\,a^6-786432\,b^6+101449728\,a^2\,b^4-3321888768\,a^4\,b^2+101449728\,a^2\,b^4\,{\mathrm {e}}^{2\,x}-3321888768\,a^4\,b^2\,{\mathrm {e}}^{2\,x}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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