Optimal. Leaf size=160 \[ -\frac {b^3 \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \text {ArcTan}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.19, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2916, 12, 908,
653, 209, 649, 266} \begin {gather*} -\frac {b \text {ArcTan}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b^3 \text {ArcTan}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b^4 \log (a+b \sinh (c+d x))}{a d \left (a^2+b^2\right )^2}+\frac {\log (\sinh (c+d x))}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 266
Rule 649
Rule 653
Rule 908
Rule 2916
Rubi steps
\begin {align*} \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {b^3 \text {Subst}\left (\int \frac {b}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^4 \text {Subst}\left (\int \frac {1}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b^4 \text {Subst}\left (\int \left (\frac {1}{a b^4 x}-\frac {1}{a \left (a^2+b^2\right )^2 (a+x)}+\frac {-b^2-a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac {-b^4-a \left (a^2+2 b^2\right ) x}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {-b^4-a \left (a^2+2 b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {-b^2-a x}{\left (b^2+x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {b^4 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^2 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac {\left (a \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {b^3 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 0.59, size = 196, normalized size = 1.22 \begin {gather*} -\frac {a b \left (a^2+b^2\right ) \text {ArcTan}(\sinh (c+d x))-2 \left (a^2+b^2\right )^2 \log (\sinh (c+d x))+a \left (a^3+2 a b^2+\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}-b \sinh (c+d x)\right )+2 b^4 \log (a+b \sinh (c+d x))+a \left (a^3+2 a b^2-\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}+b \sinh (c+d x)\right )-a^2 \left (a^2+b^2\right ) \text {sech}^2(c+d x)+a b \left (a^2+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right )^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.76, size = 230, normalized size = 1.44
method | result | size |
derivativedivides | \(\frac {-\frac {b^{4} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3}+a \,b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4}+\frac {\left (a^{2} b +3 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(230\) |
default | \(\frac {-\frac {b^{4} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3}+a \,b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4}+\frac {\left (a^{2} b +3 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(230\) |
risch | \(\frac {2 a^{3} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {2 a^{3} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 x}{a}-\frac {2 c}{a d}+\frac {2 b^{4} x}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{4} c}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d}-\frac {b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(632\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 265, normalized size = 1.66 \begin {gather*} -\frac {b^{4} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} + \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a 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. 1279 vs.
\(2 (157) = 314\).
time = 0.56, size = 1279, normalized size = 7.99 \begin {gather*} -\frac {{\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{3} b + a b^{3}\right )} \sinh \left (d x + c\right )^{3} - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (2 \, a^{4} + 2 \, a^{2} b^{2} - 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} b + 3 \, a b^{3}\right )} \sinh \left (d x + c\right )^{4} + a^{3} b + 3 \, a b^{3} + 2 \, {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} b + 3 \, a b^{3} + 3 \, {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{3} b + 3 \, a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right ) + {\left (b^{4} \cosh \left (d x + c\right )^{4} + 4 \, b^{4} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{4} \sinh \left (d x + c\right )^{4} + 2 \, b^{4} \cosh \left (d x + c\right )^{2} + b^{4} + 2 \, {\left (3 \, b^{4} \cosh \left (d x + c\right )^{2} + b^{4}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b^{4} \cosh \left (d x + c\right )^{3} + b^{4} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\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 ) + {\left ({\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (d x + c\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - {\left (a^{3} b + a b^{3} - 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d + 4 \, {\left ({\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right )^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs.
\(2 (157) = 314\).
time = 0.46, size = 343, normalized size = 2.14 \begin {gather*} -\frac {\frac {4 \, b^{5} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{5} b + 2 \, a^{3} b^{3} + a b^{5}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{2} b + 3 \, b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a} - \frac {2 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 2 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 8 \, a^{3} + 12 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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