Optimal. Leaf size=187 \[ \frac {a^3 x}{d^3}-\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d)^2 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.33, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2841, 3047,
3100, 2814, 2739, 632, 210} \begin {gather*} -\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f (c+d)^2 \sqrt {c^2-d^2}}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 f (c+d)^2 (c+d \sin (e+f x))}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{2 d f (c+d) (c+d \sin (e+f x))^2}+\frac {a^3 x}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2841
Rule 3047
Rule 3100
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^3} \, dx &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {(a+a \sin (e+f x)) (a (c-5 d)-2 a (c+d) \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {a^2 (c-5 d)+\left (a^2 (c-5 d)-2 a^2 (c+d)\right ) \sin (e+f x)-2 a^2 (c+d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {a \int \frac {a^2 (c-d) d (c+7 d)+2 a^2 (c-d) (c+d)^2 \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 (c-d) d^2 (c+d)^2}\\ &=\frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 d^3 (c+d)^2}\\ &=\frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac {\left (a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 (c+d)^2 f}\\ &=\frac {a^3 x}{d^3}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac {\left (2 a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 (c+d)^2 f}\\ &=\frac {a^3 x}{d^3}-\frac {a^3 (c-d) \left (2 c^2+6 c d+7 d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 (c+d)^2 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {a^3 (c-d) (2 c+5 d) \cos (e+f x)}{2 d^2 (c+d)^2 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.63, size = 196, normalized size = 1.05 \begin {gather*} \frac {a^3 (1+\sin (e+f x))^3 \left (2 (e+f x)-\frac {2 \left (2 c^3+4 c^2 d+c d^2-7 d^3\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d)^2 \sqrt {c^2-d^2}}-\frac {(c-d)^2 d \cos (e+f x)}{(c+d) (c+d \sin (e+f x))^2}+\frac {3 d \left (c^2+c d-2 d^2\right ) \cos (e+f x)}{(c+d)^2 (c+d \sin (e+f x))}\right )}{2 d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 353, normalized size = 1.89
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}-\frac {\frac {-\frac {d^{2} \left (c^{3}+5 c^{2} d -4 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}-\frac {d \left (2 c^{5}+4 c^{4} d -c^{3} d^{2}+7 c^{2} d^{3}-10 c \,d^{4}-2 d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}-\frac {d^{2} \left (7 c^{3}+11 c^{2} d -16 c \,d^{2}-2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}-\frac {d \left (2 c^{3}+4 c^{2} d -5 c \,d^{2}-d^{3}\right )}{2 \left (c^{2}+2 c d +d^{2}\right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 c^{3}+4 c^{2} d +c \,d^{2}-7 d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}\right )}{f}\) | \(353\) |
default | \(\frac {2 a^{3} \left (\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{d^{3}}-\frac {\frac {-\frac {d^{2} \left (c^{3}+5 c^{2} d -4 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c}-\frac {d \left (2 c^{5}+4 c^{4} d -c^{3} d^{2}+7 c^{2} d^{3}-10 c \,d^{4}-2 d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) c^{2}}-\frac {d^{2} \left (7 c^{3}+11 c^{2} d -16 c \,d^{2}-2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{2}+2 c d +d^{2}\right )}-\frac {d \left (2 c^{3}+4 c^{2} d -5 c \,d^{2}-d^{3}\right )}{2 \left (c^{2}+2 c d +d^{2}\right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 c^{3}+4 c^{2} d +c \,d^{2}-7 d^{3}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {c^{2}-d^{2}}}}{d^{3}}\right )}{f}\) | \(353\) |
risch | \(\frac {a^{3} x}{d^{3}}-\frac {i a^{3} \left (-i d^{4} {\mathrm e}^{i \left (f x +e \right )}-17 i c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}-i d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-4 i c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}+10 i c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+7 i c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+8 i c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}-2 i c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 d \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-9 d^{2} c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 d^{3} c \,{\mathrm e}^{2 i \left (f x +e \right )}-6 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 c^{2} d^{2}-3 d^{3} c +6 d^{4}\right )}{\left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{2} \left (c +d \right )^{2} f \,d^{3}}+\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right )^{3} f \,d^{3}}+\frac {3 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right )^{3} f \,d^{2}}+\frac {7 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {-i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{2 \left (c +d \right )^{3} f d}-\frac {\sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c^{2}}{\left (c +d \right )^{3} f \,d^{3}}-\frac {3 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right ) c}{\left (c +d \right )^{3} f \,d^{2}}-\frac {7 \sqrt {-\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c +\sqrt {-\left (c +d \right ) \left (c -d \right )}}{d}\right )}{2 \left (c +d \right )^{3} f d}\) | \(654\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 497 vs.
\(2 (184) = 368\).
time = 0.42, size = 1089, normalized size = 5.82 \begin {gather*} \left [\frac {4 \, {\left (a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right )} f x \cos \left (f x + e\right )^{2} - 4 \, {\left (a^{3} c^{4} + 2 \, a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right )} f x - {\left (2 \, a^{3} c^{4} + 6 \, a^{3} c^{3} d + 9 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4} - {\left (2 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} + 7 \, a^{3} c d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left ({\left (c^{2} + c d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 2 \, {\left (2 \, a^{3} c^{3} d + 4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} - a^{3} d^{4}\right )} \cos \left (f x + e\right ) - 2 \, {\left (4 \, {\left (a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right )} f x + 3 \, {\left (a^{3} c^{2} d^{2} + a^{3} c d^{3} - 2 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{3} d^{4} + 2 \, c^{2} d^{5} + c d^{6}\right )} f \sin \left (f x + e\right ) - {\left (c^{4} d^{3} + 2 \, c^{3} d^{4} + 2 \, c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right )} f\right )}}, \frac {2 \, {\left (a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right )} f x \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{3} c^{4} + 2 \, a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right )} f x - {\left (2 \, a^{3} c^{4} + 6 \, a^{3} c^{3} d + 9 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4} - {\left (2 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} + 7 \, a^{3} c d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (c \sin \left (f x + e\right ) + d\right )} \sqrt {\frac {c - d}{c + d}}}{{\left (c - d\right )} \cos \left (f x + e\right )}\right ) - {\left (2 \, a^{3} c^{3} d + 4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} - a^{3} d^{4}\right )} \cos \left (f x + e\right ) - {\left (4 \, {\left (a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right )} f x + 3 \, {\left (a^{3} c^{2} d^{2} + a^{3} c d^{3} - 2 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{3} d^{4} + 2 \, c^{2} d^{5} + c d^{6}\right )} f \sin \left (f x + e\right ) - {\left (c^{4} d^{3} + 2 \, c^{3} d^{4} + 2 \, c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right )} f\right )}}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 522 vs.
\(2 (184) = 368\).
time = 0.53, size = 522, normalized size = 2.79 \begin {gather*} \frac {\frac {{\left (f x + e\right )} a^{3}}{d^{3}} - \frac {{\left (2 \, a^{3} c^{3} + 4 \, a^{3} c^{2} d + a^{3} c d^{2} - 7 \, a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 16 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c^{5} + 4 \, a^{3} c^{4} d - 5 \, a^{3} c^{3} d^{2} - a^{3} c^{2} d^{3}}{{\left (c^{4} d^{2} + 2 \, c^{3} d^{3} + c^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.27, size = 2500, normalized size = 13.37 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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