Optimal. Leaf size=373 \[ -\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {ArcSin}(c x)}}+\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } \cos \left (\frac {6 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 b^{3/2} c^2}+\frac {d^2 \sqrt {\frac {\pi }{2}} S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } S\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {6 a}{b}\right )}{8 b^{3/2} c^2} \]
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Rubi [A]
time = 0.82, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps
used = 32, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {4799, 4753,
3393, 3387, 3386, 3432, 3385, 3433, 4809, 4491} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} d^2 \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {3 \pi } d^2 \cos \left (\frac {6 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^2}+\frac {5 \sqrt {\pi } d^2 \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{8 b^{3/2} c^2}+\frac {5 \sqrt {\pi } d^2 \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d^2 \sin \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {\sqrt {3 \pi } d^2 \sin \left (\frac {6 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^2}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {ArcSin}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4491
Rule 4753
Rule 4799
Rule 4809
Rubi steps
\begin {align*} \int \frac {x \left (d-c^2 d x^2\right )^2}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (2 d^2\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {\left (12 c d^2\right ) \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac {\left (12 d^2\right ) \text {Subst}\left (\int \frac {\cos ^4(x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {a+b x}}+\frac {\cos (2 x)}{2 \sqrt {a+b x}}+\frac {\cos (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac {\left (12 d^2\right ) \text {Subst}\left (\int \left (\frac {1}{16 \sqrt {a+b x}}+\frac {\cos (2 x)}{32 \sqrt {a+b x}}-\frac {\cos (4 x)}{16 \sqrt {a+b x}}-\frac {\cos (6 x)}{32 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {d^2 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cos (6 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {d^2 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (3 d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}+\frac {\left (d^2 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\left (3 d^2 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\left (3 d^2 \cos \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {6 a}{b}+6 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}-\frac {\left (3 d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}+\frac {\left (d^2 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\left (3 d^2 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\left (3 d^2 \sin \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {6 a}{b}+6 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}\\ &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (3 d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {\left (2 d^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2 c^2}+\frac {\left (d^2 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c^2}+\frac {\left (3 d^2 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c^2}+\frac {\left (3 d^2 \cos \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {6 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^2}-\frac {\left (3 d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^2}+\frac {\left (2 d^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2 c^2}+\frac {\left (d^2 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c^2}+\frac {\left (3 d^2 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c^2}+\frac {\left (3 d^2 \sin \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {6 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^2}\\ &=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } \cos \left (\frac {6 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 b^{3/2} c^2}+\frac {d^2 \sqrt {\frac {\pi }{2}} S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } S\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {6 a}{b}\right )}{8 b^{3/2} c^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.08, size = 509, normalized size = 1.36 \begin {gather*} \frac {d^2 \left (64 \left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {\pi }}\right )+64 \left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )+\frac {i e^{-\frac {6 i a}{b}} \left (11 \sqrt {2} e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )-11 \sqrt {2} e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {2 i (a+b \text {ArcSin}(c x))}{b}\right )-8 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {4 i (a+b \text {ArcSin}(c x))}{b}\right )+8 e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {4 i (a+b \text {ArcSin}(c x))}{b}\right )-\sqrt {6} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {6 i (a+b \text {ArcSin}(c x))}{b}\right )+\sqrt {6} e^{\frac {12 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {6 i (a+b \text {ArcSin}(c x))}{b}\right )+10 i e^{\frac {6 i a}{b}} \sin (2 \text {ArcSin}(c x))+8 i e^{\frac {6 i a}{b}} \sin (4 \text {ArcSin}(c x))+2 i e^{\frac {6 i a}{b}} \sin (6 \text {ArcSin}(c x))\right )}{b \sqrt {a+b \text {ArcSin}(c x)}}\right )}{32 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 455, normalized size = 1.22
method | result | size |
default | \(\frac {d^{2} \left (\sqrt {-\frac {6}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \FresnelC \left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \cos \left (\frac {6 a}{b}\right )-\sqrt {-\frac {6}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \mathrm {S}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sin \left (\frac {6 a}{b}\right )+8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {4 a}{b}\right )-8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {4 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+5 \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}-5 \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}+5 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )+4 \sin \left (-\frac {4 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {4 a}{b}\right )+\sin \left (-\frac {6 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {6 a}{b}\right )\right )}{16 c^{2} b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(455\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {x\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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