Optimal. Leaf size=310 \[ \frac {405405 \sqrt {\pi } \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5} \]
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Rubi [A] time = 0.36, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3416, 3311, 30, 3312, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac {405405 \sqrt {\pi } \sin (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac {429 x^{3/2} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {27027 x^{5/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}+\frac {3 x^{13/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {39 x^{11/6}}{16 b^2}+\frac {3861 x^{7/6}}{256 b^4}-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {x^{5/2}}{5} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3311
Rule 3312
Rule 3351
Rule 3352
Rule 3416
Rubi steps
\begin {align*} \int x^{3/2} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname {Subst}\left (\int x^{13/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {3}{2} \operatorname {Subst}\left (\int x^{13/2} \, dx,x,\sqrt [3]{x}\right )-\frac {429 \operatorname {Subst}\left (\int x^{9/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{16 b^2}\\ &=\frac {x^{5/2}}{5}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {27027 \operatorname {Subst}\left (\int x^{5/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{256 b^4}-\frac {429 \operatorname {Subst}\left (\int x^{9/2} \, dx,x,\sqrt [3]{x}\right )}{32 b^2}\\ &=-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \operatorname {Subst}\left (\int \sqrt {x} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{4096 b^6}+\frac {27027 \operatorname {Subst}\left (\int x^{5/2} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}\\ &=\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \operatorname {Subst}\left (\int \left (\frac {\sqrt {x}}{2}+\frac {1}{2} \sqrt {x} \cos (2 a+2 b x)\right ) \, dx,x,\sqrt [3]{x}\right )}{4096 b^6}\\ &=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \operatorname {Subst}\left (\int \sqrt {x} \cos (2 a+2 b x) \, dx,x,\sqrt [3]{x}\right )}{8192 b^6}\\ &=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {405405 \operatorname {Subst}\left (\int \frac {\sin (2 a+2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7}\\ &=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {(405405 \cos (2 a)) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7}+\frac {(405405 \sin (2 a)) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{32768 b^7}\\ &=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}+\frac {(405405 \cos (2 a)) \operatorname {Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{16384 b^7}+\frac {(405405 \sin (2 a)) \operatorname {Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{16384 b^7}\\ &=-\frac {135135 \sqrt {x}}{4096 b^6}+\frac {3861 x^{7/6}}{256 b^4}-\frac {39 x^{11/6}}{16 b^2}+\frac {x^{5/2}}{5}+\frac {135135 \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{2048 b^6}-\frac {3861 x^{7/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {39 x^{11/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {405405 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{32768 b^{15/2}}+\frac {405405 \sqrt {\pi } C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{32768 b^{15/2}}+\frac {27027 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{512 b^5}-\frac {429 x^{3/2} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{13/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {405405 \sqrt [6]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{16384 b^7}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 174, normalized size = 0.56 \[ \frac {2 \sqrt {b} \sqrt [6]{x} \left (780 \left (256 b^5 x^{5/3}-1584 b^3 x+3465 b \sqrt [3]{x}\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+15 \left (4096 b^6 x^2-36608 b^4 x^{4/3}+144144 b^2 x^{2/3}-135135\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+16384 b^7 x^{7/3}\right )+2027025 \sqrt {\pi } \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+2027025 \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{163840 b^{15/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 184, normalized size = 0.59 \[ -\frac {399360 \, b^{6} x^{\frac {11}{6}} - 2471040 \, b^{4} x^{\frac {7}{6}} - 2027025 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 2027025 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 3120 \, {\left (256 \, b^{6} x^{\frac {11}{6}} - 1584 \, b^{4} x^{\frac {7}{6}} + 3465 \, b^{2} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 60 \, {\left (36608 \, b^{5} x^{\frac {3}{2}} - 144144 \, b^{3} x^{\frac {5}{6}} - {\left (4096 \, b^{7} x^{2} - 135135 \, b\right )} x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) - 8 \, {\left (4096 \, b^{8} x^{2} - 675675 \, b^{2}\right )} \sqrt {x}}{163840 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.25, size = 224, normalized size = 0.72 \[ \frac {1}{5} \, x^{\frac {5}{2}} - \frac {3 \, {\left (4096 i \, b^{6} x^{\frac {13}{6}} - 13312 \, b^{5} x^{\frac {11}{6}} - 36608 i \, b^{4} x^{\frac {3}{2}} + 82368 \, b^{3} x^{\frac {7}{6}} + 144144 i \, b^{2} x^{\frac {5}{6}} - 180180 \, b \sqrt {x} - 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{32768 \, b^{7}} - \frac {3 \, {\left (-4096 i \, b^{6} x^{\frac {13}{6}} - 13312 \, b^{5} x^{\frac {11}{6}} + 36608 i \, b^{4} x^{\frac {3}{2}} + 82368 \, b^{3} x^{\frac {7}{6}} - 144144 i \, b^{2} x^{\frac {5}{6}} - 180180 \, b \sqrt {x} + 135135 i \, x^{\frac {1}{6}}\right )} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{32768 \, b^{7}} + \frac {405405 i \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{65536 \, b^{\frac {15}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {405405 i \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{65536 \, b^{\frac {15}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 219, normalized size = 0.71 \[ \frac {x^{\frac {5}{2}}}{5}+\frac {3 x^{\frac {13}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {39 \left (-\frac {x^{\frac {11}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {11 x^{\frac {3}{2}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {99 \left (-\frac {x^{\frac {7}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {7 x^{\frac {5}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {35 \left (-\frac {\sqrt {x}\, \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {3 x^{\frac {1}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {3 \sqrt {\pi }\, \left (\cos \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{32 b^{\frac {3}{2}}}}{b}\right )}{16 b}}{b}\right )}{16 b}}{b}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.24, size = 161, normalized size = 0.52 \[ \frac {262144 \, b^{9} x^{\frac {5}{2}} - 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (2027025 i + 2027025\right ) \, \cos \left (2 \, a\right ) + \left (2027025 i - 2027025\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (\left (2027025 i - 2027025\right ) \, \cos \left (2 \, a\right ) - \left (2027025 i + 2027025\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 12480 \, {\left (256 \, b^{7} x^{\frac {11}{6}} - 1584 \, b^{5} x^{\frac {7}{6}} + 3465 \, b^{3} \sqrt {x}\right )} \cos \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right ) + 240 \, {\left (4096 \, b^{8} x^{\frac {13}{6}} - 36608 \, b^{6} x^{\frac {3}{2}} + 144144 \, b^{4} x^{\frac {5}{6}} - 135135 \, b^{2} x^{\frac {1}{6}}\right )} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{1310720 \, b^{9}} \]
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 \[ \int x^{3/2}\,{\cos \left (a+b\,x^{1/3}\right )}^2 \,d x \]
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 \[ \int x^{\frac {3}{2}} \cos ^{2}{\left (a + b \sqrt [3]{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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