Optimal. Leaf size=198 \[ \frac {1}{6} \left (2 a^2+b^2\right ) x^3-\frac {a b x \cos \left (c+d x^2\right )}{d}+\frac {a b \sqrt {\frac {\pi }{2}} \cos (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{d^{3/2}}+\frac {b^2 \sqrt {\pi } \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )}{16 d^{3/2}}-\frac {a b \sqrt {\frac {\pi }{2}} S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)}{d^{3/2}}+\frac {b^2 \sqrt {\pi } C\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right ) \sin (2 c)}{16 d^{3/2}}-\frac {b^2 x \sin \left (2 c+2 d x^2\right )}{8 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3484, 6, 3467,
3434, 3433, 3432, 3466, 3435} \begin {gather*} \frac {1}{6} x^3 \left (2 a^2+b^2\right )+\frac {\sqrt {\frac {\pi }{2}} a b \cos (c) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {d} x\right )}{d^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} a b \sin (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{d^{3/2}}-\frac {a b x \cos \left (c+d x^2\right )}{d}+\frac {\sqrt {\pi } b^2 \sin (2 c) \text {FresnelC}\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )}{16 d^{3/2}}+\frac {\sqrt {\pi } b^2 \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )}{16 d^{3/2}}-\frac {b^2 x \sin \left (2 c+2 d x^2\right )}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 3432
Rule 3433
Rule 3434
Rule 3435
Rule 3466
Rule 3467
Rule 3484
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin \left (c+d x^2\right )\right )^2 \, dx &=\int \left (a^2 x^2+\frac {b^2 x^2}{2}-\frac {1}{2} b^2 x^2 \cos \left (2 c+2 d x^2\right )+2 a b x^2 \sin \left (c+d x^2\right )\right ) \, dx\\ &=\int \left (\left (a^2+\frac {b^2}{2}\right ) x^2-\frac {1}{2} b^2 x^2 \cos \left (2 c+2 d x^2\right )+2 a b x^2 \sin \left (c+d x^2\right )\right ) \, dx\\ &=\frac {1}{6} \left (2 a^2+b^2\right ) x^3+(2 a b) \int x^2 \sin \left (c+d x^2\right ) \, dx-\frac {1}{2} b^2 \int x^2 \cos \left (2 c+2 d x^2\right ) \, dx\\ &=\frac {1}{6} \left (2 a^2+b^2\right ) x^3-\frac {a b x \cos \left (c+d x^2\right )}{d}-\frac {b^2 x \sin \left (2 c+2 d x^2\right )}{8 d}+\frac {(a b) \int \cos \left (c+d x^2\right ) \, dx}{d}+\frac {b^2 \int \sin \left (2 c+2 d x^2\right ) \, dx}{8 d}\\ &=\frac {1}{6} \left (2 a^2+b^2\right ) x^3-\frac {a b x \cos \left (c+d x^2\right )}{d}-\frac {b^2 x \sin \left (2 c+2 d x^2\right )}{8 d}+\frac {(a b \cos (c)) \int \cos \left (d x^2\right ) \, dx}{d}+\frac {\left (b^2 \cos (2 c)\right ) \int \sin \left (2 d x^2\right ) \, dx}{8 d}-\frac {(a b \sin (c)) \int \sin \left (d x^2\right ) \, dx}{d}+\frac {\left (b^2 \sin (2 c)\right ) \int \cos \left (2 d x^2\right ) \, dx}{8 d}\\ &=\frac {1}{6} \left (2 a^2+b^2\right ) x^3-\frac {a b x \cos \left (c+d x^2\right )}{d}+\frac {a b \sqrt {\frac {\pi }{2}} \cos (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )}{d^{3/2}}+\frac {b^2 \sqrt {\pi } \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )}{16 d^{3/2}}-\frac {a b \sqrt {\frac {\pi }{2}} S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)}{d^{3/2}}+\frac {b^2 \sqrt {\pi } C\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right ) \sin (2 c)}{16 d^{3/2}}-\frac {b^2 x \sin \left (2 c+2 d x^2\right )}{8 d}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 191, normalized size = 0.96 \begin {gather*} \frac {16 a^2 d^{3/2} x^3+8 b^2 d^{3/2} x^3-48 a b \sqrt {d} x \cos \left (c+d x^2\right )+24 a b \sqrt {2 \pi } \cos (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+3 b^2 \sqrt {\pi } \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )-24 a b \sqrt {2 \pi } S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)+3 b^2 \sqrt {\pi } C\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right ) \sin (2 c)-6 b^2 \sqrt {d} x \sin \left (2 \left (c+d x^2\right )\right )}{48 d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 142, normalized size = 0.72
method | result | size |
default | \(\frac {x^{3} a^{2}}{3}+\frac {x^{3} b^{2}}{6}-\frac {b^{2} \left (\frac {x \sin \left (2 d \,x^{2}+2 c \right )}{4 d}-\frac {\sqrt {\pi }\, \left (\cos \left (2 c \right ) \mathrm {S}\left (\frac {2 x \sqrt {d}}{\sqrt {\pi }}\right )+\sin \left (2 c \right ) \FresnelC \left (\frac {2 x \sqrt {d}}{\sqrt {\pi }}\right )\right )}{8 d^{\frac {3}{2}}}\right )}{2}+2 a b \left (-\frac {x \cos \left (d \,x^{2}+c \right )}{2 d}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (c \right ) \FresnelC \left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (c \right ) \mathrm {S}\left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 d^{\frac {3}{2}}}\right )\) | \(142\) |
risch | \(\frac {i b^{2} \sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {i d}\, x \right ) {\mathrm e}^{-2 i c}}{64 d \sqrt {i d}}-\frac {i b^{2} \sqrt {\pi }\, \erf \left (\sqrt {-2 i d}\, x \right ) {\mathrm e}^{2 i c}}{32 d \sqrt {-2 i d}}+\frac {a b \sqrt {\pi }\, \erf \left (\sqrt {-i d}\, x \right ) {\mathrm e}^{i c}}{4 d \sqrt {-i d}}+\frac {a b \sqrt {\pi }\, \erf \left (\sqrt {i d}\, x \right ) {\mathrm e}^{-i c}}{4 d \sqrt {i d}}+\frac {x^{3} a^{2}}{3}+\frac {x^{3} b^{2}}{6}-\frac {a b x \cos \left (d \,x^{2}+c \right )}{d}-\frac {b^{2} x \sin \left (2 d \,x^{2}+2 c \right )}{8 d}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 171, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, a^{2} x^{3} - \frac {{\left (8 \, d^{2} x \cos \left (d x^{2} + c\right ) + \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (c\right ) + \left (i + 1\right ) \, \sin \left (c\right )\right )} \operatorname {erf}\left (\sqrt {i \, d} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (c\right ) - \left (i - 1\right ) \, \sin \left (c\right )\right )} \operatorname {erf}\left (\sqrt {-i \, d} x\right )\right )} d^{\frac {3}{2}}\right )} a b}{8 \, d^{3}} + \frac {{\left (64 \, d^{3} x^{3} - 48 \, d^{2} x \sin \left (2 \, d x^{2} + 2 \, c\right ) + 3 \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i + 1\right ) \, \cos \left (2 \, c\right ) - \left (i - 1\right ) \, \sin \left (2 \, c\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, d} x\right ) + {\left (-\left (i - 1\right ) \, \cos \left (2 \, c\right ) + \left (i + 1\right ) \, \sin \left (2 \, c\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, d} x\right )\right )} d^{\frac {3}{2}}\right )} b^{2}}{384 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 176, normalized size = 0.89 \begin {gather*} \frac {8 \, {\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{3} + 24 \, \sqrt {2} \pi a b \sqrt {\frac {d}{\pi }} \cos \left (c\right ) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) - 12 \, b^{2} d x \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) - 24 \, \sqrt {2} \pi a b \sqrt {\frac {d}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) \sin \left (c\right ) + 3 \, \pi b^{2} \sqrt {\frac {d}{\pi }} \cos \left (2 \, c\right ) \operatorname {S}\left (2 \, x \sqrt {\frac {d}{\pi }}\right ) + 3 \, \pi b^{2} \sqrt {\frac {d}{\pi }} \operatorname {C}\left (2 \, x \sqrt {\frac {d}{\pi }}\right ) \sin \left (2 \, c\right ) - 48 \, a b d x \cos \left (d x^{2} + c\right )}{48 \, d^{2}} \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 x^{2} \left (a + b \sin {\left (c + d x^{2} \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 6.19, size = 283, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, a^{2} x^{3} + \frac {1}{6} \, b^{2} x^{3} + \frac {i \, b^{2} x e^{\left (2 i \, d x^{2} + 2 i \, c\right )}}{16 \, d} - \frac {a b x e^{\left (i \, d x^{2} + i \, c\right )}}{2 \, d} - \frac {a b x e^{\left (-i \, d x^{2} - i \, c\right )}}{2 \, d} - \frac {i \, b^{2} x e^{\left (-2 i \, d x^{2} - 2 i \, c\right )}}{16 \, d} - \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}\right ) e^{\left (i \, c\right )}}{4 \, d {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}} - \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}\right ) e^{\left (-i \, c\right )}}{4 \, d {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )} \sqrt {{\left | d \right |}}} + \frac {i \, \sqrt {\pi } b^{2} \operatorname {erf}\left (-\sqrt {d} x {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )}\right ) e^{\left (2 i \, c\right )}}{32 \, d^{\frac {3}{2}} {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )}} - \frac {i \, \sqrt {\pi } b^{2} \operatorname {erf}\left (-\sqrt {d} x {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )}\right ) e^{\left (-2 i \, c\right )}}{32 \, d^{\frac {3}{2}} {\left (\frac {i \, d}{{\left | d \right |}} + 1\right )}} \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 x^2\,{\left (a+b\,\sin \left (d\,x^2+c\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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