Optimal. Leaf size=187 \[ -\frac {2 a^2+b^2}{2 x}+\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x}+2 a b \sqrt {d} \sqrt {2 \pi } \cos (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+b^2 \sqrt {d} \sqrt {\pi } \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )-2 a b \sqrt {d} \sqrt {2 \pi } S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)+b^2 \sqrt {d} \sqrt {\pi } C\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right ) \sin (2 c)-\frac {2 a b \sin \left (c+d x^2\right )}{x} \]
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Rubi [A]
time = 0.11, antiderivative size = 187, 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, 3469,
3434, 3433, 3432, 3468, 3435} \begin {gather*} -\frac {2 a^2+b^2}{2 x}+2 \sqrt {2 \pi } a b \sqrt {d} \cos (c) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {d} x\right )-2 \sqrt {2 \pi } a b \sqrt {d} \sin (c) S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2 a b \sin \left (c+d x^2\right )}{x}+\sqrt {\pi } b^2 \sqrt {d} \sin (2 c) \text {FresnelC}\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )+\sqrt {\pi } b^2 \sqrt {d} \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )+\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 3432
Rule 3433
Rule 3434
Rule 3435
Rule 3468
Rule 3469
Rule 3484
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin \left (c+d x^2\right )\right )^2}{x^2} \, dx &=\int \left (\frac {a^2}{x^2}+\frac {b^2}{2 x^2}-\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x^2}+\frac {2 a b \sin \left (c+d x^2\right )}{x^2}\right ) \, dx\\ &=\int \left (\frac {a^2+\frac {b^2}{2}}{x^2}-\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x^2}+\frac {2 a b \sin \left (c+d x^2\right )}{x^2}\right ) \, dx\\ &=-\frac {2 a^2+b^2}{2 x}+(2 a b) \int \frac {\sin \left (c+d x^2\right )}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\cos \left (2 c+2 d x^2\right )}{x^2} \, dx\\ &=-\frac {2 a^2+b^2}{2 x}+\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x}-\frac {2 a b \sin \left (c+d x^2\right )}{x}+(4 a b d) \int \cos \left (c+d x^2\right ) \, dx+\left (2 b^2 d\right ) \int \sin \left (2 c+2 d x^2\right ) \, dx\\ &=-\frac {2 a^2+b^2}{2 x}+\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x}-\frac {2 a b \sin \left (c+d x^2\right )}{x}+(4 a b d \cos (c)) \int \cos \left (d x^2\right ) \, dx+\left (2 b^2 d \cos (2 c)\right ) \int \sin \left (2 d x^2\right ) \, dx-(4 a b d \sin (c)) \int \sin \left (d x^2\right ) \, dx+\left (2 b^2 d \sin (2 c)\right ) \int \cos \left (2 d x^2\right ) \, dx\\ &=-\frac {2 a^2+b^2}{2 x}+\frac {b^2 \cos \left (2 c+2 d x^2\right )}{2 x}+2 a b \sqrt {d} \sqrt {2 \pi } \cos (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+b^2 \sqrt {d} \sqrt {\pi } \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )-2 a b \sqrt {d} \sqrt {2 \pi } S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)+b^2 \sqrt {d} \sqrt {\pi } C\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right ) \sin (2 c)-\frac {2 a b \sin \left (c+d x^2\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 184, normalized size = 0.98 \begin {gather*} \frac {-2 a^2-b^2+b^2 \cos \left (2 \left (c+d x^2\right )\right )+4 a b \sqrt {d} \sqrt {2 \pi } x \cos (c) C\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+2 b^2 \sqrt {d} \sqrt {\pi } x \cos (2 c) S\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right )-4 a b \sqrt {d} \sqrt {2 \pi } x S\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)+2 b^2 \sqrt {d} \sqrt {\pi } x C\left (\frac {2 \sqrt {d} x}{\sqrt {\pi }}\right ) \sin (2 c)-4 a b \sin \left (c+d x^2\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 137, normalized size = 0.73
method | result | size |
default | \(-\frac {a^{2}+\frac {b^{2}}{2}}{x}-\frac {b^{2} \left (-\frac {\cos \left (2 d \,x^{2}+2 c \right )}{x}-2 \sqrt {d}\, \sqrt {\pi }\, \left (\cos \left (2 c \right ) \mathrm {S}\left (\frac {2 x \sqrt {d}}{\sqrt {\pi }}\right )+\sin \left (2 c \right ) \FresnelC \left (\frac {2 x \sqrt {d}}{\sqrt {\pi }}\right )\right )\right )}{2}+2 a b \left (-\frac {\sin \left (d \,x^{2}+c \right )}{x}+\sqrt {d}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (c \right ) \FresnelC \left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (c \right ) \mathrm {S}\left (\frac {x \sqrt {d}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )\) | \(137\) |
risch | \(\frac {i b^{2} d \sqrt {\pi }\, \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {i d}\, x \right ) {\mathrm e}^{-2 i c}}{4 \sqrt {i d}}-\frac {i b^{2} d \sqrt {\pi }\, \erf \left (\sqrt {-2 i d}\, x \right ) {\mathrm e}^{2 i c}}{2 \sqrt {-2 i d}}+\frac {a b d \sqrt {\pi }\, \erf \left (\sqrt {-i d}\, x \right ) {\mathrm e}^{i c}}{\sqrt {-i d}}+\frac {a b d \sqrt {\pi }\, \erf \left (\sqrt {i d}\, x \right ) {\mathrm e}^{-i c}}{\sqrt {i d}}-\frac {a^{2}}{x}-\frac {b^{2}}{2 x}-\frac {2 a b \sin \left (d \,x^{2}+c \right )}{x}+\frac {b^{2} \cos \left (2 d \,x^{2}+2 c \right )}{2 x}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.69, size = 170, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {d x^{2}} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, d x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, d x^{2}\right )\right )} \cos \left (c\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, i \, d x^{2}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} a b}{4 \, x} - \frac {{\left (\sqrt {2} \sqrt {d x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, 2 i \, d x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -2 i \, d x^{2}\right )\right )} \cos \left (2 \, c\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, 2 i \, d x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {1}{2}, -2 i \, d x^{2}\right )\right )} \sin \left (2 \, c\right )\right )} + 8\right )} b^{2}}{16 \, x} - \frac {a^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 159, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {2} \pi a b x \sqrt {\frac {d}{\pi }} \cos \left (c\right ) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) - 2 \, \sqrt {2} \pi a b x \sqrt {\frac {d}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) \sin \left (c\right ) + \pi b^{2} x \sqrt {\frac {d}{\pi }} \cos \left (2 \, c\right ) \operatorname {S}\left (2 \, x \sqrt {\frac {d}{\pi }}\right ) + \pi b^{2} x \sqrt {\frac {d}{\pi }} \operatorname {C}\left (2 \, x \sqrt {\frac {d}{\pi }}\right ) \sin \left (2 \, c\right ) + b^{2} \cos \left (d x^{2} + c\right )^{2} - 2 \, a b \sin \left (d x^{2} + c\right ) - a^{2} - b^{2}}{x} \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 \frac {\left (a + b \sin {\left (c + d x^{2} \right )}\right )^{2}}{x^{2}}\, dx \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (d\,x^2+c\right )\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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