Optimal. Leaf size=224 \[ a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+\frac {1}{6} b d^3 f^2 \cos (c) \text {Ci}\left (\frac {d}{x}\right )+b d^2 e f \sin (c) \text {Ci}\left (\frac {d}{x}\right )-b d e^2 \cos (c) \text {Ci}\left (\frac {d}{x}\right )-\frac {1}{6} b d^3 f^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )+b d^2 e f \cos (c) \text {Si}\left (\frac {d}{x}\right )-\frac {1}{6} b d^2 f^2 x \sin \left (c+\frac {d}{x}\right )+b d e^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )+b e^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+b d e f x \cos \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )+\frac {1}{6} b d f^2 x^2 \cos \left (c+\frac {d}{x}\right ) \]
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Rubi [A] time = 0.46, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3431, 14, 3297, 3303, 3299, 3302} \[ a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+b d^2 e f \sin (c) \text {CosIntegral}\left (\frac {d}{x}\right )+\frac {1}{6} b d^3 f^2 \cos (c) \text {CosIntegral}\left (\frac {d}{x}\right )-b d e^2 \cos (c) \text {CosIntegral}\left (\frac {d}{x}\right )+b d^2 e f \cos (c) \text {Si}\left (\frac {d}{x}\right )-\frac {1}{6} b d^3 f^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )-\frac {1}{6} b d^2 f^2 x \sin \left (c+\frac {d}{x}\right )+b d e^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )+b e^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+b d e f x \cos \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )+\frac {1}{6} b d f^2 x^2 \cos \left (c+\frac {d}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 14
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3431
Rubi steps
\begin {align*} \int (e+f x)^2 \left (a+b \sin \left (c+\frac {d}{x}\right )\right ) \, dx &=-\operatorname {Subst}\left (\int \left (\frac {f^2 (a+b \sin (c+d x))}{x^4}+\frac {2 e f (a+b \sin (c+d x))}{x^3}+\frac {e^2 (a+b \sin (c+d x))}{x^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\left (e^2 \operatorname {Subst}\left (\int \frac {a+b \sin (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )\right )-(2 e f) \operatorname {Subst}\left (\int \frac {a+b \sin (c+d x)}{x^3} \, dx,x,\frac {1}{x}\right )-f^2 \operatorname {Subst}\left (\int \frac {a+b \sin (c+d x)}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\left (e^2 \operatorname {Subst}\left (\int \left (\frac {a}{x^2}+\frac {b \sin (c+d x)}{x^2}\right ) \, dx,x,\frac {1}{x}\right )\right )-(2 e f) \operatorname {Subst}\left (\int \left (\frac {a}{x^3}+\frac {b \sin (c+d x)}{x^3}\right ) \, dx,x,\frac {1}{x}\right )-f^2 \operatorname {Subst}\left (\int \left (\frac {a}{x^4}+\frac {b \sin (c+d x)}{x^4}\right ) \, dx,x,\frac {1}{x}\right )\\ &=a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3-\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )-(2 b e f) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x^3} \, dx,x,\frac {1}{x}\right )-\left (b f^2\right ) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+b e^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )-\left (b d e^2\right ) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{x} \, dx,x,\frac {1}{x}\right )-(b d e f) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{3} \left (b d f^2\right ) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+b d e f x \cos \left (c+\frac {d}{x}\right )+\frac {1}{6} b d f^2 x^2 \cos \left (c+\frac {d}{x}\right )+b e^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )+\left (b d^2 e f\right ) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (b d^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )-\left (b d e^2 \cos (c)\right ) \operatorname {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (b d e^2 \sin (c)\right ) \operatorname {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+b d e f x \cos \left (c+\frac {d}{x}\right )+\frac {1}{6} b d f^2 x^2 \cos \left (c+\frac {d}{x}\right )-b d e^2 \cos (c) \text {Ci}\left (\frac {d}{x}\right )+b e^2 x \sin \left (c+\frac {d}{x}\right )-\frac {1}{6} b d^2 f^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )+b d e^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )+\frac {1}{6} \left (b d^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (b d^2 e f \cos (c)\right ) \operatorname {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (b d^2 e f \sin (c)\right ) \operatorname {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+b d e f x \cos \left (c+\frac {d}{x}\right )+\frac {1}{6} b d f^2 x^2 \cos \left (c+\frac {d}{x}\right )-b d e^2 \cos (c) \text {Ci}\left (\frac {d}{x}\right )+b d^2 e f \text {Ci}\left (\frac {d}{x}\right ) \sin (c)+b e^2 x \sin \left (c+\frac {d}{x}\right )-\frac {1}{6} b d^2 f^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )+b d^2 e f \cos (c) \text {Si}\left (\frac {d}{x}\right )+b d e^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )+\frac {1}{6} \left (b d^3 f^2 \cos (c)\right ) \operatorname {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (b d^3 f^2 \sin (c)\right ) \operatorname {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a e^2 x+a e f x^2+\frac {1}{3} a f^2 x^3+b d e f x \cos \left (c+\frac {d}{x}\right )+\frac {1}{6} b d f^2 x^2 \cos \left (c+\frac {d}{x}\right )-b d e^2 \cos (c) \text {Ci}\left (\frac {d}{x}\right )+\frac {1}{6} b d^3 f^2 \cos (c) \text {Ci}\left (\frac {d}{x}\right )+b d^2 e f \text {Ci}\left (\frac {d}{x}\right ) \sin (c)+b e^2 x \sin \left (c+\frac {d}{x}\right )-\frac {1}{6} b d^2 f^2 x \sin \left (c+\frac {d}{x}\right )+b e f x^2 \sin \left (c+\frac {d}{x}\right )+\frac {1}{3} b f^2 x^3 \sin \left (c+\frac {d}{x}\right )+b d^2 e f \cos (c) \text {Si}\left (\frac {d}{x}\right )+b d e^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )-\frac {1}{6} b d^3 f^2 \sin (c) \text {Si}\left (\frac {d}{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.62, size = 150, normalized size = 0.67 \[ \frac {1}{6} \left (x \left (2 a \left (3 e^2+3 e f x+f^2 x^2\right )+b \sin \left (c+\frac {d}{x}\right ) \left (-f^2 \left (d^2-2 x^2\right )+6 e^2+6 e f x\right )+b d f (6 e+f x) \cos \left (c+\frac {d}{x}\right )\right )+b d \text {Ci}\left (\frac {d}{x}\right ) \left (\cos (c) \left (d^2 f^2-6 e^2\right )+6 d e f \sin (c)\right )-b d \text {Si}\left (\frac {d}{x}\right ) \left (\sin (c) \left (d^2 f^2-6 e^2\right )-6 d e f \cos (c)\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 224, normalized size = 1.00 \[ \frac {1}{3} \, a f^{2} x^{3} + a e f x^{2} + a e^{2} x + \frac {1}{12} \, {\left (12 \, b d^{2} e f \operatorname {Si}\left (\frac {d}{x}\right ) + {\left (b d^{3} f^{2} - 6 \, b d e^{2}\right )} \operatorname {Ci}\left (\frac {d}{x}\right ) + {\left (b d^{3} f^{2} - 6 \, b d e^{2}\right )} \operatorname {Ci}\left (-\frac {d}{x}\right )\right )} \cos \relax (c) + \frac {1}{6} \, {\left (b d f^{2} x^{2} + 6 \, b d e f x\right )} \cos \left (\frac {c x + d}{x}\right ) + \frac {1}{6} \, {\left (3 \, b d^{2} e f \operatorname {Ci}\left (\frac {d}{x}\right ) + 3 \, b d^{2} e f \operatorname {Ci}\left (-\frac {d}{x}\right ) - {\left (b d^{3} f^{2} - 6 \, b d e^{2}\right )} \operatorname {Si}\left (\frac {d}{x}\right )\right )} \sin \relax (c) + \frac {1}{6} \, {\left (2 \, b f^{2} x^{3} + 6 \, b e f x^{2} - {\left (b d^{2} f^{2} - 6 \, b e^{2}\right )} x\right )} \sin \left (\frac {c x + d}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.55, size = 1264, normalized size = 5.64 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 209, normalized size = 0.93 \[ -d \left (-\frac {a \,e^{2} x}{d}-\frac {a e f \,x^{2}}{d}-\frac {a \,f^{2} x^{3}}{3 d}+b \,e^{2} \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x}{d}-\Si \left (\frac {d}{x}\right ) \sin \relax (c )+\Ci \left (\frac {d}{x}\right ) \cos \relax (c )\right )+2 b d e f \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x^{2}}{2 d^{2}}-\frac {\cos \left (c +\frac {d}{x}\right ) x}{2 d}-\frac {\Si \left (\frac {d}{x}\right ) \cos \relax (c )}{2}-\frac {\Ci \left (\frac {d}{x}\right ) \sin \relax (c )}{2}\right )+b \,d^{2} f^{2} \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x^{3}}{3 d^{3}}-\frac {\cos \left (c +\frac {d}{x}\right ) x^{2}}{6 d^{2}}+\frac {\sin \left (c +\frac {d}{x}\right ) x}{6 d}+\frac {\Si \left (\frac {d}{x}\right ) \sin \relax (c )}{6}-\frac {\Ci \left (\frac {d}{x}\right ) \cos \relax (c )}{6}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.55, size = 258, normalized size = 1.15 \[ \frac {1}{3} \, a f^{2} x^{3} + a e f x^{2} - \frac {1}{2} \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, d}{x}\right ) + {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \cos \relax (c) - {\left (-i \, {\rm Ei}\left (\frac {i \, d}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \sin \relax (c)\right )} d - 2 \, x \sin \left (\frac {c x + d}{x}\right )\right )} b e^{2} + \frac {1}{2} \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, d}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \cos \relax (c) + {\left ({\rm Ei}\left (\frac {i \, d}{x}\right ) + {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \sin \relax (c)\right )} d^{2} + 2 \, d x \cos \left (\frac {c x + d}{x}\right ) + 2 \, x^{2} \sin \left (\frac {c x + d}{x}\right )\right )} b e f + \frac {1}{12} \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, d}{x}\right ) + {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \cos \relax (c) + {\left (i \, {\rm Ei}\left (\frac {i \, d}{x}\right ) - i \, {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \sin \relax (c)\right )} d^{3} + 2 \, d x^{2} \cos \left (\frac {c x + d}{x}\right ) - 2 \, {\left (d^{2} x - 2 \, x^{3}\right )} \sin \left (\frac {c x + d}{x}\right )\right )} b f^{2} + a e^{2} x \]
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 \[ \int {\left (e+f\,x\right )}^2\,\left (a+b\,\sin \left (c+\frac {d}{x}\right )\right ) \,d x \]
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 \[ \int \left (a + b \sin {\left (c + \frac {d}{x} \right )}\right ) \left (e + f x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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