Optimal. Leaf size=254 \[ a^2 e x+\frac{1}{2} a^2 f x^2+a b d^2 f \sin (c) \text{CosIntegral}\left (\frac{d}{x}\right )-2 a b d e \cos (c) \text{CosIntegral}\left (\frac{d}{x}\right )+a b d^2 f \cos (c) \text{Si}\left (\frac{d}{x}\right )+2 a b d e \sin (c) \text{Si}\left (\frac{d}{x}\right )+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+a b d f x \cos \left (c+\frac{d}{x}\right )-b^2 d^2 f \cos (2 c) \text{CosIntegral}\left (\frac{2 d}{x}\right )-b^2 d e \sin (2 c) \text{CosIntegral}\left (\frac{2 d}{x}\right )+b^2 d^2 f \sin (2 c) \text{Si}\left (\frac{2 d}{x}\right )-b^2 d e \cos (2 c) \text{Si}\left (\frac{2 d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )+b^2 d f x \sin \left (c+\frac{d}{x}\right ) \cos \left (c+\frac{d}{x}\right ) \]
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Rubi [A] time = 0.616003, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {3431, 3317, 3297, 3303, 3299, 3302, 3314, 29, 3312, 3313, 12} \[ a^2 e x+\frac{1}{2} a^2 f x^2+a b d^2 f \sin (c) \text{CosIntegral}\left (\frac{d}{x}\right )-2 a b d e \cos (c) \text{CosIntegral}\left (\frac{d}{x}\right )+a b d^2 f \cos (c) \text{Si}\left (\frac{d}{x}\right )+2 a b d e \sin (c) \text{Si}\left (\frac{d}{x}\right )+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+a b d f x \cos \left (c+\frac{d}{x}\right )-b^2 d^2 f \cos (2 c) \text{CosIntegral}\left (\frac{2 d}{x}\right )-b^2 d e \sin (2 c) \text{CosIntegral}\left (\frac{2 d}{x}\right )+b^2 d^2 f \sin (2 c) \text{Si}\left (\frac{2 d}{x}\right )-b^2 d e \cos (2 c) \text{Si}\left (\frac{2 d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )+b^2 d f x \sin \left (c+\frac{d}{x}\right ) \cos \left (c+\frac{d}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3431
Rule 3317
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rule 3314
Rule 29
Rule 3312
Rule 3313
Rule 12
Rubi steps
\begin{align*} \int (e+f x) \left (a+b \sin \left (c+\frac{d}{x}\right )\right )^2 \, dx &=-\operatorname{Subst}\left (\int \left (\frac{f (a+b \sin (c+d x))^2}{x^3}+\frac{e (a+b \sin (c+d x))^2}{x^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\left (e \operatorname{Subst}\left (\int \frac{(a+b \sin (c+d x))^2}{x^2} \, dx,x,\frac{1}{x}\right )\right )-f \operatorname{Subst}\left (\int \frac{(a+b \sin (c+d x))^2}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=-\left (e \operatorname{Subst}\left (\int \left (\frac{a^2}{x^2}+\frac{2 a b \sin (c+d x)}{x^2}+\frac{b^2 \sin ^2(c+d x)}{x^2}\right ) \, dx,x,\frac{1}{x}\right )\right )-f \operatorname{Subst}\left (\int \left (\frac{a^2}{x^3}+\frac{2 a b \sin (c+d x)}{x^3}+\frac{b^2 \sin ^2(c+d x)}{x^3}\right ) \, dx,x,\frac{1}{x}\right )\\ &=a^2 e x+\frac{1}{2} a^2 f x^2-(2 a b e) \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{x^2} \, dx,x,\frac{1}{x}\right )-\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(c+d x)}{x^2} \, dx,x,\frac{1}{x}\right )-(2 a b f) \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{x^3} \, dx,x,\frac{1}{x}\right )-\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(c+d x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=a^2 e x+\frac{1}{2} a^2 f x^2+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+b^2 d f x \cos \left (c+\frac{d}{x}\right ) \sin \left (c+\frac{d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )-(2 a b d e) \operatorname{Subst}\left (\int \frac{\cos (c+d x)}{x} \, dx,x,\frac{1}{x}\right )-\left (2 b^2 d e\right ) \operatorname{Subst}\left (\int \frac{\sin (2 c+2 d x)}{2 x} \, dx,x,\frac{1}{x}\right )-(a b d f) \operatorname{Subst}\left (\int \frac{\cos (c+d x)}{x^2} \, dx,x,\frac{1}{x}\right )-\left (b^2 d^2 f\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )+\left (2 b^2 d^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(c+d x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=a^2 e x+\frac{1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac{d}{x}\right )+b^2 d^2 f \log (x)+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+b^2 d f x \cos \left (c+\frac{d}{x}\right ) \sin \left (c+\frac{d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )-\left (b^2 d e\right ) \operatorname{Subst}\left (\int \frac{\sin (2 c+2 d x)}{x} \, dx,x,\frac{1}{x}\right )+\left (a b d^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{x} \, dx,x,\frac{1}{x}\right )+\left (2 b^2 d^2 f\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 c+2 d x)}{2 x}\right ) \, dx,x,\frac{1}{x}\right )-(2 a b d e \cos (c)) \operatorname{Subst}\left (\int \frac{\cos (d x)}{x} \, dx,x,\frac{1}{x}\right )+(2 a b d e \sin (c)) \operatorname{Subst}\left (\int \frac{\sin (d x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=a^2 e x+\frac{1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac{d}{x}\right )-2 a b d e \cos (c) \text{Ci}\left (\frac{d}{x}\right )+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+b^2 d f x \cos \left (c+\frac{d}{x}\right ) \sin \left (c+\frac{d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )+2 a b d e \sin (c) \text{Si}\left (\frac{d}{x}\right )-\left (b^2 d^2 f\right ) \operatorname{Subst}\left (\int \frac{\cos (2 c+2 d x)}{x} \, dx,x,\frac{1}{x}\right )+\left (a b d^2 f \cos (c)\right ) \operatorname{Subst}\left (\int \frac{\sin (d x)}{x} \, dx,x,\frac{1}{x}\right )-\left (b^2 d e \cos (2 c)\right ) \operatorname{Subst}\left (\int \frac{\sin (2 d x)}{x} \, dx,x,\frac{1}{x}\right )+\left (a b d^2 f \sin (c)\right ) \operatorname{Subst}\left (\int \frac{\cos (d x)}{x} \, dx,x,\frac{1}{x}\right )-\left (b^2 d e \sin (2 c)\right ) \operatorname{Subst}\left (\int \frac{\cos (2 d x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=a^2 e x+\frac{1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac{d}{x}\right )-2 a b d e \cos (c) \text{Ci}\left (\frac{d}{x}\right )+a b d^2 f \text{Ci}\left (\frac{d}{x}\right ) \sin (c)-b^2 d e \text{Ci}\left (\frac{2 d}{x}\right ) \sin (2 c)+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+b^2 d f x \cos \left (c+\frac{d}{x}\right ) \sin \left (c+\frac{d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )+a b d^2 f \cos (c) \text{Si}\left (\frac{d}{x}\right )+2 a b d e \sin (c) \text{Si}\left (\frac{d}{x}\right )-b^2 d e \cos (2 c) \text{Si}\left (\frac{2 d}{x}\right )-\left (b^2 d^2 f \cos (2 c)\right ) \operatorname{Subst}\left (\int \frac{\cos (2 d x)}{x} \, dx,x,\frac{1}{x}\right )+\left (b^2 d^2 f \sin (2 c)\right ) \operatorname{Subst}\left (\int \frac{\sin (2 d x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=a^2 e x+\frac{1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac{d}{x}\right )-2 a b d e \cos (c) \text{Ci}\left (\frac{d}{x}\right )-b^2 d^2 f \cos (2 c) \text{Ci}\left (\frac{2 d}{x}\right )+a b d^2 f \text{Ci}\left (\frac{d}{x}\right ) \sin (c)-b^2 d e \text{Ci}\left (\frac{2 d}{x}\right ) \sin (2 c)+2 a b e x \sin \left (c+\frac{d}{x}\right )+a b f x^2 \sin \left (c+\frac{d}{x}\right )+b^2 d f x \cos \left (c+\frac{d}{x}\right ) \sin \left (c+\frac{d}{x}\right )+b^2 e x \sin ^2\left (c+\frac{d}{x}\right )+\frac{1}{2} b^2 f x^2 \sin ^2\left (c+\frac{d}{x}\right )+a b d^2 f \cos (c) \text{Si}\left (\frac{d}{x}\right )+2 a b d e \sin (c) \text{Si}\left (\frac{d}{x}\right )-b^2 d e \cos (2 c) \text{Si}\left (\frac{2 d}{x}\right )+b^2 d^2 f \sin (2 c) \text{Si}\left (\frac{2 d}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.560838, size = 252, normalized size = 0.99 \[ \frac{1}{4} \left (4 a^2 e x+2 a^2 f x^2+4 a b d \text{CosIntegral}\left (\frac{d}{x}\right ) (d f \sin (c)-2 e \cos (c))+4 a b d^2 f \cos (c) \text{Si}\left (\frac{d}{x}\right )+8 a b d e \sin (c) \text{Si}\left (\frac{d}{x}\right )+8 a b e x \sin \left (c+\frac{d}{x}\right )+4 a b f x^2 \sin \left (c+\frac{d}{x}\right )+4 a b d f x \cos \left (c+\frac{d}{x}\right )-4 b^2 d \text{CosIntegral}\left (\frac{2 d}{x}\right ) (d f \cos (2 c)+e \sin (2 c))+4 b^2 d^2 f \sin (2 c) \text{Si}\left (\frac{2 d}{x}\right )-4 b^2 d e \cos (2 c) \text{Si}\left (\frac{2 d}{x}\right )-2 b^2 e x \cos \left (2 \left (c+\frac{d}{x}\right )\right )-b^2 f x^2 \cos \left (2 \left (c+\frac{d}{x}\right )\right )+2 b^2 d f x \sin \left (2 \left (c+\frac{d}{x}\right )\right )+2 b^2 e x+b^2 f x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 265, normalized size = 1. \begin{align*} -d \left ( -{\frac{{a}^{2}ex}{d}}-{\frac{{a}^{2}f{x}^{2}}{2\,d}}+2\,abe \left ( -{\frac{x}{d}\sin \left ( c+{\frac{d}{x}} \right ) }-{\it Si} \left ({\frac{d}{x}} \right ) \sin \left ( c \right ) +{\it Ci} \left ({\frac{d}{x}} \right ) \cos \left ( c \right ) \right ) +2\,abfd \left ( -1/2\,{\frac{{x}^{2}}{{d}^{2}}\sin \left ( c+{\frac{d}{x}} \right ) }-1/2\,{\frac{x}{d}\cos \left ( c+{\frac{d}{x}} \right ) }-1/2\,{\it Si} \left ({\frac{d}{x}} \right ) \cos \left ( c \right ) -1/2\,{\it Ci} \left ({\frac{d}{x}} \right ) \sin \left ( c \right ) \right ) -{\frac{{b}^{2}ex}{2\,d}}-{\frac{{b}^{2}e}{4} \left ( -2\,{\frac{x}{d}\cos \left ( 2\,{\frac{d}{x}}+2\,c \right ) }-4\,{\it Si} \left ( 2\,{\frac{d}{x}} \right ) \cos \left ( 2\,c \right ) -4\,{\it Ci} \left ( 2\,{\frac{d}{x}} \right ) \sin \left ( 2\,c \right ) \right ) }-{\frac{{b}^{2}f{x}^{2}}{4\,d}}-{\frac{{b}^{2}fd}{4} \left ( -{\frac{{x}^{2}}{{d}^{2}}\cos \left ( 2\,{\frac{d}{x}}+2\,c \right ) }+2\,{\frac{x}{d}\sin \left ( 2\,{\frac{d}{x}}+2\,c \right ) }+4\,{\it Si} \left ( 2\,{\frac{d}{x}} \right ) \sin \left ( 2\,c \right ) -4\,{\it Ci} \left ( 2\,{\frac{d}{x}} \right ) \cos \left ( 2\,c \right ) \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.41808, size = 435, normalized size = 1.71 \begin{align*} \frac{1}{2} \, a^{2} f x^{2} -{\left ({\left ({\left ({\rm Ei}\left (\frac{i \, d}{x}\right ) +{\rm Ei}\left (-\frac{i \, d}{x}\right )\right )} \cos \left (c\right ) -{\left (-i \,{\rm Ei}\left (\frac{i \, d}{x}\right ) + i \,{\rm Ei}\left (-\frac{i \, d}{x}\right )\right )} \sin \left (c\right )\right )} d - 2 \, x \sin \left (\frac{c x + d}{x}\right )\right )} a b e - \frac{1}{2} \,{\left ({\left ({\left (-i \,{\rm Ei}\left (\frac{2 i \, d}{x}\right ) + i \,{\rm Ei}\left (-\frac{2 i \, d}{x}\right )\right )} \cos \left (2 \, c\right ) +{\left ({\rm Ei}\left (\frac{2 i \, d}{x}\right ) +{\rm Ei}\left (-\frac{2 i \, d}{x}\right )\right )} \sin \left (2 \, c\right )\right )} d + x \cos \left (\frac{2 \,{\left (c x + d\right )}}{x}\right ) - x\right )} b^{2} e + \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 \left (c\right ) +{\left ({\rm Ei}\left (\frac{i \, d}{x}\right ) +{\rm Ei}\left (-\frac{i \, d}{x}\right )\right )} \sin \left (c\right )\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 )} a b f - \frac{1}{4} \,{\left ({\left (2 \,{\left ({\rm Ei}\left (\frac{2 i \, d}{x}\right ) +{\rm Ei}\left (-\frac{2 i \, d}{x}\right )\right )} \cos \left (2 \, c\right ) -{\left (-2 i \,{\rm Ei}\left (\frac{2 i \, d}{x}\right ) + 2 i \,{\rm Ei}\left (-\frac{2 i \, d}{x}\right )\right )} \sin \left (2 \, c\right )\right )} d^{2} + x^{2} \cos \left (\frac{2 \,{\left (c x + d\right )}}{x}\right ) - 2 \, d x \sin \left (\frac{2 \,{\left (c x + d\right )}}{x}\right ) - x^{2}\right )} b^{2} f + a^{2} e x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63403, size = 828, normalized size = 3.26 \begin{align*} a b d f x \cos \left (\frac{c x + d}{x}\right ) + \frac{1}{2} \,{\left (a^{2} + b^{2}\right )} f x^{2} +{\left (a^{2} + b^{2}\right )} e x - \frac{1}{2} \,{\left (b^{2} f x^{2} + 2 \, b^{2} e x\right )} \cos \left (\frac{c x + d}{x}\right )^{2} - \frac{1}{2} \,{\left (b^{2} d^{2} f \operatorname{Ci}\left (\frac{2 \, d}{x}\right ) + b^{2} d^{2} f \operatorname{Ci}\left (-\frac{2 \, d}{x}\right ) + 2 \, b^{2} d e \operatorname{Si}\left (\frac{2 \, d}{x}\right )\right )} \cos \left (2 \, c\right ) +{\left (a b d^{2} f \operatorname{Si}\left (\frac{d}{x}\right ) - a b d e \operatorname{Ci}\left (\frac{d}{x}\right ) - a b d e \operatorname{Ci}\left (-\frac{d}{x}\right )\right )} \cos \left (c\right ) + \frac{1}{2} \,{\left (2 \, b^{2} d^{2} f \operatorname{Si}\left (\frac{2 \, d}{x}\right ) - b^{2} d e \operatorname{Ci}\left (\frac{2 \, d}{x}\right ) - b^{2} d e \operatorname{Ci}\left (-\frac{2 \, d}{x}\right )\right )} \sin \left (2 \, c\right ) + \frac{1}{2} \,{\left (a b d^{2} f \operatorname{Ci}\left (\frac{d}{x}\right ) + a b d^{2} f \operatorname{Ci}\left (-\frac{d}{x}\right ) + 4 \, a b d e \operatorname{Si}\left (\frac{d}{x}\right )\right )} \sin \left (c\right ) +{\left (b^{2} d f x \cos \left (\frac{c x + d}{x}\right ) + a b f x^{2} + 2 \, a b e x\right )} \sin \left (\frac{c x + d}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (c + \frac{d}{x} \right )}\right )^{2} \left (e + f x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b \sin \left (c + \frac{d}{x}\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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