Optimal. Leaf size=233 \[ \frac{a}{e^2 \left (\frac{e}{x}+f\right )}-\frac{a f}{2 e^2 \left (\frac{e}{x}+f\right )^2}-\frac{b d^2 f \sin \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{2 e^4}-\frac{b d \cos \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{e^3}-\frac{b d^2 f \cos \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{2 e^4}+\frac{b d \sin \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{e^3}+\frac{b \sin \left (c+\frac{d}{x}\right )}{e^2 \left (\frac{e}{x}+f\right )}-\frac{b f \sin \left (c+\frac{d}{x}\right )}{2 e^2 \left (\frac{e}{x}+f\right )^2}-\frac{b d f \cos \left (c+\frac{d}{x}\right )}{2 e^3 \left (\frac{e}{x}+f\right )} \]
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Rubi [A] time = 0.487439, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3431, 3317, 3297, 3303, 3299, 3302} \[ \frac{a}{e^2 \left (\frac{e}{x}+f\right )}-\frac{a f}{2 e^2 \left (\frac{e}{x}+f\right )^2}-\frac{b d^2 f \sin \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{2 e^4}-\frac{b d \cos \left (c-\frac{d f}{e}\right ) \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{e^3}-\frac{b d^2 f \cos \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{2 e^4}+\frac{b d \sin \left (c-\frac{d f}{e}\right ) \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right )}{e^3}+\frac{b \sin \left (c+\frac{d}{x}\right )}{e^2 \left (\frac{e}{x}+f\right )}-\frac{b f \sin \left (c+\frac{d}{x}\right )}{2 e^2 \left (\frac{e}{x}+f\right )^2}-\frac{b d f \cos \left (c+\frac{d}{x}\right )}{2 e^3 \left (\frac{e}{x}+f\right )} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 3317
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{a+b \sin \left (c+\frac{d}{x}\right )}{(e+f x)^3} \, dx &=-\operatorname{Subst}\left (\int \left (-\frac{f (a+b \sin (c+d x))}{e (f+e x)^3}+\frac{a+b \sin (c+d x)}{e (f+e x)^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \sin (c+d x)}{(f+e x)^2} \, dx,x,\frac{1}{x}\right )}{e}+\frac{f \operatorname{Subst}\left (\int \frac{a+b \sin (c+d x)}{(f+e x)^3} \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{(f+e x)^2}+\frac{b \sin (c+d x)}{(f+e x)^2}\right ) \, dx,x,\frac{1}{x}\right )}{e}+\frac{f \operatorname{Subst}\left (\int \left (\frac{a}{(f+e x)^3}+\frac{b \sin (c+d x)}{(f+e x)^3}\right ) \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{a f}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{a}{e^2 \left (f+\frac{e}{x}\right )}-\frac{b \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{(f+e x)^2} \, dx,x,\frac{1}{x}\right )}{e}+\frac{(b f) \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{(f+e x)^3} \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{a f}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{a}{e^2 \left (f+\frac{e}{x}\right )}-\frac{b f \sin \left (c+\frac{d}{x}\right )}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{b \sin \left (c+\frac{d}{x}\right )}{e^2 \left (f+\frac{e}{x}\right )}-\frac{(b d) \operatorname{Subst}\left (\int \frac{\cos (c+d x)}{f+e x} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{(b d f) \operatorname{Subst}\left (\int \frac{\cos (c+d x)}{(f+e x)^2} \, dx,x,\frac{1}{x}\right )}{2 e^2}\\ &=-\frac{a f}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{a}{e^2 \left (f+\frac{e}{x}\right )}-\frac{b d f \cos \left (c+\frac{d}{x}\right )}{2 e^3 \left (f+\frac{e}{x}\right )}-\frac{b f \sin \left (c+\frac{d}{x}\right )}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{b \sin \left (c+\frac{d}{x}\right )}{e^2 \left (f+\frac{e}{x}\right )}-\frac{\left (b d^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{f+e x} \, dx,x,\frac{1}{x}\right )}{2 e^3}-\frac{\left (b d \cos \left (c-\frac{d f}{e}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{d f}{e}+d x\right )}{f+e x} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{\left (b d \sin \left (c-\frac{d f}{e}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{d f}{e}+d x\right )}{f+e x} \, dx,x,\frac{1}{x}\right )}{e^2}\\ &=-\frac{a f}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{a}{e^2 \left (f+\frac{e}{x}\right )}-\frac{b d f \cos \left (c+\frac{d}{x}\right )}{2 e^3 \left (f+\frac{e}{x}\right )}-\frac{b d \cos \left (c-\frac{d f}{e}\right ) \text{Ci}\left (\frac{d \left (f+\frac{e}{x}\right )}{e}\right )}{e^3}-\frac{b f \sin \left (c+\frac{d}{x}\right )}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{b \sin \left (c+\frac{d}{x}\right )}{e^2 \left (f+\frac{e}{x}\right )}+\frac{b d \sin \left (c-\frac{d f}{e}\right ) \text{Si}\left (\frac{d \left (f+\frac{e}{x}\right )}{e}\right )}{e^3}-\frac{\left (b d^2 f \cos \left (c-\frac{d f}{e}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{d f}{e}+d x\right )}{f+e x} \, dx,x,\frac{1}{x}\right )}{2 e^3}-\frac{\left (b d^2 f \sin \left (c-\frac{d f}{e}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{d f}{e}+d x\right )}{f+e x} \, dx,x,\frac{1}{x}\right )}{2 e^3}\\ &=-\frac{a f}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{a}{e^2 \left (f+\frac{e}{x}\right )}-\frac{b d f \cos \left (c+\frac{d}{x}\right )}{2 e^3 \left (f+\frac{e}{x}\right )}-\frac{b d \cos \left (c-\frac{d f}{e}\right ) \text{Ci}\left (\frac{d \left (f+\frac{e}{x}\right )}{e}\right )}{e^3}-\frac{b d^2 f \text{Ci}\left (\frac{d \left (f+\frac{e}{x}\right )}{e}\right ) \sin \left (c-\frac{d f}{e}\right )}{2 e^4}-\frac{b f \sin \left (c+\frac{d}{x}\right )}{2 e^2 \left (f+\frac{e}{x}\right )^2}+\frac{b \sin \left (c+\frac{d}{x}\right )}{e^2 \left (f+\frac{e}{x}\right )}-\frac{b d^2 f \cos \left (c-\frac{d f}{e}\right ) \text{Si}\left (\frac{d \left (f+\frac{e}{x}\right )}{e}\right )}{2 e^4}+\frac{b d \sin \left (c-\frac{d f}{e}\right ) \text{Si}\left (\frac{d \left (f+\frac{e}{x}\right )}{e}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 1.8573, size = 151, normalized size = 0.65 \[ -\frac{\frac{e \left (a e^3+b d f^2 x (e+f x) \cos \left (c+\frac{d}{x}\right )-b e f x (2 e+f x) \sin \left (c+\frac{d}{x}\right )\right )}{f (e+f x)^2}+b d \text{CosIntegral}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right ) \left (d f \sin \left (c-\frac{d f}{e}\right )+2 e \cos \left (c-\frac{d f}{e}\right )\right )+b d \text{Si}\left (d \left (\frac{f}{e}+\frac{1}{x}\right )\right ) \left (d f \cos \left (c-\frac{d f}{e}\right )-2 e \sin \left (c-\frac{d f}{e}\right )\right )}{2 e^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 527, normalized size = 2.3 \begin{align*} -d \left ( -{\frac{a}{{e}^{2}} \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-1}}-{\frac{a \left ( ce-df \right ) }{2\,{e}^{2}} \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-2}}+{\frac{ \left ( ce-df \right ) b}{e} \left ( -{\frac{1}{2\,e}\sin \left ( c+{\frac{d}{x}} \right ) \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-2}}+{\frac{1}{2\,e} \left ( -{\frac{1}{e}\cos \left ( c+{\frac{d}{x}} \right ) \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-1}}-{\frac{1}{e} \left ({\frac{1}{e}{\it Si} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \cos \left ({\frac{-ce+df}{e}} \right ) }-{\frac{1}{e}{\it Ci} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \sin \left ({\frac{-ce+df}{e}} \right ) } \right ) } \right ) } \right ) }+{\frac{b}{e} \left ( -{\frac{1}{e}\sin \left ( c+{\frac{d}{x}} \right ) \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-1}}+{\frac{1}{e} \left ({\frac{1}{e}{\it Si} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \sin \left ({\frac{-ce+df}{e}} \right ) }+{\frac{1}{e}{\it Ci} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \cos \left ({\frac{-ce+df}{e}} \right ) } \right ) } \right ) }+{\frac{ca}{2\,e} \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-2}}-cb \left ( -{\frac{1}{2\,e}\sin \left ( c+{\frac{d}{x}} \right ) \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-2}}+{\frac{1}{2\,e} \left ( -{\frac{1}{e}\cos \left ( c+{\frac{d}{x}} \right ) \left ( e \left ( c+{\frac{d}{x}} \right ) -ce+df \right ) ^{-1}}-{\frac{1}{e} \left ({\frac{1}{e}{\it Si} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \cos \left ({\frac{-ce+df}{e}} \right ) }-{\frac{1}{e}{\it Ci} \left ({\frac{d}{x}}+c+{\frac{-ce+df}{e}} \right ) \sin \left ({\frac{-ce+df}{e}} \right ) } \right ) } \right ) } \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b{\left (\int \frac{\sin \left (\frac{c x + d}{x}\right )}{2 \,{\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}}\,{d x} + \int \frac{\sin \left (\frac{c x + d}{x}\right )}{2 \,{\left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )} \cos \left (\frac{c x + d}{x}\right )^{2} +{\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )} \sin \left (\frac{c x + d}{x}\right )^{2}\right )}}\,{d x}\right )} - \frac{a}{2 \,{\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59395, size = 959, normalized size = 4.12 \begin{align*} -\frac{2 \, a e^{4} + 2 \,{\left ({\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname{Ci}\left (\frac{d f x + d e}{e x}\right ) +{\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname{Ci}\left (-\frac{d f x + d e}{e x}\right ) +{\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname{Si}\left (\frac{d f x + d e}{e x}\right )\right )} \cos \left (-\frac{c e - d f}{e}\right ) + 2 \,{\left (b d e f^{3} x^{2} + b d e^{2} f^{2} x\right )} \cos \left (\frac{c x + d}{x}\right ) -{\left ({\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname{Ci}\left (\frac{d f x + d e}{e x}\right ) +{\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname{Ci}\left (-\frac{d f x + d e}{e x}\right ) - 4 \,{\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname{Si}\left (\frac{d f x + d e}{e x}\right )\right )} \sin \left (-\frac{c e - d f}{e}\right ) - 2 \,{\left (b e^{2} f^{2} x^{2} + 2 \, b e^{3} f x\right )} \sin \left (\frac{c x + d}{x}\right )}{4 \,{\left (e^{4} f^{3} x^{2} + 2 \, e^{5} f^{2} x + e^{6} f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{b \sin \left (c + \frac{d}{x}\right ) + a}{{\left (f x + e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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