Optimal. Leaf size=94 \[ \frac {a}{e \left (\frac {e}{x}+f\right )}-\frac {b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^2}+\frac {b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^2}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e \left (\frac {e}{x}+f\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3431, 3317, 3297, 3303, 3299, 3302} \[ \frac {a}{e \left (\frac {e}{x}+f\right )}-\frac {b d \cos \left (c-\frac {d f}{e}\right ) \text {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^2}+\frac {b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^2}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e \left (\frac {e}{x}+f\right )} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3317
Rule 3431
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+\frac {d}{x}\right )}{(e+f x)^2} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \sin (c+d x)}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )\\ &=-\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 )\\ &=\frac {a}{e \left (f+\frac {e}{x}\right )}-b \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a}{e \left (f+\frac {e}{x}\right )}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e \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}\\ &=\frac {a}{e \left (f+\frac {e}{x}\right )}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e \left (f+\frac {e}{x}\right )}-\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}+\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}\\ &=\frac {a}{e \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^2}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e \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^2}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 85, normalized size = 0.90 \[ \frac {\frac {e \left (b f x \sin \left (c+\frac {d}{x}\right )-a e\right )}{f (e+f x)}-b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 164, normalized size = 1.74 \[ \frac {2 \, b e f x \sin \left (\frac {c x + d}{x}\right ) - 2 \, a e^{2} - 2 \, {\left (b d f^{2} x + b d e f\right )} \sin \left (-\frac {c e - d f}{e}\right ) \operatorname {Si}\left (\frac {d f x + d e}{e x}\right ) - {\left ({\left (b d f^{2} x + b d e f\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) + {\left (b d f^{2} x + b d e f\right )} \operatorname {Ci}\left (-\frac {d f x + d e}{e x}\right )\right )} \cos \left (-\frac {c e - d f}{e}\right )}{2 \, {\left (e^{2} f^{2} x + e^{3} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 347, normalized size = 3.69 \[ -\frac {b d^{3} f \cos \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) \operatorname {Ci}\left ({\left (d f - c e + \frac {{\left (c x + d\right )} e}{x}\right )} e^{\left (-1\right )}\right ) - b c d^{2} \cos \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) \operatorname {Ci}\left ({\left (d f - c e + \frac {{\left (c x + d\right )} e}{x}\right )} e^{\left (-1\right )}\right ) e + b d^{3} f \sin \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) \operatorname {Si}\left (-{\left (d f - c e + \frac {{\left (c x + d\right )} e}{x}\right )} e^{\left (-1\right )}\right ) - b c d^{2} e \sin \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) \operatorname {Si}\left (-{\left (d f - c e + \frac {{\left (c x + d\right )} e}{x}\right )} e^{\left (-1\right )}\right ) + \frac {{\left (c x + d\right )} b d^{2} \cos \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) \operatorname {Ci}\left ({\left (d f - c e + \frac {{\left (c x + d\right )} e}{x}\right )} e^{\left (-1\right )}\right ) e}{x} + \frac {{\left (c x + d\right )} b d^{2} e \sin \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) \operatorname {Si}\left (-{\left (d f - c e + \frac {{\left (c x + d\right )} e}{x}\right )} e^{\left (-1\right )}\right )}{x} - b d^{2} e \sin \left (\frac {c x + d}{x}\right ) - a d^{2} e}{{\left (d f e^{2} - c e^{3} + \frac {{\left (c x + d\right )} e^{3}}{x}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 144, normalized size = 1.53 \[ -d \left (-\frac {a}{\left (e \left (c +\frac {d}{x}\right )-c e +d f \right ) e}+b \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{\left (e \left (c +\frac {d}{x}\right )-c e +d f \right ) e}+\frac {\frac {\Si \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}+\frac {\Ci \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}}{e}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\int \frac {\sin \left (\frac {c x + d}{x}\right )}{2 \, {\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )}}\,{d x} + \int \frac {\sin \left (\frac {c x + d}{x}\right )}{2 \, {\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} \cos \left (\frac {c x + d}{x}\right )^{2} + {\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} \sin \left (\frac {c x + d}{x}\right )^{2}\right )}}\,{d x}\right )} - \frac {a}{f^{2} x + e f} \]
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 \[ \int \frac {a+b\,\sin \left (c+\frac {d}{x}\right )}{{\left (e+f\,x\right )}^2} \,d x \]
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 \[ \int \frac {a + b \sin {\left (c + \frac {d}{x} \right )}}{\left (e + f x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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