Optimal. Leaf size=233 \[ \frac {a}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a f}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {b d^2 f \sin \left (c-\frac {d f}{e}\right ) \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{2 e^4}-\frac {b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{2 e^4}+\frac {b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b d f \cos \left (c+\frac {d}{x}\right )}{2 e^3 \left (\frac {e}{x}+f\right )}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {b f \sin \left (c+\frac {d}{x}\right )}{2 e^2 \left (\frac {e}{x}+f\right )^2} \]
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Rubi [A] time = 0.49, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \frac {a}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a f}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {b d^2 f \sin \left (c-\frac {d f}{e}\right ) \text {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{2 e^4}-\frac {b d \cos \left (c-\frac {d f}{e}\right ) \text {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{2 e^4}+\frac {b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}+\frac {b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {b f \sin \left (c+\frac {d}{x}\right )}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {b d f \cos \left (c+\frac {d}{x}\right )}{2 e^3 \left (\frac {e}{x}+f\right )} \]
Antiderivative was successfully verified.
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Rule 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)^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*}
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Mathematica [A] time = 1.89, size = 151, normalized size = 0.65 \[ -\frac {\frac {e \left (a e^3+b d f^2 x (e+f x) \cos \left (c+\frac {d}{x}\right )-b e f x (2 e+f x) \sin \left (c+\frac {d}{x}\right )\right )}{f (e+f x)^2}+b d \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \left (d f \sin \left (c-\frac {d f}{e}\right )+2 e \cos \left (c-\frac {d f}{e}\right )\right )+b d \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \left (d f \cos \left (c-\frac {d f}{e}\right )-2 e \sin \left (c-\frac {d f}{e}\right )\right )}{2 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 429, normalized size = 1.84 \[ -\frac {2 \, a e^{4} + 2 \, {\left ({\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) + {\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname {Ci}\left (-\frac {d f x + d e}{e x}\right ) + {\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname {Si}\left (\frac {d f x + d e}{e x}\right )\right )} \cos \left (-\frac {c e - d f}{e}\right ) + 2 \, {\left (b d e f^{3} x^{2} + b d e^{2} f^{2} x\right )} \cos \left (\frac {c x + d}{x}\right ) - {\left ({\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) + {\left (b d^{2} f^{4} x^{2} + 2 \, b d^{2} e f^{3} x + b d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (-\frac {d f x + d e}{e x}\right ) - 4 \, {\left (b d e f^{3} x^{2} + 2 \, b d e^{2} f^{2} x + b d e^{3} f\right )} \operatorname {Si}\left (\frac {d f x + d e}{e x}\right )\right )} \sin \left (-\frac {c e - d f}{e}\right ) - 2 \, {\left (b e^{2} f^{2} x^{2} + 2 \, b e^{3} f x\right )} \sin \left (\frac {c x + d}{x}\right )}{4 \, {\left (e^{4} f^{3} x^{2} + 2 \, e^{5} f^{2} x + e^{6} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 1502, normalized size = 6.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 527, normalized size = 2.26 \[ -d \left (-\frac {a}{e^{2} \left (e \left (c +\frac {d}{x}\right )-c e +d f \right )}-\frac {a \left (c e -d f \right )}{2 e^{2} \left (e \left (c +\frac {d}{x}\right )-c e +d f \right )^{2}}+\frac {\left (c e -d f \right ) b \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (e \left (c +\frac {d}{x}\right )-c e +d f \right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (e \left (c +\frac {d}{x}\right )-c e +d f \right ) e}-\frac {\frac {\Si \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\Ci \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )}{e}+\frac {b \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{\left (e \left (c +\frac {d}{x}\right )-c e +d f \right ) e}+\frac {\frac {\Si \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}+\frac {\Ci \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}}{e}\right )}{e}+\frac {c a}{2 \left (e \left (c +\frac {d}{x}\right )-c e +d f \right )^{2} e}-c b \left (-\frac {\sin \left (c +\frac {d}{x}\right )}{2 \left (e \left (c +\frac {d}{x}\right )-c e +d f \right )^{2} e}+\frac {-\frac {\cos \left (c +\frac {d}{x}\right )}{\left (e \left (c +\frac {d}{x}\right )-c e +d f \right ) e}-\frac {\frac {\Si \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \cos \left (\frac {-c e +d f}{e}\right )}{e}-\frac {\Ci \left (\frac {d}{x}+c +\frac {-c e +d f}{e}\right ) \sin \left (\frac {-c e +d f}{e}\right )}{e}}{e}}{2 e}\right )\right ) \]
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 \[ b {\left (\int \frac {\sin \left (\frac {c x + d}{x}\right )}{2 \, {\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )}}\,{d x} + \int \frac {\sin \left (\frac {c x + d}{x}\right )}{2 \, {\left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )} \cos \left (\frac {c x + d}{x}\right )^{2} + {\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )} \sin \left (\frac {c x + d}{x}\right )^{2}\right )}}\,{d x}\right )} - \frac {a}{2 \, {\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \]
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 \[ \int \frac {a+b\,\sin \left (c+\frac {d}{x}\right )}{{\left (e+f\,x\right )}^3} \,d x \]
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 \[ \int \frac {a + b \sin {\left (c + \frac {d}{x} \right )}}{\left (e + f x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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