Optimal. Leaf size=123 \[ -\frac {a}{2 d (c+d x)^2}-\frac {a f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {a f^2 \text {Ci}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {a \sin (e+f x)}{2 d (c+d x)^2}-\frac {a f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{2 d^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378,
3384, 3380, 3383} \begin {gather*} -\frac {a f^2 \text {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {a f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {a f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {a \sin (e+f x)}{2 d (c+d x)^2}-\frac {a}{2 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3398
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{(c+d x)^3} \, dx &=\int \left (\frac {a}{(c+d x)^3}+\frac {a \sin (e+f x)}{(c+d x)^3}\right ) \, dx\\ &=-\frac {a}{2 d (c+d x)^2}+a \int \frac {\sin (e+f x)}{(c+d x)^3} \, dx\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a \sin (e+f x)}{2 d (c+d x)^2}+\frac {(a f) \int \frac {\cos (e+f x)}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {a \sin (e+f x)}{2 d (c+d x)^2}-\frac {\left (a f^2\right ) \int \frac {\sin (e+f x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {a \sin (e+f x)}{2 d (c+d x)^2}-\frac {\left (a f^2 \cos \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sin \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}-\frac {\left (a f^2 \sin \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cos \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}\\ &=-\frac {a}{2 d (c+d x)^2}-\frac {a f \cos (e+f x)}{2 d^2 (c+d x)}-\frac {a f^2 \text {Ci}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {a \sin (e+f x)}{2 d (c+d x)^2}-\frac {a f^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 104, normalized size = 0.85 \begin {gather*} -\frac {a \left (f^2 (c+d x)^2 \text {Ci}\left (f \left (\frac {c}{d}+x\right )\right ) \sin \left (e-\frac {c f}{d}\right )+d (f (c+d x) \cos (e+f x)+d (1+\sin (e+f x)))+f^2 (c+d x)^2 \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{2 d^3 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 177, normalized size = 1.44
method | result | size |
derivativedivides | \(\frac {a \,f^{3} \left (-\frac {\sin \left (f x +e \right )}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\sinIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\cosineIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}}{2 d}\right )-\frac {a \,f^{3}}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}}{f}\) | \(177\) |
default | \(\frac {a \,f^{3} \left (-\frac {\sin \left (f x +e \right )}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {\cos \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {\frac {\sinIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\cosineIntegral \left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}}{d}}{2 d}\right )-\frac {a \,f^{3}}{2 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}}{f}\) | \(177\) |
risch | \(-\frac {a}{2 d \left (d x +c \right )^{2}}+\frac {i f^{2} a \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \expIntegral \left (1, i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{4 d^{3}}-\frac {i f^{2} a \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \expIntegral \left (1, -i f x -i e -\frac {i c f -i d e}{d}\right )}{4 d^{3}}+\frac {i a \left (-2 i d^{3} f^{3} x^{3}-6 i c \,d^{2} f^{3} x^{2}-6 i c^{2} d \,f^{3} x -2 i c^{3} f^{3}\right ) \cos \left (f x +e \right )}{4 d^{2} \left (d x +c \right )^{2} \left (-d^{2} x^{2} f^{2}-2 c d \,f^{2} x -c^{2} f^{2}\right )}-\frac {a \left (-2 d^{2} x^{2} f^{2}-4 c d \,f^{2} x -2 c^{2} f^{2}\right ) \sin \left (f x +e \right )}{4 d \left (d x +c \right )^{2} \left (-d^{2} x^{2} f^{2}-2 c d \,f^{2} x -c^{2} f^{2}\right )}\) | \(292\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.41, size = 279, normalized size = 2.27 \begin {gather*} -\frac {\frac {a f^{3}}{{\left (f x + e\right )}^{2} d^{3} + c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}} - \frac {{\left (f^{3} {\left (-i \, E_{3}\left (\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right ) + i \, E_{3}\left (-\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right )\right )} \cos \left (\frac {c f - d e}{d}\right ) + f^{3} {\left (E_{3}\left (\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right ) + E_{3}\left (-\frac {i \, {\left (f x + e\right )} d + i \, c f - i \, d e}{d}\right )\right )} \sin \left (\frac {c f - d e}{d}\right )\right )} a}{{\left (f x + e\right )}^{2} d^{3} + c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 231, normalized size = 1.88 \begin {gather*} -\frac {2 \, a d^{2} \sin \left (f x + e\right ) + 2 \, a d^{2} + 2 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} \cos \left (-\frac {c f - d e}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) + 2 \, {\left (a d^{2} f x + a c d f\right )} \cos \left (f x + e\right ) + {\left ({\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) + {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )} \operatorname {Ci}\left (-\frac {d f x + c f}{d}\right )\right )} \sin \left (-\frac {c f - d e}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end {gather*}
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 \begin {gather*} a \left (\int \frac {\sin {\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {1}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.98, size = 6157, normalized size = 50.06 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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