Optimal. Leaf size=189 \[ -\frac {d \cos (c+d x)}{2 a x}-\frac {b d \cos (c) \text {Ci}(d x)}{a^2}+\frac {b^2 \text {Ci}(d x) \sin (c)}{a^3}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a}-\frac {b^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{2 a x^2}+\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}+\frac {b d \sin (c) \text {Si}(d x)}{a^2}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3} \]
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Rubi [A]
time = 0.34, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378,
3384, 3380, 3383} \begin {gather*} \frac {b^2 \sin (c) \text {CosIntegral}(d x)}{a^3}-\frac {b^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {b d \cos (c) \text {CosIntegral}(d x)}{a^2}+\frac {b d \sin (c) \text {Si}(d x)}{a^2}+\frac {b \sin (c+d x)}{a^2 x}-\frac {d^2 \sin (c) \text {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 6874
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x^3 (a+b x)} \, dx &=\int \left (\frac {\sin (c+d x)}{a x^3}-\frac {b \sin (c+d x)}{a^2 x^2}+\frac {b^2 \sin (c+d x)}{a^3 x}-\frac {b^3 \sin (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^3} \, dx}{a}-\frac {b \int \frac {\sin (c+d x)}{x^2} \, dx}{a^2}+\frac {b^2 \int \frac {\sin (c+d x)}{x} \, dx}{a^3}-\frac {b^3 \int \frac {\sin (c+d x)}{a+b x} \, dx}{a^3}\\ &=-\frac {\sin (c+d x)}{2 a x^2}+\frac {b \sin (c+d x)}{a^2 x}+\frac {d \int \frac {\cos (c+d x)}{x^2} \, dx}{2 a}-\frac {(b d) \int \frac {\cos (c+d x)}{x} \, dx}{a^2}+\frac {\left (b^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{a^3}-\frac {\left (b^3 \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^3}+\frac {\left (b^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{a^3}-\frac {\left (b^3 \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^3}\\ &=-\frac {d \cos (c+d x)}{2 a x}+\frac {b^2 \text {Ci}(d x) \sin (c)}{a^3}-\frac {b^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{2 a x^2}+\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{2 a}-\frac {(b d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{a^2}+\frac {(b d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{a^2}\\ &=-\frac {d \cos (c+d x)}{2 a x}-\frac {b d \cos (c) \text {Ci}(d x)}{a^2}+\frac {b^2 \text {Ci}(d x) \sin (c)}{a^3}-\frac {b^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{2 a x^2}+\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}+\frac {b d \sin (c) \text {Si}(d x)}{a^2}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{2 a}-\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{2 a}\\ &=-\frac {d \cos (c+d x)}{2 a x}-\frac {b d \cos (c) \text {Ci}(d x)}{a^2}+\frac {b^2 \text {Ci}(d x) \sin (c)}{a^3}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a}-\frac {b^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{2 a x^2}+\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}+\frac {b d \sin (c) \text {Si}(d x)}{a^2}-\frac {b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 176, normalized size = 0.93 \begin {gather*} -\frac {a^2 d x \cos (c+d x)+x^2 \text {Ci}(d x) \left (2 a b d \cos (c)+\left (-2 b^2+a^2 d^2\right ) \sin (c)\right )+2 b^2 x^2 \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+a^2 \sin (c+d x)-2 a b x \sin (c+d x)-2 b^2 x^2 \cos (c) \text {Si}(d x)+a^2 d^2 x^2 \cos (c) \text {Si}(d x)-2 a b d x^2 \sin (c) \text {Si}(d x)+2 b^2 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 202, normalized size = 1.07
method | result | size |
derivativedivides | \(d^{2} \left (-\frac {b \left (-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )\right )}{a^{2} d}+\frac {b^{2} \left (\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )\right )}{a^{3} d^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}}{a}-\frac {b^{3} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2} a^{3}}\right )\) | \(202\) |
default | \(d^{2} \left (-\frac {b \left (-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )\right )}{a^{2} d}+\frac {b^{2} \left (\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )\right )}{a^{3} d^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}}{a}-\frac {b^{3} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d^{2} a^{3}}\right )\) | \(202\) |
risch | \(-\frac {i d^{2} {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{4 a}+\frac {d b \,{\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}-\frac {i b^{2} {\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{3}}+\frac {i b^{2} {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{3}}+\frac {i b^{2} {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a^{3}}+\frac {d \,{\mathrm e}^{-i c} \expIntegral \left (1, i d x \right ) b}{2 a^{2}}+\frac {i d^{2} {\mathrm e}^{-i c} \expIntegral \left (1, i d x \right )}{4 a}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, i d x \right ) b^{2}}{2 a^{3}}-\frac {d \cos \left (d x +c \right )}{2 a x}-\frac {\left (-4 b x +2 a \right ) \sin \left (d x +c \right )}{4 a^{2} x^{2}}\) | \(263\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 245, normalized size = 1.30 \begin {gather*} -\frac {4 \, b^{2} x^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + 2 \, a^{2} d x \cos \left (d x + c\right ) + 2 \, {\left (a b d x^{2} \operatorname {Ci}\left (d x\right ) + a b d x^{2} \operatorname {Ci}\left (-d x\right ) + {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) - 2 \, {\left (2 \, a b x - a^{2}\right )} \sin \left (d x + c\right ) - {\left (4 \, a b d x^{2} \operatorname {Si}\left (d x\right ) - {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Ci}\left (d x\right ) - {\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right ) - 2 \, {\left (b^{2} x^{2} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + b^{2} x^{2} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, a^{3} x^{2}} \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*} \int \frac {\sin {\left (c + d x \right )}}{x^{3} \left (a + b x\right )}\, dx \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 = 3.89, size = 4565, normalized size = 24.15 \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 {\sin \left (c+d\,x\right )}{x^3\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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