Optimal. Leaf size=188 \[ \frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {2 b \text {Ci}(d x) \sin (c)}{a^3}+\frac {2 b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2} \]
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Rubi [A]
time = 0.35, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 16, 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 {2 b \sin (c) \text {CosIntegral}(d x)}{a^3}+\frac {2 b \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {d \cos (c) \text {CosIntegral}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 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^2 (a+b x)^2} \, dx &=\int \left (\frac {\sin (c+d x)}{a^2 x^2}-\frac {2 b \sin (c+d x)}{a^3 x}+\frac {b^2 \sin (c+d x)}{a^2 (a+b x)^2}+\frac {2 b^2 \sin (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^2} \, dx}{a^2}-\frac {(2 b) \int \frac {\sin (c+d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{a^3}+\frac {b^2 \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{a^2}+\frac {(b d) \int \frac {\cos (c+d x)}{a+b x} \, dx}{a^2}-\frac {(2 b \cos (c)) \int \frac {\sin (d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2 \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 {(2 b \sin (c)) \int \frac {\cos (d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2 \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 {2 b \text {Ci}(d x) \sin (c)}{a^3}+\frac {2 b \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)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{a^2}+\frac {\left (b d \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^2}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{a^2}-\frac {\left (b d \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^2}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {2 b \text {Ci}(d x) \sin (c)}{a^3}+\frac {2 b \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)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 1.22, size = 184, normalized size = 0.98 \begin {gather*} -\frac {-a d \cos (c) \text {Ci}(d x)-a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )+\frac {a (a+2 b x) \cos (d x) \sin (c)}{x (a+b x)}+2 b \text {Ci}(d x) \sin (c)-2 b \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+\frac {a (a+2 b x) \cos (c) \sin (d x)}{x (a+b x)}+2 b \cos (c) \text {Si}(d x)+a d \sin (c) \text {Si}(d x)-2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 256, normalized size = 1.36
method | result | size |
derivativedivides | \(d \left (\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )}{a^{2}}-\frac {2 b \left (\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )\right )}{d \,a^{3}}+\frac {2 b^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d \,a^{3}}\right )\) | \(256\) |
default | \(d \left (\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{d x}-\sinIntegral \left (d x \right ) \sin \left (c \right )+\cosineIntegral \left (d x \right ) \cos \left (c \right )}{a^{2}}-\frac {2 b \left (\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )\right )}{d \,a^{3}}+\frac {2 b^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d \,a^{3}}\right )\) | \(256\) |
risch | \(-\frac {d \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{2}}-\frac {d \,{\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}+\frac {i b \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{a^{3}}-\frac {i b \,{\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{a^{3}}-\frac {d \,{\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a^{2}}-\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right ) b}{a^{3}}-\frac {d \,{\mathrm e}^{-i c} \expIntegral \left (1, i d x \right )}{2 a^{2}}+\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, i d x \right ) b}{a^{3}}+\frac {\left (-4 b x -2 a \right ) \sin \left (d x +c \right )}{2 a^{2} \left (b x +a \right ) x}\) | \(299\) |
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 = 0.38, size = 355, normalized size = 1.89 \begin {gather*} \frac {{\left ({\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (d x\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (-d x\right ) - 4 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Si}\left (d x\right )\right )} \cos \left (c\right ) + {\left ({\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + 4 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (2 \, a b x + a^{2}\right )} \sin \left (d x + c\right ) - 2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (d x\right ) + {\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (-d x\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Si}\left (d x\right )\right )} \sin \left (c\right ) - 2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{3} b x^{2} + a^{4} x\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*} \int \frac {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3180 vs.
\(2 (191) = 382\).
time = 3.94, size = 3180, normalized size = 16.91 \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^2\,{\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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