Optimal. Leaf size=124 \[ -\frac {a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}+\frac {a \sin (c+d x)}{b^2 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6874, 3378,
3384, 3380, 3383} \begin {gather*} -\frac {a d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}+\frac {a \sin (c+d x)}{b^2 (a+b 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 {x \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (-\frac {a \sin (c+d x)}{b (a+b x)^2}+\frac {\sin (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{a+b x} \, dx}{b}-\frac {a \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{b}\\ &=\frac {a \sin (c+d x)}{b^2 (a+b x)}-\frac {(a d) \int \frac {\cos (c+d x)}{a+b x} \, dx}{b^2}+\frac {\cos \left (c-\frac {a d}{b}\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}+\frac {\sin \left (c-\frac {a d}{b}\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}+\frac {a \sin (c+d x)}{b^2 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {\left (a 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}{b^2}+\frac {\left (a 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}{b^2}\\ &=-\frac {a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}+\frac {a \sin (c+d x)}{b^2 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2}+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 96, normalized size = 0.77 \begin {gather*} \frac {\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-a d \cos \left (c-\frac {a d}{b}\right )+b \sin \left (c-\frac {a d}{b}\right )\right )+\frac {a b \sin (c+d x)}{a+b x}+\left (b \cos \left (c-\frac {a d}{b}\right )+a d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs.
\(2(130)=260\).
time = 0.06, size = 315, normalized size = 2.54
method | result | size |
risch | \(\frac {\left (-2 a b d x -2 a^{2} d \right ) \sin \left (d x +c \right )}{2 b^{2} \left (b x +a \right ) \left (-d x b -d a \right )}+\frac {\cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}+\frac {\cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}-\frac {i \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}+\frac {i \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}+\frac {i \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}-\frac {i \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a d}{2 b^{3}}+\frac {\sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}+\frac {\sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right )}{2 b^{2}}\) | \(313\) |
derivativedivides | \(\frac {-\frac {d^{2} \left (d a -c b \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b}+\frac {d^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-d^{2} c \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{d^{2}}\) | \(315\) |
default | \(\frac {-\frac {d^{2} \left (d a -c b \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b}+\frac {d^{2} \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}-d^{2} c \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{d^{2}}\) | \(315\) |
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 = 0.36, size = 208, normalized size = 1.68 \begin {gather*} \frac {2 \, a b \sin \left (d x + c\right ) - {\left ({\left (a b d x + a^{2} d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b d x + a^{2} d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 2 \, {\left (b^{2} x + a b\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left ({\left (b^{2} x + a b\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{2} x + a b\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + 2 \, {\left (a b d x + a^{2} d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{4} x + a b^{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*} \int \frac {x \sin {\left (c + d x \right )}}{\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. 951 vs.
\(2 (130) = 260\).
time = 5.02, size = 951, normalized size = 7.67 \begin {gather*} -\frac {{\left ({\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - a b c d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a^{2} d^{3} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + {\left (b x + a\right )} a {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - a b c d^{2} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a^{2} d^{3} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - b^{2} c d \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) + a b d^{2} \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - {\left (b x + a\right )} b {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + b^{2} c d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - a b d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a b d^{2} \sin \left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )\right )} b}{{\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \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 {x\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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