Integrand size = 17, antiderivative size = 512 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}} \]
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Time = 0.57 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3422, 3415, 3378, 3384, 3380, 3383} \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d^2 \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3415
Rule 3422
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \frac {\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b} \\ & = -\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \left (-\frac {b \cos (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cos (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \cos (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 b} \\ & = -\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a}-\frac {d \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a}-\frac {d \int \frac {\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \int \left (-\frac {\sqrt {-a} \cos (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cos (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a}-\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b}+\frac {\left (d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}+\frac {\left (d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b}-\frac {\left (d^2 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {\left (d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}-\frac {\left (d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b} \\ & = -\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.53 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.62 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {i d e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (\sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+\left (\sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )-i d e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (-\left (\left (\sqrt {b}-\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\left (\sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 \sqrt {a} b \cos (d x) \left (d x \left (a+b x^2\right ) \cos (c)-2 a \sin (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 \sqrt {a} b \left (2 a \cos (c)+d x \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^2\right )^2}}{32 a^{3/2} b^2} \]
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Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\frac {i d^{2} {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{2}}+\frac {i d^{2} {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{2}}-\frac {i d \,{\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \sqrt {a b}\, \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a^{2} b^{2}}+\frac {i d \sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{2} b^{2}}-\frac {i d^{2} {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{2}}-\frac {i d^{2} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}}}{32 a \,b^{2}}+\frac {i d \sqrt {a b}\, {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a^{2} b^{2}}-\frac {i d \sqrt {a b}\, \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}}}{32 a^{2} b^{2}}-\frac {d^{3} x \left (d^{2} x^{2} b +a \,d^{2}\right ) \cos \left (d x +c \right )}{8 a b \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}+\frac {d^{4} \sin \left (d x +c \right )}{4 b \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\) | \(628\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1360\) |
default | \(\text {Expression too large to display}\) | \(1360\) |
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.94 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {8 \, a^{2} b \sin \left (d x + c\right ) + {\left (i \, a b^{2} d^{2} x^{4} + 2 i \, a^{2} b d^{2} x^{2} + i \, a^{3} d^{2} - {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (i \, a b^{2} d^{2} x^{4} + 2 i \, a^{2} b d^{2} x^{2} + i \, a^{3} d^{2} - {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-i \, a b^{2} d^{2} x^{4} - 2 i \, a^{2} b d^{2} x^{2} - i \, a^{3} d^{2} - {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-i \, a b^{2} d^{2} x^{4} - 2 i \, a^{2} b d^{2} x^{2} - i \, a^{3} d^{2} - {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, {\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right )}{32 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]
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Timed out. \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
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\[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {x\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]
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