Integrand size = 19, antiderivative size = 431 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {-a} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}} \]
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Time = 0.46 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3424, 3426, 3384, 3380, 3383, 3427, 2717, 3415} \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {-a} d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sin (c+d x)}{2 b^2} \]
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Rule 2717
Rule 3380
Rule 3383
Rule 3384
Rule 3415
Rule 3424
Rule 3426
Rule 3427
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x \sin (c+d x)}{a+b x^2} \, dx}{b}+\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^2} \, dx}{2 b} \\ & = -\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\int \left (-\frac {\sin (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{b}+\frac {d \int \left (\frac {\cos (c+d x)}{b}-\frac {a \cos (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b} \\ & = -\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {d \int \cos (c+d x) \, dx}{2 b^2}-\frac {(a d) \int \frac {\cos (c+d x)}{a+b x^2} \, dx}{2 b^2} \\ & = \frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {(a d) \int \left (\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}-\frac {\sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}} \\ & = \frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}-\frac {\left (\sqrt {-a} d\right ) \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (\sqrt {-a} d\right ) \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2} \\ & = \frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}-\frac {\left (\sqrt {-a} 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}{4 b^2}-\frac {\left (\sqrt {-a} 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}{4 b^2}+\frac {\left (\sqrt {-a} 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}{4 b^2}-\frac {\left (\sqrt {-a} 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}{4 b^2} \\ & = \frac {\sqrt {-a} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {\sqrt {-a} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {\sin (c+d x)}{2 b^2}-\frac {x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {\sqrt {-a} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.82 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.66 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {i e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (2 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\left (2 \sqrt {b}-\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )-i e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (2 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\left (2 \sqrt {b}-\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 a \sqrt {b} \cos (d x) \sin (c)}{a+b x^2}+\frac {4 a \sqrt {b} \cos (c) \sin (d x)}{a+b x^2}}{8 b^{5/2}} \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.98
method | result | size |
risch | \(\frac {i \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 ) d}{8 b^{3}}-\frac {i \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 ) d}{8 b^{3}}+\frac {i {\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 )}{4 b^{2}}+\frac {i {\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 )}{4 b^{2}}+\frac {i {\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 ) \sqrt {a b}\, d}{8 b^{3}}-\frac {i {\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 ) \sqrt {a b}\, d}{8 b^{3}}-\frac {i {\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 )}{4 b^{2}}-\frac {i {\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 )}{4 b^{2}}+\frac {\left (\frac {\left (3 i \left (i d x +i c \right ) a b c \,d^{2}-i \left (i d x +i c \right ) b^{2} c^{3}+a^{2} d^{4}-b^{2} c^{4}\right ) d^{2}}{2 \left (2 i \left (i d x +i c \right ) b c -b \left (i d x +i c \right )^{2}+a \,d^{2}+c^{2} b \right ) b^{2} a}+\frac {c^{3} d^{3} x}{2 \left (-2 i \left (i d x +i c \right ) b c +b \left (i d x +i c \right )^{2}-a \,d^{2}-c^{2} b \right ) a}-\frac {3 c^{2} d^{2} \left (i \left (i d x +i c \right ) b c +a \,d^{2}+c^{2} b \right )}{2 a b \left (2 i \left (i d x +i c \right ) b c -b \left (i d x +i c \right )^{2}+a \,d^{2}+c^{2} b \right )}-\frac {3 i c \,d^{2} \left (-i a c \,d^{2}-i b \,c^{3}-\left (i d x +i c \right ) a \,d^{2}+\left (i d x +i c \right ) b \,c^{2}\right )}{2 a b \left (-2 i \left (i d x +i c \right ) b c +b \left (i d x +i c \right )^{2}-a \,d^{2}-c^{2} b \right )}\right ) \sin \left (d x +c \right )}{d^{4}}\) | \(852\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2531\) |
default | \(\text {Expression too large to display}\) | \(2531\) |
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.68 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} + 2 i \, a\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-2 i \, b x^{2} - {\left (i \, b x^{2} + i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-2 i \, b x^{2} - {\left (-i \, b x^{2} - i \, a\right )} \sqrt {\frac {a d^{2}}{b}} - 2 i \, a\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, a \sin \left (d x + c\right )}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]
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\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{3} \sin {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^3\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]
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