Integrand size = 16, antiderivative size = 856 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}-\frac {3 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^2 b}-\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}+\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\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)^{3/2} b^{3/2}}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}-\frac {3 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^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}+\frac {3 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^2 b} \]
[Out]
Time = 0.81 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3414, 3378, 3384, 3380, 3383} \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {\operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} 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 ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d^2}{16 (-a)^{3/2} b^{3/2}}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cos (c+d x) d}{16 (-a)^{3/2} b \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) d}{16 a^2 b}+\frac {3 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) d}{16 a^2 b}-\frac {3 \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )^2}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}} \]
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3414
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}-b x\right )^3}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b^{3/2} \sin (c+d x)}{8 (-a)^{3/2} \left (\sqrt {-a} \sqrt {b}+b x\right )^3}-\frac {3 b \sin (c+d x)}{16 a^2 \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {3 b \sin (c+d x)}{8 a^2 \left (-a b-b^2 x^2\right )}\right ) \, dx \\ & = -\frac {(3 b) \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^2}-\frac {(3 b) \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^2}-\frac {(3 b) \int \frac {\sin (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^2}-\frac {b^{3/2} \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^3} \, dx}{8 (-a)^{3/2}}-\frac {b^{3/2} \int \frac {\sin (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^3} \, dx}{8 (-a)^{3/2}} \\ & = -\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {(3 b) \int \left (-\frac {\sqrt {-a} \sin (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \sin (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a^2}+\frac {(3 d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^2}-\frac {(3 d) \int \frac {\cos (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^2}+\frac {\left (\sqrt {b} d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 (-a)^{3/2}}-\frac {\left (\sqrt {b} d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 (-a)^{3/2}} \\ & = \frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{5/2}}-\frac {3 \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{5/2}}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 (-a)^{3/2} \sqrt {b}}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 (-a)^{3/2} \sqrt {b}}-\frac {\left (3 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}+b x} \, dx}{16 a^2}+\frac {\left (3 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}-b x} \, dx}{16 a^2}+\frac {\left (3 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}+b x} \, dx}{16 a^2}+\frac {\left (3 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}-b x} \, dx}{16 a^2} \\ & = \frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}-\frac {3 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^2 b}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}+\frac {3 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^2 b}-\frac {\left (3 \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} x} \, dx}{16 (-a)^{5/2}}+\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)^{3/2} \sqrt {b}}+\frac {\left (3 \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} x} \, dx}{16 (-a)^{5/2}}-\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)^{3/2} \sqrt {b}}-\frac {\left (3 \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} x} \, dx}{16 (-a)^{5/2}}+\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)^{3/2} \sqrt {b}}-\frac {\left (3 \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} x} \, dx}{16 (-a)^{5/2}}+\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)^{3/2} \sqrt {b}} \\ & = \frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 (-a)^{3/2} b \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 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^2 b}-\frac {3 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^2 b}-\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}+\frac {3 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{5/2} \sqrt {b}}-\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)^{3/2} b^{3/2}}-\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )^2}-\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sin (c+d x)}{16 (-a)^{3/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )^2}+\frac {3 \sin (c+d x)}{16 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}-\frac {3 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^2 b}-\frac {3 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{5/2} \sqrt {b}}+\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)^{3/2} b^{3/2}}+\frac {3 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^2 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.99 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.44 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 b-3 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\left (3 b+3 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )+e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 b-3 \sqrt {a} \sqrt {b} d+a d^2\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\left (3 b+3 \sqrt {a} \sqrt {b} d+a d^2\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )+\frac {4 \sqrt {a} \sqrt {b} \cos (d x) \left (a d \left (a+b x^2\right ) \cos (c)+b x \left (5 a+3 b x^2\right ) \sin (c)\right )}{\left (a+b x^2\right )^2}-\frac {4 \sqrt {a} \sqrt {b} \left (-b x \left (5 a+3 b x^2\right ) \cos (c)+a d \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^2\right )^2}}{32 a^{5/2} b^{3/2}} \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(d^{5} \left (-\frac {\sin \left (d x +c \right ) \left (5 a c \,d^{2}-5 a \,d^{2} \left (d x +c \right )+3 c^{3} b -9 b \,c^{2} \left (d x +c \right )+9 b c \left (d x +c \right )^{2}-3 b \left (d x +c \right )^{3}\right )}{8 a^{2} d^{4} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )^{2}}+\frac {\cos \left (d x +c \right )}{8 a b \,d^{2} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {3 \left (-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b}-\frac {3 \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b}\right )\) | \(598\) |
default | \(d^{5} \left (-\frac {\sin \left (d x +c \right ) \left (5 a c \,d^{2}-5 a \,d^{2} \left (d x +c \right )+3 c^{3} b -9 b \,c^{2} \left (d x +c \right )+9 b c \left (d x +c \right )^{2}-3 b \left (d x +c \right )^{3}\right )}{8 a^{2} d^{4} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )^{2}}+\frac {\cos \left (d x +c \right )}{8 a b \,d^{2} \left (a \,d^{2}+c^{2} b -2 b c \left (d x +c \right )+b \left (d x +c \right )^{2}\right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (-\frac {d \sqrt {-a b}+c b}{b}+c \right )}-\frac {\left (a \,d^{2}+3 b \right ) \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b^{2} \left (\frac {d \sqrt {-a b}-c b}{b}+c \right )}-\frac {3 \left (-\operatorname {Si}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\operatorname {Ci}\left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{16 a^{2} d^{4} b}-\frac {3 \left (\operatorname {Si}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )+\operatorname {Ci}\left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{16 a^{2} d^{4} b}\right )\) | \(598\) |
risch | \(-\frac {d^{2} \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 {d^{2} \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 {3 d \,{\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}+\frac {3 d \,{\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}-\frac {3 \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^{3} b}+\frac {3 \,{\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}}{32 a^{3} b}+\frac {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 ) \sqrt {a b}}{32 a^{2} b^{2}}-\frac {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 ) \sqrt {a b}}{32 a^{2} b^{2}}+\frac {3 d \,{\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}+\frac {3 d \,{\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}+\frac {3 \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^{3} b}-\frac {3 \,{\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}}{32 a^{3} b}-\frac {d^{3} \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 \left (3 d^{3} x^{3} b +5 a \,d^{3} x \right ) \sin \left (d x +c \right )}{8 a^{2} \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\) | \(902\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 611, normalized size of antiderivative = 0.71 \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} - {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} + {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} - {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (3 \, a b^{2} d^{2} x^{4} + 6 \, a^{2} b d^{2} x^{2} + 3 \, a^{3} d^{2} + {\left (a^{3} d^{2} + {\left (a b^{2} d^{2} + 3 \, b^{3}\right )} x^{4} + 3 \, a^{2} b + 2 \, {\left (a^{2} b d^{2} + 3 \, a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 4 \, {\left (a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cos \left (d x + c\right ) - 4 \, {\left (3 \, a b^{2} d x^{3} + 5 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{3} b^{3} d x^{4} + 2 \, a^{4} b^{2} d x^{2} + a^{5} b d\right )}} \]
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Timed out. \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
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\[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]
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