Optimal. Leaf size=791 \[ -\frac {d \cos (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {9 \sqrt {b} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {3 b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}-\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}-\frac {9 \sqrt {b} 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)^{7/2}}+\frac {3 b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^4}+\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}-\frac {9 \sqrt {b} 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)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.38, antiderivative size = 791, normalized size of antiderivative = 1.00, number of steps
used = 46, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3426, 3378,
3384, 3380, 3383, 3422, 3415} \begin {gather*} -\frac {3 b \sin (c) \text {CosIntegral}(d x)}{a^4}+\frac {3 b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^4}+\frac {3 b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {3 b \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}+\frac {3 b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a^3}+\frac {d^2 \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a^3}-\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^3}+\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^3}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d^2 \sin (c) \text {CosIntegral}(d x)}{2 a^3}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {9 \sqrt {b} d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}-\frac {9 \sqrt {b} 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)^{7/2}}-\frac {9 \sqrt {b} 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)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3415
Rule 3422
Rule 3426
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x^3 \left (a+b x^2\right )^3} \, dx &=\int \left (\frac {\sin (c+d x)}{a^3 x^3}-\frac {3 b \sin (c+d x)}{a^4 x}+\frac {b^2 x \sin (c+d x)}{a^2 \left (a+b x^2\right )^3}+\frac {2 b^2 x \sin (c+d x)}{a^3 \left (a+b x^2\right )^2}+\frac {3 b^2 x \sin (c+d x)}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^3} \, dx}{a^3}-\frac {(3 b) \int \frac {\sin (c+d x)}{x} \, dx}{a^4}+\frac {\left (3 b^2\right ) \int \frac {x \sin (c+d x)}{a+b x^2} \, dx}{a^4}+\frac {\left (2 b^2\right ) \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{a^3}+\frac {b^2 \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}+\frac {\left (3 b^2\right ) \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}{a^4}+\frac {d \int \frac {\cos (c+d x)}{x^2} \, dx}{2 a^3}+\frac {(b d) \int \frac {\cos (c+d x)}{a+b x^2} \, dx}{a^3}+\frac {(b d) \int \frac {\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 a^2}-\frac {(3 b \cos (c)) \int \frac {\sin (d x)}{x} \, dx}{a^4}-\frac {(3 b \sin (c)) \int \frac {\cos (d x)}{x} \, dx}{a^4}\\ &=-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {\left (3 b^{3/2}\right ) \int \frac {\sin (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 a^4}+\frac {\left (3 b^{3/2}\right ) \int \frac {\sin (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 a^4}+\frac {(b 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}{a^3}+\frac {(b 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 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{2 a^3}\\ &=-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}+\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 (-a)^{7/2}}+\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 (-a)^{7/2}}-\frac {\left (b^2 d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a^3}-\frac {\left (b^2 d\right ) \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a^3}-\frac {\left (b^2 d\right ) \int \frac {\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a^3}-\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{2 a^3}+\frac {\left (3 b^{3/2} \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} x} \, dx}{2 a^4}+\frac {\left (3 b^{3/2} \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} x} \, dx}{2 a^4}-\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{2 a^3}+\frac {\left (3 b^{3/2} \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} x} \, dx}{2 a^4}-\frac {\left (3 b^{3/2} \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} x} \, dx}{2 a^4}\\ &=-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {3 b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}+\frac {3 b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^4}-\frac {\left (b^2 d\right ) \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^3}-\frac {\left (b d^2\right ) \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a^3}+\frac {\left (b d^2\right ) \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a^3}+\frac {\left (b 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}{2 (-a)^{7/2}}+\frac {\left (b 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}{2 (-a)^{7/2}}-\frac {\left (b 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}{2 (-a)^{7/2}}+\frac {\left (b 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}{2 (-a)^{7/2}}\\ &=-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sqrt {b} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}+\frac {\sqrt {b} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {3 b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}-\frac {\sqrt {b} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}+\frac {3 b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^4}-\frac {\sqrt {b} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}+\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{7/2}}+\frac {(b d) \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{7/2}}+\frac {\left (b 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}+\frac {\left (b 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}+\frac {\left (b 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}-\frac {\left (b 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}\\ &=-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sqrt {b} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}+\frac {\sqrt {b} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {3 b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}-\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}-\frac {\sqrt {b} d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 (-a)^{7/2}}+\frac {3 b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^4}+\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}-\frac {\sqrt {b} d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 (-a)^{7/2}}+\frac {\left (b 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)^{7/2}}+\frac {\left (b 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)^{7/2}}-\frac {\left (b 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)^{7/2}}+\frac {\left (b 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)^{7/2}}\\ &=-\frac {d \cos (c+d x)}{2 a^3 x}-\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {b} d \cos (c+d x)}{16 a^3 \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {9 \sqrt {b} d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{7/2}}+\frac {9 \sqrt {b} d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{7/2}}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}+\frac {3 b \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^4}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a^3}-\frac {\sin (c+d x)}{2 a^3 x^2}-\frac {b \sin (c+d x)}{4 a^2 \left (a+b x^2\right )^2}-\frac {b \sin (c+d x)}{a^3 \left (a+b x^2\right )}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^3}-\frac {3 b \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^4}-\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}-\frac {9 \sqrt {b} 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)^{7/2}}+\frac {3 b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^4}+\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}-\frac {9 \sqrt {b} 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)^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.72, size = 995, normalized size = 1.26 \begin {gather*} \frac {-\frac {2 a \cos (d x) \left (d x \left (4 a^2+7 a b x^2+3 b^2 x^4\right ) \cos (c)+2 \left (2 a^2+9 a b x^2+6 b^2 x^4\right ) \sin (c)\right )}{x^2 \left (a+b x^2\right )^2}+\frac {2 a \left (-2 \left (2 a^2+9 a b x^2+6 b^2 x^4\right ) \cos (c)+d x \left (4 a^2+7 a b x^2+3 b^2 x^4\right ) \sin (c)\right ) \sin (d x)}{x^2 \left (a+b x^2\right )^2}-8 \left (6 b+a d^2\right ) (\text {Ci}(d x) \sin (c)+\cos (c) \text {Si}(d x))+24 b \cos (c) \left (i \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-i \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+a d^2 \cos (c) \left (i \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-i \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+9 \sqrt {a} \sqrt {b} d \cos (c) \left (-i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (-\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )-9 \sqrt {a} \sqrt {b} d \sin (c) \left (\text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right ) \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+24 b \sin (c) \left (\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )+a d^2 \sin (c) \left (\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )}{16 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.84, size = 697, normalized size = 0.88 Too large to display
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 [C] Result contains complex when optimal does not.
time = 0.47, size = 768, normalized size = 0.97 \begin {gather*} -\frac {8 \, {\left (-i \, {\left (a b^{2} d^{2} + 6 \, b^{3}\right )} x^{6} - 2 i \, {\left (a^{2} b d^{2} + 6 \, a b^{2}\right )} x^{4} - i \, {\left (a^{3} d^{2} + 6 \, a^{2} b\right )} x^{2}\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 8 \, {\left (i \, {\left (a b^{2} d^{2} + 6 \, b^{3}\right )} x^{6} + 2 i \, {\left (a^{2} b d^{2} + 6 \, a b^{2}\right )} x^{4} + i \, {\left (a^{3} d^{2} + 6 \, a^{2} b\right )} x^{2}\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} + 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} + i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{4} - i \, a^{2} b 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 (i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} + 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} + i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{4} + i \, a^{2} b 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 (-i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} - 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} - i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{4} + i \, a^{2} b 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 (-i \, {\left (a b^{2} d^{2} + 24 \, b^{3}\right )} x^{6} - 2 i \, {\left (a^{2} b d^{2} + 24 \, a b^{2}\right )} x^{4} - i \, {\left (a^{3} d^{2} + 24 \, a^{2} b\right )} x^{2} + 9 \, {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{4} - i \, a^{2} b 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 (3 \, a b^{2} d x^{5} + 7 \, a^{2} b d x^{3} + 4 \, a^{3} d x\right )} \cos \left (d x + c\right ) + 8 \, {\left (6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{32 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{x^3\,{\left (b\,x^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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