Optimal. Leaf size=1141 \[ \frac {\text {Ci}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \text {Ci}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {\sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{5/3} b^{4/3}}-\frac {\cos (c+d x) d}{18 b^2 x^3 \left (b x^3+a\right )}+\frac {\cos (c+d x) d}{18 a b^2 x^3}-\frac {2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{27 a^2 b}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{27 a^2 b}+\frac {2 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}+\frac {2 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{27 a^2 b}-\frac {2 \text {Ci}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {2 (-1)^{2/3} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sqrt [3]{-1} \text {Ci}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sin (c+d x)}{9 a^2 b x}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (b x^3+a\right )}-\frac {\sin (c+d x)}{6 b x \left (b x^3+a\right )^2}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 (-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}} \]
[Out]
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Rubi [A] time = 3.12, antiderivative size = 1141, normalized size of antiderivative = 1.00, number of steps used = 89, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346, 3344, 3333} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3333
Rule 3343
Rule 3344
Rule 3345
Rule 3346
Rubi steps
\begin {align*} \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}-\frac {\int \frac {\sin (c+d x)}{x^2 \left (a+b x^3\right )^2} \, dx}{6 b}+\frac {d \int \frac {\cos (c+d x)}{x \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \frac {\sin (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {d \int \frac {\cos (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac {d \int \frac {\cos (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{6 b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \left (\frac {\sin (c+d x)}{a x^5}-\frac {b \sin (c+d x)}{a^2 x^2}+\frac {b^2 x \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^4}-\frac {b \cos (c+d x)}{a^2 x}+\frac {b^2 x^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^4}-\frac {b \cos (c+d x)}{a^2 x}+\frac {b^2 x^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{a x^3}-\frac {b \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \frac {x \sin (c+d x)}{a+b x^3} \, dx}{9 a^2}+\frac {2 \int \frac {\sin (c+d x)}{x^5} \, dx}{9 a b^2}-\frac {2 \int \frac {\sin (c+d x)}{x^2} \, dx}{9 a^2 b}-\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^4} \, dx}{18 a b^2}-\frac {d \int \frac {\cos (c+d x)}{x^4} \, dx}{6 a b^2}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{18 a^2 b}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{6 a^2 b}-\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{18 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=\frac {2 d \cos (c+d x)}{27 a b^2 x^3}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {d^2 \sin (c+d x)}{36 a b^2 x^2}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 \int \left (-\frac {\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac {d \int \left (\frac {\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}-\frac {d \int \left (\frac {\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^4} \, dx}{18 a b^2}-\frac {(2 d) \int \frac {\cos (c+d x)}{x} \, dx}{9 a^2 b}+\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{54 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{18 a b^2}+\frac {d^2 \int \left (-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{36 a b^2}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{6 a^2 b}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{6 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^3}+\frac {d^3 \cos (c+d x)}{36 a b^2 x}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac {2 d \cos (c) \text {Ci}(d x)}{9 a^2 b}-\frac {\sin (c+d x)}{18 a b^2 x^4}-\frac {d^2 \sin (c+d x)}{108 a b^2 x^2}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {2 d \sin (c) \text {Si}(d x)}{9 a^2 b}-\frac {2 \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 \sqrt [3]{-1}\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 (-1)^{2/3}\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{x^3} \, dx}{54 a b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{108 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{36 a b^2}+\frac {d^4 \int \frac {\sin (c+d x)}{x} \, dx}{36 a b^2}-\frac {(2 d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{9 a^2 b}+\frac {(2 d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{9 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^3}-\frac {d^3 \cos (c+d x)}{108 a b^2 x}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {d^3 \int \frac {\cos (c+d x)}{x^2} \, dx}{108 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x} \, dx}{108 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x} \, dx}{36 a b^2}+\frac {\left (d^4 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{36 a b^2}-\frac {\left (2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {\left (d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {\left (2 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\frac {\left (d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {\left (2 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {\left (d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac {\left (d^4 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{36 a b^2}-\frac {\left (2 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {\left (d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {\left (d^2 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac {\left (2 \sqrt [3]{-1} \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {\left (d^2 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac {\left (2 (-1)^{2/3} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {\left (d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac {\left (d^2 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^3}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {2 d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {d^4 \text {Ci}(d x) \sin (c)}{36 a b^2}-\frac {2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {2 (-1)^{2/3} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}+\frac {2 \sqrt [3]{-1} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {d^4 \cos (c) \text {Si}(d x)}{36 a b^2}+\frac {2 (-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac {\sqrt [3]{-1} d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac {2 d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {2 \sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {d^4 \int \frac {\sin (c+d x)}{x} \, dx}{108 a b^2}-\frac {\left (d^4 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{108 a b^2}-\frac {\left (d^4 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{36 a b^2}-\frac {\left (d^4 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{108 a b^2}-\frac {\left (d^4 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{36 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^3}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {2 d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {d^4 \text {Ci}(d x) \sin (c)}{108 a b^2}-\frac {2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {2 (-1)^{2/3} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}+\frac {2 \sqrt [3]{-1} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {d^4 \cos (c) \text {Si}(d x)}{108 a b^2}+\frac {2 (-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac {\sqrt [3]{-1} d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac {2 d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {2 \sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {\left (d^4 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{108 a b^2}+\frac {\left (d^4 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{108 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^3}-\frac {d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac {2 d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {2 d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}-\frac {2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {2 (-1)^{2/3} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}+\frac {2 \sqrt [3]{-1} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{5/3} b^{4/3}}-\frac {\sin (c+d x)}{18 a b^2 x^4}+\frac {2 \sin (c+d x)}{9 a^2 b x}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {2 (-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac {\sqrt [3]{-1} d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{5/3} b^{4/3}}-\frac {2 d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac {2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac {2 \sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac {2 d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}\\ \end {align*}
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Mathematica [C] time = 0.59, size = 698, normalized size = 0.61 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-4 i \text {$\#$1}^2 b d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+4 \text {$\#$1}^2 b d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-4 \text {$\#$1}^2 b d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 i \text {$\#$1}^2 b d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-a d^2 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i a d^2 \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i a d^2 \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-a d^2 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 \text {$\#$1} b \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-4 i \text {$\#$1} b \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+4 i \text {$\#$1} b \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 \text {$\#$1} b \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {4 i \text {$\#$1}^2 b d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+4 \text {$\#$1}^2 b d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-4 \text {$\#$1}^2 b d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+4 i \text {$\#$1}^2 b d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-a d^2 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i a d^2 \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i a d^2 \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-a d^2 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 \text {$\#$1} b \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+4 i \text {$\#$1} b \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-4 i \text {$\#$1} b \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 \text {$\#$1} b \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]-\frac {6 b \cos (d x) \left (a d \cos (c) \left (a+b x^3\right )+b x^2 \sin (c) \left (7 a+4 b x^3\right )\right )}{\left (a+b x^3\right )^2}-\frac {6 b \sin (d x) \left (b x^2 \cos (c) \left (7 a+4 b x^3\right )-a d \sin (c) \left (a+b x^3\right )\right )}{\left (a+b x^3\right )^2}}{108 a^2 b^2} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 1.12, size = 1321, normalized size = 1.16 \[ \text {result too large to display} \]
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 \[ \int \frac {x \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 845, normalized size = 0.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x\,\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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