Optimal. Leaf size=357 \[ -\frac {\sqrt [3]{a} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{a} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac {\cos (c+d x)}{b d} \]
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Rubi [A] time = 0.68, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3345, 2638, 3333, 3303, 3299, 3302} \[ -\frac {\sqrt [3]{a} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{a} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac {\cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3299
Rule 3302
Rule 3303
Rule 3333
Rule 3345
Rubi steps
\begin {align*} \int \frac {x^3 \sin (c+d x)}{a+b x^3} \, dx &=\int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {\int \sin (c+d x) \, dx}{b}-\frac {a \int \frac {\sin (c+d x)}{a+b x^3} \, dx}{b}\\ &=-\frac {\cos (c+d x)}{b d}-\frac {a \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}{b}\\ &=-\frac {\cos (c+d x)}{b d}+\frac {\sqrt [3]{a} \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 b}+\frac {\sqrt [3]{a} \int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b}+\frac {\sqrt [3]{a} \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b}\\ &=-\frac {\cos (c+d x)}{b d}+\frac {\left (\sqrt [3]{a} \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}{3 b}-\frac {\left (\sqrt [3]{a} \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}{3 b}+\frac {\left (\sqrt [3]{a} \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}{3 b}+\frac {\left (\sqrt [3]{a} \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}{3 b}+\frac {\left (\sqrt [3]{a} \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}{3 b}+\frac {\left (\sqrt [3]{a} \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}{3 b}\\ &=-\frac {\cos (c+d x)}{b d}-\frac {\sqrt [3]{a} \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 )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \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 )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \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 )}{3 b^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{a} \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 )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \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 )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \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 )}{3 b^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.36, size = 216, normalized size = 0.61 \[ -\frac {i a d \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-i \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-\sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-i \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]-i a d \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {i \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-\sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+i \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]+6 b \cos (c+d x)}{6 b^2 d} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.93, size = 393, normalized size = 1.10 \[ \frac {\left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} + 2 \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} + 2 \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 12 \, \cos \left (d x + c\right )}{12 \, b d} \]
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 \[ \int \frac {x^{3} \sin \left (d x + c\right )}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 392, normalized size = 1.10 \[ \frac {-\frac {d^{3} \cos \left (d x +c \right )}{b}+\frac {d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (3 \textit {\_R1}^{2} b c -3 \textit {\_R1} b \,c^{2}-a \,d^{3}+b \,c^{3}\right ) \left (-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}-\frac {c \,d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} \left (-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{b}+\frac {c^{2} d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\textit {\_R1} \left (-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{b}-\frac {c^{3} d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}}{d^{4}} \]
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 \[ \int \frac {x^3\,\sin \left (c+d\,x\right )}{b\,x^3+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sin {\left (c + d x \right )}}{a + b x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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