Optimal. Leaf size=343 \[ \frac {\text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{2/3} \sqrt [3]{b}} \]
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Rubi [A]
time = 0.26, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3414, 3384,
3380, 3383} \begin {gather*} \frac {\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 a^{2/3} \sqrt [3]{b}}-\frac {\sqrt [3]{-1} \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 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \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 a^{2/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \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 a^{2/3} \sqrt [3]{b}}+\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 )}{3 a^{2/3} \sqrt [3]{b}}+\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 )}{3 a^{2/3} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 3414
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{a+b x^3} \, dx &=\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\\ &=-\frac {\int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac {\int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=-\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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 a^{2/3}}+\frac {\cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 a^{2/3}}-\frac {\cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 a^{2/3}}-\frac {\sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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 a^{2/3}}-\frac {\sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 a^{2/3}}-\frac {\sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 a^{2/3}}\\ &=\frac {\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 a^{2/3} \sqrt [3]{b}}-\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 )}{3 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \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 a^{2/3} \sqrt [3]{b}}+\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 )}{3 a^{2/3} \sqrt [3]{b}}+\frac {\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 a^{2/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} \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 a^{2/3} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 0.10, size = 196, normalized size = 0.57 \begin {gather*} \frac {i \left (\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]-\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]\right )}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.05, size = 85, normalized size = 0.25
method | result | size |
derivativedivides | \(\frac {d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}\) | \(85\) |
default | \(\frac {d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b}\) | \(85\) |
risch | \(-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {{\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{6 b}+\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {{\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{6 b}\) | \(166\) |
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 = 0.41, size = 385, normalized size = 1.12 \begin {gather*} \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 \, a d} \end {gather*}
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 \begin {gather*} \int \frac {\sin {\left (c + d x \right )}}{a + b x^{3}}\, dx \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 )}{b\,x^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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