Optimal. Leaf size=234 \[ \cos \left (a+b \sqrt [3]{c}\right ) \text {Ci}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Ci}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Ci}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right ) \]
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Rubi [A] time = 0.47, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3432, 3303, 3299, 3302} \[ \cos \left (a+b \sqrt [3]{c}\right ) \text {CosIntegral}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {CosIntegral}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {CosIntegral}\left (b \sqrt [3]{c+d x}+\sqrt [3]{-1} b \sqrt [3]{c}\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} \sqrt [3]{c} b+\sqrt [3]{c+d x} b\right ) \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3432
Rubi steps
\begin {align*} \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{x} \, dx &=\frac {3 \operatorname {Subst}\left (\int \left (-\frac {d \cos (a+b x)}{3 \left (\sqrt [3]{c}-x\right )}-\frac {d \cos (a+b x)}{3 \left (-\sqrt [3]{-1} \sqrt [3]{c}-x\right )}-\frac {d \cos (a+b x)}{3 \left ((-1)^{2/3} \sqrt [3]{c}-x\right )}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=-\operatorname {Subst}\left (\int \frac {\cos (a+b x)}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname {Subst}\left (\int \frac {\cos (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\operatorname {Subst}\left (\int \frac {\cos (a+b x)}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=-\left (\cos \left (a+b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\right )-\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\sqrt [3]{-1} b \sqrt [3]{c}+b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\cos \left ((-1)^{2/3} b \sqrt [3]{c}-b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sin \left (a+b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (b \sqrt [3]{c}-b x\right )}{\sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )+\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\sqrt [3]{-1} b \sqrt [3]{c}+b x\right )}{-\sqrt [3]{-1} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )-\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {\sin \left ((-1)^{2/3} b \sqrt [3]{c}-b x\right )}{(-1)^{2/3} \sqrt [3]{c}-x} \, dx,x,\sqrt [3]{c+d x}\right )\\ &=\cos \left (a+b \sqrt [3]{c}\right ) \text {Ci}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Ci}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\cos \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Ci}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )+\sin \left (a+b \sqrt [3]{c}\right ) \text {Si}\left (b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )+\sin \left (a+(-1)^{2/3} b \sqrt [3]{c}\right ) \text {Si}\left ((-1)^{2/3} b \sqrt [3]{c}-b \sqrt [3]{c+d x}\right )-\sin \left (a-\sqrt [3]{-1} b \sqrt [3]{c}\right ) \text {Si}\left (\sqrt [3]{-1} b \sqrt [3]{c}+b \sqrt [3]{c+d x}\right )\\ \end {align*}
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Mathematica [C] time = 0.43, size = 243, normalized size = 1.04 \[ \frac {1}{2} \left (\text {RootSum}\left [c-\text {$\#$1}^3\& ,-i \sin (\text {$\#$1} b+a) \text {Ci}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cos (\text {$\#$1} b+a) \text {Ci}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\sin (\text {$\#$1} b+a) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-i \cos (\text {$\#$1} b+a) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\& \right ]+\text {RootSum}\left [c-\text {$\#$1}^3\& ,i \sin (\text {$\#$1} b+a) \text {Ci}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+\cos (\text {$\#$1} b+a) \text {Ci}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )-\sin (\text {$\#$1} b+a) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )+i \cos (\text {$\#$1} b+a) \text {Si}\left (b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\& \right ]\right ) \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 1.09, size = 287, normalized size = 1.23 \[ \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-i \, b^{3} c\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, a\right )} + \frac {1}{2} \, {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (i \, b^{3} c\right )^{\frac {1}{3}}\right ) e^{\left (i \, a - \left (i \, b^{3} c\right )^{\frac {1}{3}}\right )} + \frac {1}{2} \, {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (-i \, b^{3} c\right )^{\frac {1}{3}}\right ) e^{\left (-i \, a - \left (-i \, b^{3} c\right )^{\frac {1}{3}}\right )} \]
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 \[ \int \frac {\cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 279, normalized size = 1.19 \[ \frac {b^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (-c \,b^{3}+\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1}^{2} \left (\Si \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\Ci \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )-2 b^{3} a \left (\munderset {\textit {\_R1} =\RootOf \left (-c \,b^{3}+\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\textit {\_R1} \left (\Si \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\Ci \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )+a^{2} b^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (-c \,b^{3}+\textit {\_Z}^{3}-3 a \,\textit {\_Z}^{2}+3 a^{2} \textit {\_Z} -a^{3}\right )}{\sum }\frac {\Si \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \sin \left (\textit {\_R1} \right )+\Ci \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \cos \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{b^{3}} \]
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 \[ \int \frac {\cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{x}\,{d x} \]
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 \[ \int \frac {\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{x} \,d x \]
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 \[ \int \frac {\cos {\left (a + b \sqrt [3]{c + d x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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