Optimal. Leaf size=434 \[ -\frac {3 \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{f}+\frac {\text {Ci}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}+\frac {\text {Ci}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}+\frac {\text {Ci}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}-\frac {3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f} \]
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Rubi [A]
time = 1.32, antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3512, 3384,
3380, 3383, 3426} \begin {gather*} \frac {\sin \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {CosIntegral}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {3 \sin (a) \text {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 3426
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx &=-\frac {3 \text {Subst}\left (\int \left (\frac {d \sin (a+b x)}{f x}+\frac {d (-d e+c f) x^2 \sin (a+b x)}{f \left (f+(d e-c f) x^3\right )}\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {(3 (d e-c f)) \text {Subst}\left (\int \frac {x^2 \sin (a+b x)}{f+(d e-c f) x^3} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}\\ &=\frac {(3 (d e-c f)) \text {Subst}\left (\int \left (\frac {\sin (a+b x)}{3 (d e-c f)^{2/3} \left (\sqrt [3]{f}+\sqrt [3]{d e-c f} x\right )}+\frac {\sin (a+b x)}{3 (d e-c f)^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{f}+\sqrt [3]{d e-c f} x\right )}+\frac {\sin (a+b x)}{3 (d e-c f)^{2/3} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{d e-c f} x\right )}\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}-\frac {(3 \cos (a)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}-\frac {(3 \sin (a)) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}\\ &=-\frac {3 \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{f}-\frac {3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\sqrt [3]{d e-c f} \text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\sqrt [3]{d e-c f} \text {Subst}\left (\int \frac {\sin (a+b x)}{-\sqrt [3]{-1} \sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\sqrt [3]{d e-c f} \text {Subst}\left (\int \frac {\sin (a+b x)}{(-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}\\ &=-\frac {3 \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{f}-\frac {3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\left (\sqrt [3]{d e-c f} \cos \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+b x\right )}{\sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}-\frac {\left (\sqrt [3]{d e-c f} \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-b x\right )}{-\sqrt [3]{-1} \sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\left (\sqrt [3]{d e-c f} \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+b x\right )}{(-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\left (\sqrt [3]{d e-c f} \sin \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+b x\right )}{\sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\left (\sqrt [3]{d e-c f} \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-b x\right )}{-\sqrt [3]{-1} \sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\left (\sqrt [3]{d e-c f} \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+b x\right )}{(-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{d e-c f} x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{f}\\ &=-\frac {3 \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{f}+\frac {\text {Ci}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}+\frac {\text {Ci}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}+\frac {\text {Ci}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}-\frac {3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 0.48, size = 170, normalized size = 0.39 \begin {gather*} \frac {i \left (\left (-3 \text {Ei}\left (-\frac {i b}{\sqrt [3]{c+d x}}\right )+\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{-\frac {i b}{\text {$\#$1}}} \text {Ei}\left (-i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )\&\right ]\right ) (\cos (a)-i \sin (a))+\left (3 \text {Ei}\left (\frac {i b}{\sqrt [3]{c+d x}}\right )-\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{\frac {i b}{\text {$\#$1}}} \text {Ei}\left (i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )\&\right ]\right ) (\cos (a)+i \sin (a))\right )}{2 f} \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 = 0.06, size = 156, normalized size = 0.36
method | result | size |
derivativedivides | \(-3 b^{3} \left (-\frac {\munderset {\textit {\_R1} =\RootOf \left (\left (c f -d e \right ) \textit {\_Z}^{3}+\left (-3 a c f +3 a d e \right ) \textit {\_Z}^{2}+\left (3 a^{2} c f -3 a^{2} d e \right ) \textit {\_Z} -a^{3} c f +a^{3} d e -f \,b^{3}\right )}{\sum }\left (-\sinIntegral \left (-\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{3 f \,b^{3}}+\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )+\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{f \,b^{3}}\right )\) | \(156\) |
default | \(-3 b^{3} \left (-\frac {\munderset {\textit {\_R1} =\RootOf \left (\left (c f -d e \right ) \textit {\_Z}^{3}+\left (-3 a c f +3 a d e \right ) \textit {\_Z}^{2}+\left (3 a^{2} c f -3 a^{2} d e \right ) \textit {\_Z} -a^{3} c f +a^{3} d e -f \,b^{3}\right )}{\sum }\left (-\sinIntegral \left (-\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{3 f \,b^{3}}+\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )+\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{f \,b^{3}}\right )\) | \(156\) |
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 = 0.42, size = 566, normalized size = 1.30 \begin {gather*} \frac {-i \, {\rm Ei}\left (\frac {2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (-i \, d x - i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, a\right )} + i \, {\rm Ei}\left (\frac {-2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (-i \, d x - i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, a\right )} - i \, {\rm Ei}\left (\frac {2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (i \, d x + i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, a\right )} + i \, {\rm Ei}\left (\frac {-2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (i \, d x + i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, a\right )} + 3 i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) e^{\left (i \, a\right )} - 3 i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) e^{\left (-i \, a\right )} - i \, {\rm Ei}\left (\frac {i \, {\left (d x + c\right )}^{\frac {2}{3}} b + \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x + c\right )}}{d x + c}\right ) e^{\left (i \, a - \left (\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}}\right )} + i \, {\rm Ei}\left (\frac {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b + \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}} {\left (d x + c\right )}}{d x + c}\right ) e^{\left (-i \, a - \left (-\frac {i \, b^{3} f}{c f - d e}\right )^{\frac {1}{3}}\right )}}{2 \, f} \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 (a + \frac {b}{\sqrt [3]{c + d x}} \right )}}{e + f x}\, 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 (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{e+f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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