Optimal. Leaf size=419 \[ \frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {b^3 (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^6 f \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{240 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^6 f \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac {b^3 (d e-c f) \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2} \]
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Rubi [A]
time = 0.34, antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3512, 3378,
3384, 3380, 3383} \begin {gather*} \frac {b^6 f \sin (a) \text {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b^6 f \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b^3 \cos (a) (d e-c f) \text {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 \sin (a) (d e-c f) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}-\frac {b^2 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {(c+d x) (d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {b (c+d x)^{2/3} (d e-c f) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3512
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx &=-\frac {3 \text {Subst}\left (\int \left (\frac {f \sin (a+b x)}{d x^7}+\frac {(d e-c f) \sin (a+b x)}{d x^4}\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {(3 f) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^7} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {(3 (d e-c f)) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {(b f) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^6} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {(b (d e-c f)) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^3} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^5} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {\left (b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {\left (b^3 f\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {\left (b^3 (d e-c f)\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^3} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {\left (b^3 (d e-c f) \cos (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {\left (b^3 (d e-c f) \sin (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {b^3 (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 (d e-c f) \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {\left (b^5 f\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {b^3 (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 (d e-c f) \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {\left (b^6 f\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {b^3 (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 (d e-c f) \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {\left (b^6 f \cos (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {\left (b^6 f \sin (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {b^3 (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^6 f \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{240 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^6 f \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac {b^3 (d e-c f) \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 540, normalized size = 1.29 \begin {gather*} \frac {e \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b \sqrt [3]{c+d x} \cos (a)-b^2 \sin (a)+2 (c+d x)^{2/3} \sin (a)\right )}{2 d}+\frac {f \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b^5 \cos (a)-120 b c \sqrt [3]{c+d x} \cos (a)-2 b^3 (c+d x)^{2/3} \cos (a)+24 b (c+d x)^{4/3} \cos (a)+120 b^2 c \sin (a)+b^4 \sqrt [3]{c+d x} \sin (a)-240 c (c+d x)^{2/3} \sin (a)-6 b^2 (c+d x) \sin (a)+120 (c+d x)^{5/3} \sin (a)\right )}{240 d^2}+\frac {e \sqrt [3]{c+d x} \left (-b^2 \cos (a)+2 (c+d x)^{2/3} \cos (a)-b \sqrt [3]{c+d x} \sin (a)\right ) \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d}+\frac {f \sqrt [3]{c+d x} \left (120 b^2 c \cos (a)+b^4 \sqrt [3]{c+d x} \cos (a)-240 c (c+d x)^{2/3} \cos (a)-6 b^2 (c+d x) \cos (a)+120 (c+d x)^{5/3} \cos (a)-b^5 \sin (a)+120 b c \sqrt [3]{c+d x} \sin (a)+2 b^3 (c+d x)^{2/3} \sin (a)-24 b (c+d x)^{4/3} \sin (a)\right ) \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b^3 e \left (\cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )-\sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{2 d}+\frac {b^3 f \left (-120 c \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+b^3 \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)+b^3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+120 c \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{240 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 391, normalized size = 0.93
method | result | size |
derivativedivides | \(-\frac {3 b^{3} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )+f \,b^{3} \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{2}}{6 b^{6}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {5}{3}}}{30 b^{5}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {4}{3}}}{120 b^{4}}+\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{360 b^{3}}-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{720 b^{2}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{720 b}-\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{720}-\frac {\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{720}\right )\right )}{d^{2}}\) | \(391\) |
default | \(-\frac {3 b^{3} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )+f \,b^{3} \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{2}}{6 b^{6}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {5}{3}}}{30 b^{5}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {4}{3}}}{120 b^{4}}+\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{360 b^{3}}-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{720 b^{2}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{720 b}-\frac {\sinIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{720}-\frac {\cosineIntegral \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{720}\right )\right )}{d^{2}}\) | \(391\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.56, size = 460, normalized size = 1.10 \begin {gather*} -\frac {\frac {120 \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) + {\left (i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{3} + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{2} - 2 \, d x - 2 \, c\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} c f}{d} - 120 \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) + {\left (i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{3} + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{2} - 2 \, d x - 2 \, c\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} e - \frac {{\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{6} + 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{5} - 2 \, {\left (d x + c\right )} b^{3} + 24 \, {\left (d x + c\right )}^{\frac {5}{3}} b\right )} \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{4} - 6 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} + 120 \, {\left (d x + c\right )}^{2}\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} f}{d}}{480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 305, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{5} f - 2 \, b^{3} d f x - 2 \, b^{3} c f + 24 \, {\left (b d f x - 4 \, b c f + 5 \, b d e\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) + {\left (b^{6} f \sin \left (a\right ) - 120 \, {\left (b^{3} c f - b^{3} d e\right )} \cos \left (a\right )\right )} \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\left (b^{6} f \sin \left (a\right ) - 120 \, {\left (b^{3} c f - b^{3} d e\right )} \cos \left (a\right )\right )} \operatorname {Ci}\left (-\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{4} f + 120 \, d^{2} f x^{2} - 120 \, c^{2} f + 240 \, {\left (d^{2} x + c d\right )} e - 6 \, {\left (b^{2} d f x - 19 \, b^{2} c f + 20 \, b^{2} d e\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) + 2 \, {\left (b^{6} f \cos \left (a\right ) + 120 \, {\left (b^{3} c f - b^{3} d e\right )} \sin \left (a\right )\right )} \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{480 \, d^{2}} \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 \left (e + f x\right ) \sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3728 vs.
\(2 (363) = 726\).
time = 4.93, size = 3728, normalized size = 8.90 \begin {gather*} \text {Too large to display} \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 \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )\,\left (e+f\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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