Optimal. Leaf size=85 \[ -\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d \sqrt [3]{e (c+d x)}}+\frac {3 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{e (c+d x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3512, 15, 3377,
2717} \begin {gather*} \frac {3 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{e (c+d x)}}-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d \sqrt [3]{e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 2717
Rule 3377
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c e+d e x}} \, dx &=\frac {3 \text {Subst}\left (\int \frac {x^2 \sin (a+b x)}{\sqrt [3]{e x^3}} \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d \sqrt [3]{e (c+d x)}}+\frac {\left (3 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d \sqrt [3]{e (c+d x)}}+\frac {3 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 70, normalized size = 0.82 \begin {gather*} \frac {-3 b (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )+3 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.46, size = 128, normalized size = 1.51 \begin {gather*} -\frac {3 \, {\left ({\left (-i \, \Gamma \left (2, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (2, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (2, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (2, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (2, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (2, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (2, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (2, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {1}{3}\right )}}{4 \, b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 66, normalized size = 0.78 \begin {gather*} -\frac {3 \, {\left ({\left (d x + c\right )}^{\frac {4}{3}} b \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\frac {2}{3}} - {\left (d x + c\right )} e^{\frac {2}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} e^{\left (-1\right )}}{b^{2} d^{2} x + b^{2} c 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 (a + b \sqrt [3]{c + d x} \right )}}{\sqrt [3]{e \left (c + d x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.92, size = 83, normalized size = 0.98 \begin {gather*} -\frac {3 \, {\left (\frac {{\left (d x e + c e\right )}^{\frac {1}{3}} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\frac {1}{3}}}{b} - \frac {e^{\frac {2}{3}} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{2}}\right )} e^{\left (-1\right )}}{d} \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 = -0.01 \begin {gather*} \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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