Optimal. Leaf size=267 \[ \frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d} \]
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Rubi [A]
time = 0.20, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3516, 3498,
3496, 3466, 3467, 3435, 3433, 3432} \begin {gather*} -\frac {45 \sqrt {\pi } e \cos (a) \sqrt [3]{e (c+d x)} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 \sqrt {\pi } e \sin (a) \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3435
Rule 3466
Rule 3467
Rule 3496
Rule 3498
Rule 3516
Rubi steps
\begin {align*} \int (c e+d e x)^{4/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {\text {Subst}\left (\int (e x)^{4/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^{4/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=\frac {\left (3 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^6 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {\left (15 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^4 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)} \cos (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}+\frac {\left (45 e \sqrt [3]{e (c+d x)} \sin (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 175, normalized size = 0.66 \begin {gather*} -\frac {3 (e (c+d x))^{4/3} \left (15 \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )-15 \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+2 \sqrt {b} \left (\sqrt [3]{c+d x} \left (-15+4 b^2 (c+d x)^{4/3}\right ) \cos \left (a+b (c+d x)^{2/3}\right )-10 b (c+d x) \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{16 b^{7/2} d (c+d x)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )\, dx\]
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
4.
time = 0.70, size = 425, normalized size = 1.59 \begin {gather*} \frac {3 \, {\left ({\left ({\left (i \, e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) - {\left ({\left (e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (-i \, e \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, e \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (-i \, e \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, e \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\frac {1}{3}}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 3.49, size = 713, normalized size = 2.67 \begin {gather*} -\frac {3 \, {\left (8 \, {\left (-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}}\right )} c^{2} e + {\left (-\frac {8 i \, \sqrt {\pi } c^{2} \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}} d^{2}} + \frac {8 i \, \sqrt {\pi } c^{2} \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}} d^{2}} + \frac {-\frac {2 i \, {\left (4 i \, {\left (d x e + c e\right )}^{\frac {5}{3}} b^{2} e^{\left (-\frac {4}{3}\right )} - 10 \, {\left (d x e + c e\right )} b e^{\left (-\frac {2}{3}\right )} - 15 i \, {\left (d x e + c e\right )}^{\frac {1}{3}}\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a\right )}}{b^{3}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}} b^{3}}}{d^{2}} + \frac {-\frac {2 i \, {\left (4 i \, {\left (d x e + c e\right )}^{\frac {5}{3}} b^{2} e^{\left (-\frac {4}{3}\right )} + 10 \, {\left (d x e + c e\right )} b e^{\left (-\frac {2}{3}\right )} - 15 i \, {\left (d x e + c e\right )}^{\frac {1}{3}}\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a\right )}}{b^{3}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}} b^{3}}}{d^{2}} - \frac {16 i \, {\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b c e^{\left (-\frac {2}{3}\right )} + c\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {1}{3}\right )}}{b^{2} d^{2}} - \frac {16 i \, {\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b c e^{\left (-\frac {2}{3}\right )} - c\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {1}{3}\right )}}{b^{2} d^{2}}\right )} d^{2} e + 16 \, {\left (\frac {i \, \sqrt {\pi } c \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a + 1\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} - \frac {i \, \sqrt {\pi } c \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a + 1\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}} - \frac {i \, {\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - 1\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {4}{3}\right )}}{b^{2}} - \frac {i \, {\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + 1\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {4}{3}\right )}}{b^{2}}\right )} c\right )}}{32 \, 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 = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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