Optimal. Leaf size=130 \[ -\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d}-\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d} \]
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Rubi [A]
time = 0.05, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3444, 3466,
3435, 3433, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d}-\frac {3 \sqrt [3]{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 3444
Rule 3466
Rubi steps
\begin {align*} \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {3 \text {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=-\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {3 \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d}\\ &=-\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {(3 \cos (a)) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d}-\frac {(3 \sin (a)) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d}\\ &=-\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d}-\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 114, normalized size = 0.88 \begin {gather*} -\frac {3 \left (2 \sqrt {b} \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )-\sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)\right )}{4 b^{3/2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 86, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {-\frac {3 \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d}\) | \(86\) |
default | \(\frac {-\frac {3 \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.33, size = 92, normalized size = 0.71 \begin {gather*} -\frac {3 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{16 \, b^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 98, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) - \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{4 \, b^{2} 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 \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \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 = 3.69, size = 170, normalized size = 1.31 \begin {gather*} -\frac {3 \, {\left (\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b}\right )}}{8 \, 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 \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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