Optimal. Leaf size=243 \[ \frac {3 f \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^2}-\frac {3 (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}-\frac {3 f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}+\frac {3 (d e-c f) \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^2}-\frac {3 (d e-c f) \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^2}+\frac {3 f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3514, 3466,
3435, 3433, 3432, 3460, 3377, 2718} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) (d e-c f) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^2}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) (d e-c f) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^2}+\frac {3 f \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^2}+\frac {3 f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^2}-\frac {3 \sqrt [3]{c+d x} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}-\frac {3 f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3432
Rule 3433
Rule 3435
Rule 3460
Rule 3466
Rule 3514
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {3 \text {Subst}\left (\int \left ((d e-c f) x^2 \sin \left (a+b x^2\right )+f x^5 \sin \left (a+b x^2\right )\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=\frac {(3 f) \text {Subst}\left (\int x^5 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}+\frac {(3 (d e-c f)) \text {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac {3 (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}+\frac {(3 f) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 d^2}+\frac {(3 (d e-c f)) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^2}\\ &=-\frac {3 (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}-\frac {3 f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}+\frac {(3 f) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{b d^2}+\frac {(3 (d e-c f) \cos (a)) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^2}-\frac {(3 (d e-c f) \sin (a)) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^2}\\ &=-\frac {3 (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}-\frac {3 f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}+\frac {3 (d e-c f) \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^2}-\frac {3 (d e-c f) \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^2}+\frac {3 f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^2}-\frac {(3 f) \text {Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{b^2 d^2}\\ &=\frac {3 f \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^2}-\frac {3 (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}-\frac {3 f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}+\frac {3 (d e-c f) \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^2}-\frac {3 (d e-c f) \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^2}+\frac {3 f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 213, normalized size = 0.88 \begin {gather*} \frac {3 \left (4 f \cos \left (a+b (c+d x)^{2/3}\right )-2 b^2 d e \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )-2 b^2 d f x \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )+b^{3/2} (d e-c f) \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )-b^{3/2} (d e-c f) \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+4 b f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )\right )}{4 b^3 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 175, normalized size = 0.72
method | result | size |
derivativedivides | \(\frac {-\frac {3 f \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 f \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}+\frac {3 \left (c f -d e \right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {3 \left (c f -d e \right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{2}}\) | \(175\) |
default | \(\frac {-\frac {3 f \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 f \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}+\frac {3 \left (c f -d e \right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {3 \left (c f -d e \right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{2}}\) | \(175\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 249, normalized size = 1.02 \begin {gather*} \frac {3 \, {\left (\frac {{\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 )} c f}{b^{3} d} - \frac {{\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 )} e}{b^{3}} + \frac {8 \, {\left (2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {4}{3}} b^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} f}{b^{3} d}\right )}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 164, normalized size = 0.67 \begin {gather*} -\frac {3 \, {\left (\sqrt {2} {\left (\pi b c f - \pi b d e\right )} \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} {\left (\pi b c f - \pi b d e\right )} \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 ) - 4 \, {\left (d x + c\right )}^{\frac {2}{3}} b f \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) + 2 \, {\left ({\left (b^{2} d f x + b^{2} d e\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, f\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{4 \, b^{3} 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 + 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 = 4.38, size = 407, normalized size = 1.67 \begin {gather*} -\frac {3 \, {\left ({\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 )} e - \frac {{\left (\frac {\sqrt {2} \sqrt {\pi } c \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 } c \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 i \, {\left (i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} - i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} c - 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b - 2 i\right )} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b^{3}} + \frac {2 i \, {\left (i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} - i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} c + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b - 2 i\right )} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b^{3}}\right )} f}{d}\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 = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,\left (e+f\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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