Optimal. Leaf size=318 \[ \frac {2 \sqrt {2 \pi } b^{3/2} \sin (a) (d e-c f) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {2 \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {b^3 f \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}-\frac {b^3 f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}-\frac {b^2 f (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {(c+d x) (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {2 b \sqrt [3]{c+d x} (d e-c f) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}+\frac {b f (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2} \]
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Rubi [A] time = 0.38, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3433, 3409, 3387, 3388, 3353, 3352, 3351, 3379, 3297, 3303, 3299, 3302} \[ \frac {b^3 f \cos (a) \text {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {2 \sqrt {2 \pi } b^{3/2} \sin (a) (d e-c f) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {2 \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^3 f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}-\frac {b^2 f (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {(c+d x) (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {2 b \sqrt [3]{c+d x} (d e-c f) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}+\frac {b f (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3351
Rule 3352
Rule 3353
Rule 3379
Rule 3387
Rule 3388
Rule 3409
Rule 3433
Rubi steps
\begin {align*} \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx &=\frac {3 \operatorname {Subst}\left (\int \left ((d e-c f) x^2 \sin \left (a+\frac {b}{x^2}\right )+f x^5 \sin \left (a+\frac {b}{x^2}\right )\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=\frac {(3 f) \operatorname {Subst}\left (\int x^5 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}+\frac {(3 (d e-c f)) \operatorname {Subst}\left (\int x^2 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac {(3 f) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^4} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^2}-\frac {(3 (d e-c f)) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{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}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}-\frac {(b f) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^3} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^2}-\frac {(2 b (d e-c f)) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac {2 b (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {b f (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}+\frac {\left (b^2 f\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {\left (4 b^2 (d e-c f)\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac {2 b (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {b f (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}-\frac {b^2 f (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}+\frac {\left (b^3 f\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {\left (4 b^2 (d e-c f) \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {\left (4 b^2 (d e-c f) \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac {2 b (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {b f (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {2 b^{3/2} (d e-c f) \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {2 b^{3/2} (d e-c f) \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^2}-\frac {b^2 f (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}+\frac {\left (b^3 f \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d^2}-\frac {\left (b^3 f \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d^2}\\ &=\frac {2 b (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {b f (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {b^3 f \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {2 b^{3/2} (d e-c f) \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac {2 b^{3/2} (d e-c f) \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^2}-\frac {b^2 f (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^2}-\frac {b^3 f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2}\\ \end {align*}
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Mathematica [A] time = 1.18, size = 378, normalized size = 1.19 \[ \frac {8 \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+8 \sqrt {2 \pi } b^{3/2} d e \sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )-8 \sqrt {2 \pi } b^{3/2} c f \sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+b^3 f \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )-b^3 f \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )-b^2 f (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )-2 c^2 f \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+4 d^2 e x \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+2 d^2 f x^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+4 c d e \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+8 b d e \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )-7 b c f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b d f x \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 267, normalized size = 0.84 \[ \frac {b^{3} f \cos \relax (a) \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + b^{3} f \cos \relax (a) \operatorname {Ci}\left (-\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 2 \, b^{3} f \sin \relax (a) \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 16 \, \sqrt {2} \pi {\left (b d e - b c f\right )} \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 16 \, \sqrt {2} \pi {\left (b d e - b c f\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) \sin \relax (a) + 2 \, {\left (b d f x + 8 \, b d e - 7 \, b c f\right )} {\left (d x + c\right )}^{\frac {1}{3}} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) + 2 \, {\left (2 \, d^{2} f x^{2} + 4 \, d^{2} e x - {\left (d x + c\right )}^{\frac {2}{3}} b^{2} f + 4 \, c d e - 2 \, c^{2} f\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{8 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 225, normalized size = 0.71 \[ \frac {-\left (c f -d e \right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )+2 \left (c f -d e \right ) b \left (-\left (d x +c \right )^{\frac {1}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )+\sin \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )+\frac {f \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-f b \left (-\frac {\left (d x +c \right )^{\frac {4}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{4}-\frac {b \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{2}+b \left (\frac {\cos \relax (a ) \Ci \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\frac {\sin \relax (a ) \Si \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}\right )\right )}{2}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.77, size = 584, normalized size = 1.84 \[ \text {result too large to display} \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,\left (e+f\,x\right ) \,d x \]
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 \[ \int \left (e + f x\right ) \sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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