Optimal. Leaf size=175 \[ -\frac {3 b^2 \sin (a) (c+d x)^{2/3} \text {Ci}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 \cos (a) (c+d x)^{2/3} \text {Si}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}} \]
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Rubi [A] time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3431, 15, 3297, 3303, 3299, 3302} \[ -\frac {3 b^2 \sin (a) (c+d x)^{2/3} \text {CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 \cos (a) (c+d x)^{2/3} \text {Si}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3431
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac {3 \operatorname {Subst}\left (\int \frac {x^2 \sin (a+b x)}{\left (e x^3\right )^{5/3}} \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^3} \, dx,x,\sqrt [3]{c+d x}\right )}{d e (e (c+d x))^{2/3}}\\ &=-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}+\frac {\left (3 b (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {\left (3 b^2 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {\left (3 b^2 (c+d x)^{2/3} \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {\left (3 b^2 (c+d x)^{2/3} \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 (c+d x)^{2/3} \text {Ci}\left (b \sqrt [3]{c+d x}\right ) \sin (a)}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 (c+d x)^{2/3} \cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 115, normalized size = 0.66 \[ -\frac {3 \left (b^2 \sin (a) (c+d x)^{2/3} \text {Ci}\left (b \sqrt [3]{c+d x}\right )+b^2 \cos (a) (c+d x)^{2/3} \text {Si}\left (b \sqrt [3]{c+d x}\right )+\sin \left (a+b \sqrt [3]{c+d x}\right )+b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )\right )}{2 d e (e (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d e x + c e\right )}^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
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 \[ \int \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.62, size = 129, normalized size = 0.74 \[ \frac {{\left ({\left (3 i \, \Gamma \left (-2, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 3 i \, \Gamma \left (-2, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 3 i \, \Gamma \left (-2, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) - 3 i \, \Gamma \left (-2, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \relax (a) + 3 \, {\left (\Gamma \left (-2, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-2, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-2, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (-2, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \relax (a)\right )} b^{2}}{4 \, d e^{\frac {5}{3}}} \]
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 \[ \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{5/3}} \,d x \]
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 \[ \int \frac {\sin {\left (a + b \sqrt [3]{c + d x} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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