Optimal. Leaf size=261 \[ \frac {360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]
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Rubi [A] time = 0.23, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3432, 3296, 2637, 2638} \[ -\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3432
Rubi steps
\begin {align*} \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \operatorname {Subst}\left (\int \left (-\frac {c x^2 \cos (a+b x)}{d}+\frac {x^5 \cos (a+b x)}{d}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {3 \operatorname {Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}-\frac {(3 c) \operatorname {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {15 \operatorname {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}+\frac {(6 c) \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}\\ &=-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 \operatorname {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {(6 c) \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}\\ &=-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {180 \operatorname {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^2}\\ &=-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac {360 \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^2}\\ &=-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {360 \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^2}\\ &=\frac {360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 117, normalized size = 0.45 \[ \frac {3 \left (b \left (b^4 d x (c+d x)^{2/3}-2 b^2 (9 c+10 d x)+120 \sqrt [3]{c+d x}\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )+\left (b^4 \sqrt [3]{c+d x} (3 c+5 d x)-60 b^2 (c+d x)^{2/3}+120\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^6 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 110, normalized size = 0.42 \[ -\frac {3 \, {\left ({\left (60 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} - {\left (5 \, b^{4} d x + 3 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{5} d x - 20 \, b^{3} d x - 18 \, b^{3} c + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 370, normalized size = 1.42 \[ -\frac {3 \, {\left (\frac {{\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} b^{3} c - 2 \, a b^{3} c - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a - 30 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{2} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{3} - 5 \, a^{4} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 120 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + 60 \, a^{2} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}} + \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} b^{3} c - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} a - 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{3} - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{4} + a^{5} - 2 \, b^{3} c + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{2} - 20 \, a^{3} - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}}\right )}}{b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 655, normalized size = 2.51 \[ \frac {-3 c \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+6 a c \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 a^{2} c \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\frac {3 \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{5} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+5 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-20 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-60 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {15 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+4 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-12 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+24 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-24 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {30 a^{2} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {30 a^{3} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {15 a^{4} \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {3 a^{5} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3}}}{d^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 523, normalized size = 2.00 \[ -\frac {3 \, {\left (a^{2} c \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a c + \frac {a^{5} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{3}} - \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{4}}{b^{3}} + {\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} c + \frac {10 \, {\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{3}}{b^{3}} - \frac {10 \, {\left (3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 6 \, a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{2}}{b^{3}} + \frac {5 \, {\left (4 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 6 \, a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 12 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 24\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a}{b^{3}} - \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 12 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 24\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 120 \, a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{3}}\right )}}{b^{3} d^{2}} \]
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 \[ \int x\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \]
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 \[ \int x \cos {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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