Integrand size = 18, antiderivative size = 537 \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {120960 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]
[Out]
Time = 0.83 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3513, 3377, 2717, 2718} \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {120960 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac {60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]
[In]
[Out]
Rule 2717
Rule 2718
Rule 3377
Rule 3513
Rubi steps \begin{align*} \text {integral}& = \frac {3 \text {Subst}\left (\int \left (\frac {c^2 x^2 \cos (a+b x)}{d^2}-\frac {2 c x^5 \cos (a+b x)}{d^2}+\frac {x^8 \cos (a+b x)}{d^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d} \\ & = \frac {3 \text {Subst}\left (\int x^8 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}-\frac {(6 c) \text {Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3} \\ & = \frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {24 \text {Subst}\left (\int x^7 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac {(30 c) \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}-\frac {\left (6 c^2\right ) \text {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3} \\ & = \frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 \text {Subst}\left (\int x^6 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {(120 c) \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {\left (6 c^2\right ) \text {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3} \\ & = \frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {1008 \text {Subst}\left (\int x^5 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {(360 c) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3} \\ & = \frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {5040 \text {Subst}\left (\int x^4 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {(720 c) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3} \\ & = \frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {20160 \text {Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {(720 c) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3} \\ & = -\frac {720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {60480 \text {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3} \\ & = -\frac {720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120960 \text {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3} \\ & = -\frac {720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120960 \text {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3} \\ & = -\frac {720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {120960 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.71 \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 e^{-i \left (a+b \sqrt [3]{c+d x}\right )} \left (-40320 i \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )-40320 b \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) \sqrt [3]{c+d x}+20160 i b^2 \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) (c+d x)^{2/3}-i b^8 d^2 \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) x^2 (c+d x)^{2/3}+2 b^7 d \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) x \sqrt [3]{c+d x} (3 c+4 d x)-240 i b^4 \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) \sqrt [3]{c+d x} (6 c+7 d x)-24 b^5 \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) (c+d x)^{2/3} (9 c+14 d x)+240 b^3 \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) (27 c+28 d x)+2 i b^6 \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) \left (9 c^2+36 c d x+28 d^2 x^2\right )\right )}{2 b^9 d^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1808\) vs. \(2(477)=954\).
Time = 1.62 (sec) , antiderivative size = 1809, normalized size of antiderivative = 3.37
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1809\) |
default | \(\text {Expression too large to display}\) | \(1809\) |
parts | \(\text {Expression too large to display}\) | \(2944\) |
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none
Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.34 \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (2 \, {\left (3360 \, b^{3} d x + 3240 \, b^{3} c - 12 \, {\left (14 \, b^{5} d x + 9 \, b^{5} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} + {\left (4 \, b^{7} d^{2} x^{2} + 3 \, b^{7} c d x - 20160 \, b\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (56 \, b^{6} d^{2} x^{2} + 72 \, b^{6} c d x + 18 \, b^{6} c^{2} - {\left (b^{8} d^{2} x^{2} - 20160 \, b^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}} - 240 \, {\left (7 \, b^{4} d x + 6 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 40320\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{9} d^{3}} \]
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\[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x^{2} \cos {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1349 vs. \(2 (477) = 954\).
Time = 0.40 (sec) , antiderivative size = 1349, normalized size of antiderivative = 2.51 \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (477) = 954\).
Time = 0.28 (sec) , antiderivative size = 1104, normalized size of antiderivative = 2.06 \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x^2\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \]
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