Optimal. Leaf size=537 \[ \frac {120960 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]
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Rubi [A] time = 0.70, antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5365, 1593, 5287, 3296, 2637, 2638} \[ \frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {120960 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 2637
Rule 2638
Rule 3296
Rule 5287
Rule 5365
Rubi steps
\begin {align*} \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (-c+x)^2 \cosh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 \operatorname {Subst}\left (\int \left (-c x+x^4\right )^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {3 \operatorname {Subst}\left (\int x^2 \left (-c+x^3\right )^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {3 \operatorname {Subst}\left (\int \left (c^2 x^2 \cosh (a+b x)-2 c x^5 \cosh (a+b x)+x^8 \cosh (a+b x)\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {3 \operatorname {Subst}\left (\int x^8 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}-\frac {(6 c) \operatorname {Subst}\left (\int x^5 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {24 \operatorname {Subst}\left (\int x^7 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac {(30 c) \operatorname {Subst}\left (\int x^4 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}-\frac {\left (6 c^2\right ) \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}\\ &=-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 \operatorname {Subst}\left (\int x^6 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {(120 c) \operatorname {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {\left (6 c^2\right ) \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}\\ &=-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {1008 \operatorname {Subst}\left (\int x^5 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {(360 c) \operatorname {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}\\ &=-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {5040 \operatorname {Subst}\left (\int x^4 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {(720 c) \operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}\\ &=-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {20160 \operatorname {Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {(720 c) \operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}\\ &=\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {60480 \operatorname {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3}\\ &=\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120960 \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3}\\ &=\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120960 \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3}\\ &=\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {120960 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 381, normalized size = 0.71 \[ \frac {3 (\sinh (a)+\cosh (a)) \left (b^8 d^2 x^2 (c+d x)^{2/3}-2 b^7 d x \sqrt [3]{c+d x} (3 c+4 d x)+2 b^6 \left (9 c^2+36 c d x+28 d^2 x^2\right )-24 b^5 (c+d x)^{2/3} (9 c+14 d x)+240 b^4 \sqrt [3]{c+d x} (6 c+7 d x)-240 b^3 (27 c+28 d x)+20160 b^2 (c+d x)^{2/3}-40320 b \sqrt [3]{c+d x}+40320\right ) \left (\sinh \left (b \sqrt [3]{c+d x}\right )+\cosh \left (b \sqrt [3]{c+d x}\right )\right )+\left (b^8 d^2 x^2 (c+d x)^{2/3}+2 b^7 d x \sqrt [3]{c+d x} (3 c+4 d x)+2 b^6 \left (9 c^2+36 c d x+28 d^2 x^2\right )+24 b^5 (c+d x)^{2/3} (9 c+14 d x)+240 b^4 \sqrt [3]{c+d x} (6 c+7 d x)+240 b^3 (27 c+28 d x)+20160 b^2 (c+d x)^{2/3}+40320 b \sqrt [3]{c+d x}+40320\right ) \left (3 \sinh \left (a+b \sqrt [3]{c+d x}\right )-3 \cosh \left (a+b \sqrt [3]{c+d x}\right )\right )}{2 b^9 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 181, normalized size = 0.34 \[ -\frac {3 \, {\left (2 \, {\left (3360 \, b^{3} d x + 3240 \, b^{3} c + 12 \, {\left (14 \, b^{5} d x + 9 \, b^{5} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} + {\left (4 \, b^{7} d^{2} x^{2} + 3 \, b^{7} c d x + 20160 \, b\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (56 \, b^{6} d^{2} x^{2} + 72 \, b^{6} c d x + 18 \, b^{6} c^{2} + {\left (b^{8} d^{2} x^{2} + 20160 \, b^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}} + 240 \, {\left (7 \, b^{4} d x + 6 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} + 40320\right )} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{9} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 2163, normalized size = 4.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 1815, normalized size = 3.38 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 642, normalized size = 1.20 \[ \frac {2 \, d^{3} x^{3} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left (\frac {c^{3} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{b} + \frac {c^{3} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b} - \frac {3 \, {\left ({\left (d x + c\right )} b^{3} e^{a} - 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 6 \, e^{a}\right )} c^{2} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{4}} - \frac {3 \, {\left ({\left (d x + c\right )} b^{3} + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 6\right )} c^{2} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{4}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} e^{a} - 120 \, {\left (d x + c\right )} b^{3} e^{a} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} - 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} + 720 \, e^{a}\right )} c e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{7}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} + 120 \, {\left (d x + c\right )} b^{3} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 720\right )} c e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{9} e^{a} - 9 \, {\left (d x + c\right )}^{\frac {8}{3}} b^{8} e^{a} + 72 \, {\left (d x + c\right )}^{\frac {7}{3}} b^{7} e^{a} - 504 \, {\left (d x + c\right )}^{2} b^{6} e^{a} + 3024 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} e^{a} - 15120 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} e^{a} + 60480 \, {\left (d x + c\right )} b^{3} e^{a} - 181440 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 362880 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 362880 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{10}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{9} + 9 \, {\left (d x + c\right )}^{\frac {8}{3}} b^{8} + 72 \, {\left (d x + c\right )}^{\frac {7}{3}} b^{7} + 504 \, {\left (d x + c\right )}^{2} b^{6} + 3024 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} + 15120 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} + 60480 \, {\left (d x + c\right )} b^{3} + 181440 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 362880 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 362880\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{10}}\right )} b}{6 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \]
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 \[ \int x^{2} \cosh {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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