Optimal. Leaf size=597 \[ \frac {a^2 x^3}{3}+\frac {945 a b \text {Li}_9\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}-\frac {1890 i a b \sqrt [3]{x} \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}-\frac {1890 a b x^{2/3} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}+\frac {2}{3} i a b x^3+\frac {945 i b^2 \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}+\frac {1890 b^2 \sqrt [3]{x} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}-\frac {1890 i b^2 x^{2/3} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac {3 i b^2 x^{8/3}}{d}-\frac {b^2 x^3}{3} \]
[Out]
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Rubi [A] time = 0.83, antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3747, 3722, 3719, 2190, 2531, 6609, 2282, 6589, 3720, 30} \[ \frac {a^2 x^3}{3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {1890 a b x^{2/3} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}-\frac {1890 i a b \sqrt [3]{x} \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac {945 a b \text {Li}_9\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}+\frac {2}{3} i a b x^3-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {1890 i b^2 x^{2/3} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {1890 b^2 \sqrt [3]{x} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac {945 i b^2 \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac {3 i b^2 x^{8/3}}{d}-\frac {b^2 x^3}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3719
Rule 3720
Rule 3722
Rule 3747
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^2 \left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2 \, dx &=3 \operatorname {Subst}\left (\int x^8 (a+b \tan (c+d x))^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (a^2 x^8+2 a b x^8 \tan (c+d x)+b^2 x^8 \tan ^2(c+d x)\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^2 x^3}{3}+(6 a b) \operatorname {Subst}\left (\int x^8 \tan (c+d x) \, dx,x,\sqrt [3]{x}\right )+\left (3 b^2\right ) \operatorname {Subst}\left (\int x^8 \tan ^2(c+d x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-(12 i a b) \operatorname {Subst}\left (\int \frac {e^{2 i (c+d x)} x^8}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt [3]{x}\right )-\left (3 b^2\right ) \operatorname {Subst}\left (\int x^8 \, dx,x,\sqrt [3]{x}\right )-\frac {\left (24 b^2\right ) \operatorname {Subst}\left (\int x^7 \tan (c+d x) \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}+\frac {(48 a b) \operatorname {Subst}\left (\int x^7 \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (48 i b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i (c+d x)} x^7}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac {(168 i a b) \operatorname {Subst}\left (\int x^6 \text {Li}_2\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^2}-\frac {\left (168 b^2\right ) \operatorname {Subst}\left (\int x^6 \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^2}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}+\frac {(504 a b) \operatorname {Subst}\left (\int x^5 \text {Li}_3\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^3}+\frac {\left (504 i b^2\right ) \operatorname {Subst}\left (\int x^5 \text {Li}_2\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^3}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}+\frac {(1260 i a b) \operatorname {Subst}\left (\int x^4 \text {Li}_4\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (1260 b^2\right ) \operatorname {Subst}\left (\int x^4 \text {Li}_3\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac {(2520 a b) \operatorname {Subst}\left (\int x^3 \text {Li}_5\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^5}-\frac {\left (2520 i b^2\right ) \operatorname {Subst}\left (\int x^3 \text {Li}_4\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^5}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}-\frac {(3780 i a b) \operatorname {Subst}\left (\int x^2 \text {Li}_6\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^6}+\frac {\left (3780 b^2\right ) \operatorname {Subst}\left (\int x^2 \text {Li}_5\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^6}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {1890 i b^2 x^{2/3} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}-\frac {1890 a b x^{2/3} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}+\frac {(3780 a b) \operatorname {Subst}\left (\int x \text {Li}_7\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^7}+\frac {\left (3780 i b^2\right ) \operatorname {Subst}\left (\int x \text {Li}_6\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^7}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {1890 i b^2 x^{2/3} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {1890 b^2 \sqrt [3]{x} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}-\frac {1890 a b x^{2/3} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}-\frac {1890 i a b \sqrt [3]{x} \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}+\frac {(1890 i a b) \operatorname {Subst}\left (\int \text {Li}_8\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^8}-\frac {\left (1890 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_7\left (-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^8}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {1890 i b^2 x^{2/3} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {1890 b^2 \sqrt [3]{x} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}-\frac {1890 a b x^{2/3} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}-\frac {1890 i a b \sqrt [3]{x} \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}+\frac {(945 a b) \operatorname {Subst}\left (\int \frac {\text {Li}_8(-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}+\frac {\left (945 i b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_7(-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}\\ &=-\frac {3 i b^2 x^{8/3}}{d}+\frac {a^2 x^3}{3}+\frac {2}{3} i a b x^3-\frac {b^2 x^3}{3}+\frac {24 b^2 x^{7/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}-\frac {6 a b x^{8/3} \log \left (1+e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {84 i b^2 x^2 \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {24 i a b x^{7/3} \text {Li}_2\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {252 b^2 x^{5/3} \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {84 a b x^2 \text {Li}_3\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^3}+\frac {630 i b^2 x^{4/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {252 i a b x^{5/3} \text {Li}_4\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^4}-\frac {1260 b^2 x \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {630 a b x^{4/3} \text {Li}_5\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^5}-\frac {1890 i b^2 x^{2/3} \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {1260 i a b x \text {Li}_6\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^6}+\frac {1890 b^2 \sqrt [3]{x} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}-\frac {1890 a b x^{2/3} \text {Li}_7\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^7}+\frac {945 i b^2 \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}-\frac {1890 i a b \sqrt [3]{x} \text {Li}_8\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^8}+\frac {945 a b \text {Li}_9\left (-e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^9}+\frac {3 b^2 x^{8/3} \tan \left (c+d \sqrt [3]{x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 4.89, size = 599, normalized size = 1.00 \[ \frac {1}{3} \left (x^3 \left (a^2+2 a b \tan (c)-b^2\right )+b \left (\frac {9 a \left (-8 i d^7 x^{7/3} \text {Li}_2\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-28 d^6 x^2 \text {Li}_3\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+84 i d^5 x^{5/3} \text {Li}_4\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+105 \left (2 d^4 x^{4/3} \text {Li}_5\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-4 i d^3 x \text {Li}_6\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-6 d^2 x^{2/3} \text {Li}_7\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+6 i d \sqrt [3]{x} \text {Li}_8\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+3 \text {Li}_9\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )\right )\right )}{d^9}-\frac {18 a x^{8/3} \log \left (1+e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d}-\frac {4 i a x^3}{1+e^{2 i c}}+\frac {72 b x^{7/3} \log \left (1+e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )}{d^2}+\frac {63 b \left (4 i d^6 x^2 \text {Li}_2\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+12 d^5 x^{5/3} \text {Li}_3\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-15 i \left (2 d^4 x^{4/3} \text {Li}_4\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-4 i d^3 x \text {Li}_5\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )-6 d^2 x^{2/3} \text {Li}_6\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+6 i d \sqrt [3]{x} \text {Li}_7\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )+3 \text {Li}_8\left (-e^{-2 i \left (c+d \sqrt [3]{x}\right )}\right )\right )\right )}{d^9}+\frac {18 i b x^{8/3}}{d+e^{2 i c} d}\right )+\frac {9 b^2 x^{8/3} \sec (c) \sin \left (d \sqrt [3]{x}\right ) \sec \left (c+d \sqrt [3]{x}\right )}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \tan \left (d x^{\frac {1}{3}} + c\right )^{2} + 2 \, a b x^{2} \tan \left (d x^{\frac {1}{3}} + c\right ) + a^{2} x^{2}, x\right ) \]
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 \[ \int {\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.42, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.16, size = 4673, normalized size = 7.83 \[ \text {result too large to display} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2 \,d x \]
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 \[ \int x^{2} \left (a + b \tan {\left (c + d \sqrt [3]{x} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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