Optimal. Leaf size=176 \[ \frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac {b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{16 c^{8/3}}-\frac {3 b x^5}{40 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5033, 321, 295, 634, 618, 204, 628, 203} \[ \frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac {b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{16 c^{8/3}}-\frac {3 b x^5}{40 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 204
Rule 295
Rule 321
Rule 618
Rule 628
Rule 634
Rule 5033
Rubi steps
\begin {align*} \int x^7 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{8} (3 b c) \int \frac {x^{10}}{1+c^2 x^6} \, dx\\ &=-\frac {3 b x^5}{40 c}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {(3 b) \int \frac {x^4}{1+c^2 x^6} \, dx}{8 c}\\ &=-\frac {3 b x^5}{40 c}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \int \frac {1}{1+c^{2/3} x^2} \, dx}{8 c^{7/3}}+\frac {b \int \frac {-\frac {1}{2}+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{7/3}}+\frac {b \int \frac {-\frac {1}{2}-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{7/3}}\\ &=-\frac {3 b x^5}{40 c}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {\left (\sqrt {3} b\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{8/3}}-\frac {\left (\sqrt {3} b\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{8/3}}+\frac {b \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{7/3}}+\frac {b \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{7/3}}\\ &=-\frac {3 b x^5}{40 c}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{16 \sqrt {3} c^{8/3}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{16 \sqrt {3} c^{8/3}}\\ &=-\frac {3 b x^5}{40 c}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac {1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac {b \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 181, normalized size = 1.03 \[ \frac {a x^8}{8}+\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac {b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{16 c^{8/3}}-\frac {3 b x^5}{40 c}+\frac {1}{8} b x^8 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 439, normalized size = 2.49 \[ \frac {20 \, b c x^{8} \arctan \left (c x^{3}\right ) + 20 \, a c x^{8} - 12 \, b x^{5} - 5 \, \sqrt {3} c \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} \log \left (\sqrt {3} b^{5} c^{13} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {5}{6}} + b^{6} c^{10} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {2}{3}} + b^{10} x^{2}\right ) + 5 \, \sqrt {3} c \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} \log \left (-\sqrt {3} b^{5} c^{13} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {5}{6}} + b^{6} c^{10} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {2}{3}} + b^{10} x^{2}\right ) - 20 \, c \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, b^{5} c^{3} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} + \sqrt {3} b^{6} - 2 \, \sqrt {\sqrt {3} b^{5} c^{13} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {5}{6}} + b^{6} c^{10} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {2}{3}} + b^{10} x^{2}} c^{3} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}}}{b^{6}}\right ) - 20 \, c \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, b^{5} c^{3} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} - \sqrt {3} b^{6} - 2 \, \sqrt {-\sqrt {3} b^{5} c^{13} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {5}{6}} + b^{6} c^{10} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {2}{3}} + b^{10} x^{2}} c^{3} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}}}{b^{6}}\right ) - 40 \, c \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} \arctan \left (-\frac {b^{5} c^{3} x \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}} - \sqrt {b^{6} c^{10} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {2}{3}} + b^{10} x^{2}} c^{3} \left (\frac {b^{6}}{c^{16}}\right )^{\frac {1}{6}}}{b^{6}}\right )}{160 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 2.99, size = 171, normalized size = 0.97 \[ -\frac {1}{32} \, b c^{15} {\left (\frac {\sqrt {3} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{16} {\left | c \right |}^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{16} {\left | c \right |}^{\frac {5}{3}}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{18}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{18}} - \frac {4 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{18}}\right )} + \frac {5 \, b c x^{8} \arctan \left (c x^{3}\right ) + 5 \, a c x^{8} - 3 \, b x^{5}}{40 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 167, normalized size = 0.95 \[ \frac {x^{8} a}{8}+\frac {x^{8} b \arctan \left (c \,x^{3}\right )}{8}-\frac {3 b \,x^{5}}{40 c}+\frac {b \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{8 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {b \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{32 c}+\frac {b \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{16 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {b \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{32 c}+\frac {b \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{16 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 152, normalized size = 0.86 \[ \frac {1}{8} \, a x^{8} + \frac {1}{160} \, {\left (20 \, x^{8} \arctan \left (c x^{3}\right ) - {\left (\frac {12 \, x^{5}}{c^{2}} + \frac {5 \, {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}}{c^{2}}\right )} c\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.00, size = 122, normalized size = 0.69 \[ \frac {a\,x^8}{8}-\frac {3\,b\,x^5}{40\,c}-\frac {b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{16\,c^{8/3}}+\frac {b\,x^8\,\mathrm {atan}\left (c\,x^3\right )}{8}+\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )\,1{}\mathrm {i}}{16\,c^{8/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 149.06, size = 320, normalized size = 1.82 \[ \begin {cases} \frac {a x^{8}}{8} + \frac {b x^{8} \operatorname {atan}{\left (c x^{3} \right )}}{8} - \frac {3 b x^{5}}{40 c} - \frac {3 \left (-1\right )^{\frac {5}{6}} b \log {\left (4 x^{2} - 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{32 c^{3} \sqrt [6]{\frac {1}{c^{2}}}} + \frac {3 \left (-1\right )^{\frac {5}{6}} b \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{32 c^{3} \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\left (-1\right )^{\frac {5}{6}} \sqrt {3} b \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{16 c^{3} \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\left (-1\right )^{\frac {5}{6}} \sqrt {3} b \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{16 c^{3} \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\sqrt [3]{-1} b \operatorname {atan}{\left (c x^{3} \right )}}{8 c^{4} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}} & \text {for}\: c \neq 0 \\\frac {a x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________