Optimal. Leaf size=192 \[ \frac {(d+e x)^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{2 e}-\frac {b e \log \left (c^2 x^4+1\right )}{4 c}-\frac {b d^2 \tan ^{-1}\left (c x^2\right )}{2 e}-\frac {b d \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b d \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b d \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b d \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}} \]
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Rubi [A] time = 0.21, antiderivative size = 191, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6742, 5027, 297, 1162, 617, 204, 1165, 628, 5033, 260} \[ \frac {a (d+e x)^2}{2 e}-\frac {b e \log \left (c^2 x^4+1\right )}{4 c}-\frac {b d \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b d \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \sqrt {c}}+b d x \tan ^{-1}\left (c x^2\right )+\frac {b d \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b d \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )}{\sqrt {2} \sqrt {c}}+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 260
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5027
Rule 5033
Rule 6742
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\int \left (a (d+e x)+b (d+e x) \tan ^{-1}\left (c x^2\right )\right ) \, dx\\ &=\frac {a (d+e x)^2}{2 e}+b \int (d+e x) \tan ^{-1}\left (c x^2\right ) \, dx\\ &=\frac {a (d+e x)^2}{2 e}+b \int \left (d \tan ^{-1}\left (c x^2\right )+e x \tan ^{-1}\left (c x^2\right )\right ) \, dx\\ &=\frac {a (d+e x)^2}{2 e}+(b d) \int \tan ^{-1}\left (c x^2\right ) \, dx+(b e) \int x \tan ^{-1}\left (c x^2\right ) \, dx\\ &=\frac {a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-(2 b c d) \int \frac {x^2}{1+c^2 x^4} \, dx-(b c e) \int \frac {x^3}{1+c^2 x^4} \, dx\\ &=\frac {a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-\frac {b e \log \left (1+c^2 x^4\right )}{4 c}+(b d) \int \frac {1-c x^2}{1+c^2 x^4} \, dx-(b d) \int \frac {1+c x^2}{1+c^2 x^4} \, dx\\ &=\frac {a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-\frac {b e \log \left (1+c^2 x^4\right )}{4 c}-\frac {(b d) \int \frac {1}{\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{2 c}-\frac {(b d) \int \frac {1}{\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{2 c}-\frac {(b d) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}+2 x}{-\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{2 \sqrt {2} \sqrt {c}}-\frac {(b d) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}-2 x}{-\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{2 \sqrt {2} \sqrt {c}}\\ &=\frac {a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )-\frac {b d \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b d \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}}-\frac {b e \log \left (1+c^2 x^4\right )}{4 c}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}\\ &=\frac {a (d+e x)^2}{2 e}+b d x \tan ^{-1}\left (c x^2\right )+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right )+\frac {b d \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b d \tan ^{-1}\left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \sqrt {c}}-\frac {b d \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}}+\frac {b d \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \sqrt {c}}-\frac {b e \log \left (1+c^2 x^4\right )}{4 c}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 153, normalized size = 0.80 \[ a d x+\frac {1}{2} a e x^2-\frac {b e \log \left (c^2 x^4+1\right )}{4 c}+b d x \tan ^{-1}\left (c x^2\right )-\frac {b d \left (\log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )-\log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )-2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )+2 \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )\right )}{2 \sqrt {2} \sqrt {c}}+\frac {1}{2} b e x^2 \tan ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 534, normalized size = 2.78 \[ \frac {2 \, a b^{4} c d^{4} e x^{2} + 4 \, a b^{4} c d^{5} x + 4 \, \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {1}{4}} b^{4} c d^{4} \arctan \left (-\frac {b^{8} d^{8} + \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {5}{4}} b^{3} c^{3} d^{3} x - \sqrt {2} \sqrt {b^{6} d^{6} x^{2} + \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {3}{4}} b^{3} c d^{3} x + \sqrt {\frac {b^{4} d^{4}}{c^{2}}} b^{4} d^{4}} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {5}{4}} c^{3}}{b^{8} d^{8}}\right ) + 4 \, \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {1}{4}} b^{4} c d^{4} \arctan \left (\frac {b^{8} d^{8} - \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {5}{4}} b^{3} c^{3} d^{3} x + \sqrt {2} \sqrt {b^{6} d^{6} x^{2} - \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {3}{4}} b^{3} c d^{3} x + \sqrt {\frac {b^{4} d^{4}}{c^{2}}} b^{4} d^{4}} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {5}{4}} c^{3}}{b^{8} d^{8}}\right ) + 2 \, {\left (b^{5} c d^{4} e x^{2} + 2 \, b^{5} c d^{5} x\right )} \arctan \left (c x^{2}\right ) - {\left (b^{5} d^{4} e - \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {1}{4}} b^{4} c d^{4}\right )} \log \left (b^{6} d^{6} x^{2} + \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {3}{4}} b^{3} c d^{3} x + \sqrt {\frac {b^{4} d^{4}}{c^{2}}} b^{4} d^{4}\right ) - {\left (b^{5} d^{4} e + \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {1}{4}} b^{4} c d^{4}\right )} \log \left (b^{6} d^{6} x^{2} - \sqrt {2} \left (\frac {b^{4} d^{4}}{c^{2}}\right )^{\frac {3}{4}} b^{3} c d^{3} x + \sqrt {\frac {b^{4} d^{4}}{c^{2}}} b^{4} d^{4}\right )}{4 \, b^{4} c d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.10, size = 201, normalized size = 1.05 \[ -\frac {1}{4} \, b c^{3} d {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{4}} + \frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{4}} - \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2} {\left | c \right |}^{\frac {3}{2}}} + \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{4}}\right )} + \frac {2 \, b c x^{2} \arctan \left (c x^{2}\right ) e + 4 \, b c d x \arctan \left (c x^{2}\right ) + 2 \, a c x^{2} e + 4 \, a c d x - b e \log \left (c^{2} x^{4} + 1\right )}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 167, normalized size = 0.87 \[ \frac {a \,x^{2} e}{2}+a d x +\frac {b \arctan \left (c \,x^{2}\right ) x^{2} e}{2}+b \arctan \left (c \,x^{2}\right ) d x -\frac {b d \sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b d \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b d \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b e \ln \left (c^{2} x^{4}+1\right )}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 168, normalized size = 0.88 \[ \frac {1}{2} \, a e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b d + a d x + \frac {{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b e}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 203, normalized size = 1.06 \[ a\,d\,x+\frac {a\,e\,x^2}{2}+b\,d\,x\,\mathrm {atan}\left (c\,x^2\right )-\frac {b\,e\,\ln \left (x\,\sqrt {-c\,1{}\mathrm {i}}-1\right )}{4\,c}-\frac {b\,e\,\ln \left (x\,\sqrt {-c\,1{}\mathrm {i}}+1\right )}{4\,c}-\frac {b\,e\,\ln \left (x\,\sqrt {c\,1{}\mathrm {i}}-1\right )}{4\,c}-\frac {b\,e\,\ln \left (x\,\sqrt {c\,1{}\mathrm {i}}+1\right )}{4\,c}+\frac {b\,e\,x^2\,\mathrm {atan}\left (c\,x^2\right )}{2}-\frac {b\,d\,\ln \left (x\,\sqrt {-c\,1{}\mathrm {i}}-1\right )\,\sqrt {-c\,1{}\mathrm {i}}}{2\,c}+\frac {b\,d\,\ln \left (x\,\sqrt {-c\,1{}\mathrm {i}}+1\right )\,\sqrt {-c\,1{}\mathrm {i}}}{2\,c}-\frac {b\,d\,\ln \left (x\,\sqrt {c\,1{}\mathrm {i}}-1\right )\,\sqrt {c\,1{}\mathrm {i}}}{2\,c}+\frac {b\,d\,\ln \left (x\,\sqrt {c\,1{}\mathrm {i}}+1\right )\,\sqrt {c\,1{}\mathrm {i}}}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.15, size = 1734, normalized size = 9.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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