Optimal. Leaf size=328 \[ \frac {b c^2 d^3 \text {ArcTan}\left (c x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {a+b \text {ArcTan}\left (c x^2\right )}{e (d+e x)}+\frac {b \sqrt {c} \left (c d^2-e^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {b \sqrt {c} \left (c d^2-e^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}-\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b c d e \log \left (1+c^2 x^4\right )}{2 \left (c^2 d^4+e^4\right )} \]
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Rubi [A]
time = 0.35, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {4980, 6857,
1890, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 209, 266} \begin {gather*} -\frac {a+b \text {ArcTan}\left (c x^2\right )}{e (d+e x)}+\frac {b c^2 d^3 \text {ArcTan}\left (c x^2\right )}{e \left (c^2 d^4+e^4\right )}+\frac {b \sqrt {c} \text {ArcTan}\left (1-\sqrt {2} \sqrt {c} x\right ) \left (c d^2-e^2\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {b \sqrt {c} \text {ArcTan}\left (\sqrt {2} \sqrt {c} x+1\right ) \left (c d^2-e^2\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b c d e \log \left (c^2 x^4+1\right )}{2 \left (c^2 d^4+e^4\right )}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}-\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 266
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1890
Rule 4980
Rule 6857
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}\left (c x^2\right )}{(d+e x)^2} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {b \int \frac {2 c x}{(d+e x) \left (1+c^2 x^4\right )} \, dx}{e}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {(2 b c) \int \frac {x}{(d+e x) \left (1+c^2 x^4\right )} \, dx}{e}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {(2 b c) \int \left (-\frac {d e^3}{\left (c^2 d^4+e^4\right ) (d+e x)}+\frac {e^3+c^2 d^3 x-c^2 d^2 e x^2+c^2 d e^2 x^3}{\left (c^2 d^4+e^4\right ) \left (1+c^2 x^4\right )}\right ) \, dx}{e}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}+\frac {(2 b c) \int \frac {e^3+c^2 d^3 x-c^2 d^2 e x^2+c^2 d e^2 x^3}{1+c^2 x^4} \, dx}{e \left (c^2 d^4+e^4\right )}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}+\frac {(2 b c) \int \left (\frac {e^3-c^2 d^2 e x^2}{1+c^2 x^4}+\frac {x \left (c^2 d^3+c^2 d e^2 x^2\right )}{1+c^2 x^4}\right ) \, dx}{e \left (c^2 d^4+e^4\right )}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}+\frac {(2 b c) \int \frac {e^3-c^2 d^2 e x^2}{1+c^2 x^4} \, dx}{e \left (c^2 d^4+e^4\right )}+\frac {(2 b c) \int \frac {x \left (c^2 d^3+c^2 d e^2 x^2\right )}{1+c^2 x^4} \, dx}{e \left (c^2 d^4+e^4\right )}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}+\frac {(b c) \text {Subst}\left (\int \frac {c^2 d^3+c^2 d e^2 x}{1+c^2 x^2} \, dx,x,x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {\left (b \left (c d^2-e^2\right )\right ) \int \frac {c+c^2 x^2}{1+c^2 x^4} \, dx}{c^2 d^4+e^4}+\frac {\left (b \left (c d^2+e^2\right )\right ) \int \frac {c-c^2 x^2}{1+c^2 x^4} \, dx}{c^2 d^4+e^4}\\ &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}+\frac {\left (b c^3 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^2\right )}{e \left (c^2 d^4+e^4\right )}+\frac {\left (b c^3 d e\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^2} \, dx,x,x^2\right )}{c^2 d^4+e^4}-\frac {\left (b \left (c d^2-e^2\right )\right ) \int \frac {1}{\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{2 \left (c^2 d^4+e^4\right )}-\frac {\left (b \left (c d^2-e^2\right )\right ) \int \frac {1}{\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx}{2 \left (c^2 d^4+e^4\right )}-\frac {\left (b \sqrt {c} \left (c d^2+e^2\right )\right ) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}+2 x}{-\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {\left (b \sqrt {c} \left (c d^2+e^2\right )\right ) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}-2 x}{-\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}\\ &=\frac {b c^2 d^3 \tan ^{-1}\left (c x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}-\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b c d e \log \left (1+c^2 x^4\right )}{2 \left (c^2 d^4+e^4\right )}-\frac {\left (b \sqrt {c} \left (c d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {\left (b \sqrt {c} \left (c d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}\\ &=\frac {b c^2 d^3 \tan ^{-1}\left (c x^2\right )}{e \left (c^2 d^4+e^4\right )}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{e (d+e x)}+\frac {b \sqrt {c} \left (c d^2-e^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {b \sqrt {c} \left (c d^2-e^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {c} x\right )}{\sqrt {2} \left (c^2 d^4+e^4\right )}-\frac {2 b c d e \log (d+e x)}{c^2 d^4+e^4}-\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b \sqrt {c} \left (c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{2 \sqrt {2} \left (c^2 d^4+e^4\right )}+\frac {b c d e \log \left (1+c^2 x^4\right )}{2 \left (c^2 d^4+e^4\right )}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 321, normalized size = 0.98 \begin {gather*} -\frac {4 a \left (c^2 d^4+e^4\right )+4 b \left (c^2 d^4+e^4\right ) \text {ArcTan}\left (c x^2\right )+2 b \sqrt {c} \left (2 c^{3/2} d^3-\sqrt {2} c d^2 e+\sqrt {2} e^3\right ) (d+e x) \text {ArcTan}\left (1-\sqrt {2} \sqrt {c} x\right )+2 b \sqrt {c} \left (2 c^{3/2} d^3+\sqrt {2} c d^2 e-\sqrt {2} e^3\right ) (d+e x) \text {ArcTan}\left (1+\sqrt {2} \sqrt {c} x\right )+8 b c d e^2 (d+e x) \log (d+e x)+\sqrt {2} b \sqrt {c} e \left (c d^2+e^2\right ) (d+e x) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )-\sqrt {2} b \sqrt {c} e \left (c d^2+e^2\right ) (d+e x) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )-2 b c d e^2 (d+e x) \log \left (1+c^2 x^4\right )}{4 e \left (c^2 d^4+e^4\right ) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 433, normalized size = 1.32
method | result | size |
default | \(-\frac {a}{\left (e x +d \right ) e}-\frac {b \arctan \left (c \,x^{2}\right )}{\left (e x +d \right ) e}-\frac {2 b c d e \ln \left (e x +d \right )}{c^{2} d^{4}+e^{4}}+\frac {b \,e^{2} c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )}{2 c^{2} d^{4}+2 e^{4}}+\frac {b \,e^{2} c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )}{2 c^{2} d^{4}+2 e^{4}}+\frac {b \,e^{2} c \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )}{4 c^{2} d^{4}+4 e^{4}}+\frac {b \,c^{3} d^{3} \arctan \left (x^{2} \sqrt {c^{2}}\right )}{e \left (c^{2} d^{4}+e^{4}\right ) \sqrt {c^{2}}}-\frac {b c \,d^{2} \sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )}{4 \left (c^{2} d^{4}+e^{4}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b c \,d^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )}{2 \left (c^{2} d^{4}+e^{4}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b c \,d^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )}{2 \left (c^{2} d^{4}+e^{4}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+\frac {b c d e \ln \left (c^{2} x^{4}+1\right )}{2 c^{2} d^{4}+2 e^{4}}\) | \(433\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 286, normalized size = 0.87 \begin {gather*} -\frac {1}{4} \, {\left ({\left (\frac {8 \, d e \log \left (x e + d\right )}{c^{2} d^{4} + e^{4}} - \frac {\frac {\sqrt {2} {\left (c d^{2} e + \sqrt {2} \sqrt {c} d e^{2} + e^{3}\right )} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}} - \frac {\sqrt {2} {\left (c d^{2} e - \sqrt {2} \sqrt {c} d e^{2} + e^{3}\right )} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}} - \frac {2 \, {\left (2 \, c^{2} d^{3} + \sqrt {2} c^{\frac {3}{2}} d^{2} e - \sqrt {2} \sqrt {c} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c} + \frac {2 \, {\left (2 \, c^{2} d^{3} - \sqrt {2} c^{\frac {3}{2}} d^{2} e + \sqrt {2} \sqrt {c} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c}}{c^{2} d^{4} e + e^{5}}\right )} c + \frac {4 \, \arctan \left (c x^{2}\right )}{x e^{2} + d e}\right )} b - \frac {a}{x e^{2} + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 883, normalized size = 2.69 \begin {gather*} \left (\sum _{k=1}^4\ln \left (\frac {{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^4\,c^8\,e^9\,x\,320-{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^4\,c^{10}\,d^5\,e^4\,128+16\,b^4\,c^{10}\,e\,x-\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )\,b^3\,c^9\,e^3\,8+{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^4\,c^8\,d\,e^8\,384+\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )\,b^3\,c^{11}\,d^3\,x\,8-{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^3\,b\,c^9\,d^2\,e^5\,320-{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^4\,c^{10}\,d^4\,e^5\,x\,192+{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^3\,b\,c^{11}\,d^5\,e^2\,x\,32+{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^2\,b^2\,c^{10}\,d^2\,e^3\,x\,64-{\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )}^3\,b\,c^9\,d\,e^6\,x\,416}{e^2}\right )\,\mathrm {root}\left (16\,c^2\,d^4\,e^4\,z^4+16\,e^8\,z^4-32\,b\,c\,d\,e^5\,z^3+8\,b^2\,c^2\,d^2\,e^2\,z^2+b^4\,c^2,z,k\right )\right )-\frac {a}{x\,e^2+d\,e}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{x\,e^2+d\,e}-\frac {2\,b\,c\,d\,e\,\ln \left (d+e\,x\right )}{c^2\,d^4+e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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