Integrand size = 17, antiderivative size = 393 \[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=-\frac {\sqrt {b} c^2 \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a+b c^4\right )}+\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\log (x)}{a+b c^4}-\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\log \left (a+b (c+d x)^4\right )}{4 \left (a+b c^4\right )} \]
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Time = 0.35 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {378, 6857, 1890, 1262, 649, 211, 266, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\sqrt {b} c^2 \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a+b c^4\right )}-\frac {\log \left (a+b (c+d x)^4\right )}{4 \left (a+b c^4\right )}+\frac {\log (x)}{a+b c^4} \]
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Rule 210
Rule 211
Rule 266
Rule 378
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1890
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(-c+x) \left (a+b x^4\right )} \, dx,x,c+d x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {1}{\left (a+b c^4\right ) (c-x)}-\frac {b \left (c^3+c^2 x+c x^2+x^3\right )}{\left (a+b c^4\right ) \left (a+b x^4\right )}\right ) \, dx,x,c+d x\right ) \\ & = \frac {\log (x)}{a+b c^4}-\frac {b \text {Subst}\left (\int \frac {c^3+c^2 x+c x^2+x^3}{a+b x^4} \, dx,x,c+d x\right )}{a+b c^4} \\ & = \frac {\log (x)}{a+b c^4}-\frac {b \text {Subst}\left (\int \left (\frac {x \left (c^2+x^2\right )}{a+b x^4}+\frac {c^3+c x^2}{a+b x^4}\right ) \, dx,x,c+d x\right )}{a+b c^4} \\ & = \frac {\log (x)}{a+b c^4}-\frac {b \text {Subst}\left (\int \frac {x \left (c^2+x^2\right )}{a+b x^4} \, dx,x,c+d x\right )}{a+b c^4}-\frac {b \text {Subst}\left (\int \frac {c^3+c x^2}{a+b x^4} \, dx,x,c+d x\right )}{a+b c^4} \\ & = \frac {\log (x)}{a+b c^4}-\frac {b \text {Subst}\left (\int \frac {c^2+x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 \left (a+b c^4\right )}+\frac {\left (c \left (1-\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \left (a+b c^4\right )}-\frac {\left (c \left (1+\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \left (a+b c^4\right )} \\ & = \frac {\log (x)}{a+b c^4}-\frac {b \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 \left (a+b c^4\right )}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 \left (a+b c^4\right )}-\frac {\left (\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\left (\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\left (c \left (1+\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \left (a+b c^4\right )}-\frac {\left (c \left (1+\frac {\sqrt {b} c^2}{\sqrt {a}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \left (a+b c^4\right )} \\ & = -\frac {\sqrt {b} c^2 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a+b c^4\right )}+\frac {\log (x)}{a+b c^4}-\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\log \left (a+b (c+d x)^4\right )}{4 \left (a+b c^4\right )}-\frac {\left (\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\left (\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )} \\ & = -\frac {\sqrt {b} c^2 \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a+b c^4\right )}+\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} c^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\log (x)}{a+b c^4}-\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}+\frac {\sqrt [4]{b} c \left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )}-\frac {\log \left (a+b (c+d x)^4\right )}{4 \left (a+b c^4\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=-\frac {-4 \log (x)+\text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}+6 b c^2 d^2 \text {$\#$1}^2+4 b c d^3 \text {$\#$1}^3+b d^4 \text {$\#$1}^4\&,\frac {4 c^3 \log (x-\text {$\#$1})+6 c^2 d \log (x-\text {$\#$1}) \text {$\#$1}+4 c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+d^3 \log (x-\text {$\#$1}) \text {$\#$1}^3}{c^3+3 c^2 d \text {$\#$1}+3 c d^2 \text {$\#$1}^2+d^3 \text {$\#$1}^3}\&\right ]}{4 \left (a+b c^4\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\ln \left (x \right )}{b \,c^{4}+a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\left (d^{3} \textit {\_R}^{3}+4 c \,d^{2} \textit {\_R}^{2}+6 c^{2} d \textit {\_R} +4 c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 \left (b \,c^{4}+a \right )}\) | \(139\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (a^{3} b \,c^{4}+a^{4}\right ) \textit {\_Z}^{4}+4 \textit {\_Z}^{3} a^{3}+6 a^{2} \textit {\_Z}^{2}+4 a \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-3 a^{2} b \,c^{4} d +5 a^{3} d \right ) \textit {\_R}^{3}+\left (-3 a b \,c^{4} d +15 d \,a^{2}\right ) \textit {\_R}^{2}+\left (-b \,c^{4} d +15 d a \right ) \textit {\_R} +5 d \right ) x +\left (-a^{2} b \,c^{5}-a^{3} c \right ) \textit {\_R}^{3}+\left (-2 a b \,c^{5}+2 c \,a^{2}\right ) \textit {\_R}^{2}+\left (-b \,c^{5}+7 a c \right ) \textit {\_R} +4 c \right )\right )}{4}+\frac {\ln \left (x \right )}{b \,c^{4}+a}\) | \(176\) |
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Result contains complex when optimal does not.
Time = 4.19 (sec) , antiderivative size = 307773, normalized size of antiderivative = 783.14 \[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x} \,d x } \]
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\[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x} \,d x } \]
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Time = 9.29 (sec) , antiderivative size = 882, normalized size of antiderivative = 2.24 \[ \int \frac {1}{x \left (a+b (c+d x)^4\right )} \, dx=\frac {\ln \left (x\right )}{b\,c^4+a}+\left (\sum _{k=1}^4\ln \left (-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^2\,b^5\,c^5\,d^{15}\,4+\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )\,b^4\,c\,d^{15}\,4+\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )\,b^4\,d^{16}\,x\,5-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^4\,a^2\,b^5\,c^5\,d^{15}\,64+{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^2\,a\,b^4\,c\,d^{15}\,28+{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^2\,a\,b^4\,d^{16}\,x\,60+{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^3\,a^2\,b^4\,c\,d^{15}\,32-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^4\,a^3\,b^4\,c\,d^{15}\,64-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^3\,a\,b^5\,c^5\,d^{15}\,32+{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^3\,a^2\,b^4\,d^{16}\,x\,240+{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^4\,a^3\,b^4\,d^{16}\,x\,320-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^2\,b^5\,c^4\,d^{16}\,x\,4-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^3\,a\,b^5\,c^4\,d^{16}\,x\,48-{\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )}^4\,a^2\,b^5\,c^4\,d^{16}\,x\,192\right )\,\mathrm {root}\left (256\,a^3\,b\,c^4\,z^4+256\,a^4\,z^4+256\,a^3\,z^3+96\,a^2\,z^2+16\,a\,z+1,z,k\right )\right ) \]
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