Integrand size = 13, antiderivative size = 221 \[ \int \frac {1}{a+b (c+d x)^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d}-\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d}+\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d} \]
1/4*arctan(-1+b^(1/4)*(d*x+c)*2^(1/2)/a^(1/4))/a^(3/4)/b^(1/4)/d*2^(1/2)+1 /4*arctan(1+b^(1/4)*(d*x+c)*2^(1/2)/a^(1/4))/a^(3/4)/b^(1/4)/d*2^(1/2)-1/8 *ln(-a^(1/4)*b^(1/4)*(d*x+c)*2^(1/2)+a^(1/2)+(d*x+c)^2*b^(1/2))/a^(3/4)/b^ (1/4)/d*2^(1/2)+1/8*ln(a^(1/4)*b^(1/4)*(d*x+c)*2^(1/2)+a^(1/2)+(d*x+c)^2*b ^(1/2))/a^(3/4)/b^(1/4)/d*2^(1/2)
Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.73 \[ \int \frac {1}{a+b (c+d x)^4} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )-\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )+\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d} \]
(-2*ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)] + 2*ArcTan[1 + (Sqrt[2 ]*b^(1/4)*(c + d*x))/a^(1/4)] - Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2] + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*b^(1/4)*d)
Time = 0.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {239, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{a+b (c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 239 |
| \(\displaystyle \frac {\int \frac {1}{b (c+d x)^4+a}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{b (c+d x)^4+a}d(c+d x)}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} (c+d x)^2+\sqrt {a}}{b (c+d x)^4+a}d(c+d x)}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{(c+d x)^2-\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d(c+d x)}{2 \sqrt {b}}+\frac {\int \frac {1}{(c+d x)^2+\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d(c+d x)}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{b (c+d x)^4+a}d(c+d x)}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{b (c+d x)^4+a}d(c+d x)}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} (c+d x)^2}{b (c+d x)^4+a}d(c+d x)}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} (c+d x)}{\sqrt [4]{b} \left ((c+d x)^2-\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d(c+d x)}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} (c+d x)+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left ((c+d x)^2+\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d(c+d x)}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} (c+d x)}{\sqrt [4]{b} \left ((c+d x)^2-\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d(c+d x)}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} (c+d x)+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left ((c+d x)^2+\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d(c+d x)}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} (c+d x)}{(c+d x)^2-\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d(c+d x)}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} (c+d x)+\sqrt [4]{a}}{(c+d x)^2+\frac {\sqrt {2} \sqrt [4]{a} (c+d x)}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d(c+d x)}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{d}\) |
((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/ 4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^ (1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x ) + Sqrt[b]*(c + d*x)^2]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2] *a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2]/(2*Sqrt[2]*a^(1/4)*b^(1/ 4)))/(2*Sqrt[a]))/d
3.2.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Simp[1/Coefficient[v, x, 1 ] Subst[Int[(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, n, p}, x] && Lin earQ[v, x] && NeQ[v, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.43
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 b d}\) | \(94\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 b d}\) | \(94\) |
1/4/b/d*sum(1/(_R^3*d^3+3*_R^2*c*d^2+3*_R*c^2*d+c^3)*ln(x-_R),_R=RootOf(_Z ^4*b*d^4+4*_Z^3*b*c*d^3+6*_Z^2*b*c^2*d^2+4*_Z*b*c^3*d+b*c^4+a))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.69 \[ \int \frac {1}{a+b (c+d x)^4} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (a d \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} + d x + c\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (i \, a d \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} + d x + c\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a d \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} + d x + c\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left (-a d \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} + d x + c\right ) \]
1/4*(-1/(a^3*b*d^4))^(1/4)*log(a*d*(-1/(a^3*b*d^4))^(1/4) + d*x + c) + 1/4 *I*(-1/(a^3*b*d^4))^(1/4)*log(I*a*d*(-1/(a^3*b*d^4))^(1/4) + d*x + c) - 1/ 4*I*(-1/(a^3*b*d^4))^(1/4)*log(-I*a*d*(-1/(a^3*b*d^4))^(1/4) + d*x + c) - 1/4*(-1/(a^3*b*d^4))^(1/4)*log(-a*d*(-1/(a^3*b*d^4))^(1/4) + d*x + c)
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{a+b (c+d x)^4} \, dx=\frac {\operatorname {RootSum} {\left (256 t^{4} a^{3} b + 1, \left ( t \mapsto t \log {\left (x + \frac {4 t a + c}{d} \right )} \right )\right )}}{d} \]
\[ \int \frac {1}{a+b (c+d x)^4} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{4} b + a} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {1}{a+b (c+d x)^4} \, dx=-\frac {1}{2} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b d x + b c}{\left (-a b^{3}\right )^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left ({\left | b d x + b c + \left (-a b^{3}\right )^{\frac {1}{4}} \right |}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b d^{4}}\right )^{\frac {1}{4}} \log \left ({\left | -b d x - b c + \left (-a b^{3}\right )^{\frac {1}{4}} \right |}\right ) \]
-1/2*(-1/(a^3*b*d^4))^(1/4)*arctan(-(b*d*x + b*c)/(-a*b^3)^(1/4)) + 1/4*(- 1/(a^3*b*d^4))^(1/4)*log(abs(b*d*x + b*c + (-a*b^3)^(1/4))) - 1/4*(-1/(a^3 *b*d^4))^(1/4)*log(abs(-b*d*x - b*c + (-a*b^3)^(1/4)))
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.27 \[ \int \frac {1}{a+b (c+d x)^4} \, dx=-\frac {\mathrm {atan}\left (\frac {b^{1/4}\,c}{{\left (-a\right )}^{1/4}}+\frac {b^{1/4}\,d\,x}{{\left (-a\right )}^{1/4}}\right )+\mathrm {atanh}\left (\frac {b^{1/4}\,c}{{\left (-a\right )}^{1/4}}+\frac {b^{1/4}\,d\,x}{{\left (-a\right )}^{1/4}}\right )}{2\,{\left (-a\right )}^{3/4}\,b^{1/4}\,d} \]