Optimal. Leaf size=147 \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^5}{5 a^5}+\frac {\left (16 \sqrt {\frac {1-a x}{a x+1}}+5\right ) (a x+1)^4}{20 a^5}-\frac {\left (17 \sqrt {\frac {1-a x}{a x+1}}+15\right ) (a x+1)^3}{15 a^5}+\frac {\left (4 \sqrt {\frac {1-a x}{a x+1}}+9\right ) (a x+1)^2}{6 a^5}-\frac {x}{a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.62, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6337, 1804, 1814, 12, 261} \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^5}{5 a^5}+\frac {\left (16 \sqrt {\frac {1-a x}{a x+1}}+5\right ) (a x+1)^4}{20 a^5}-\frac {\left (17 \sqrt {\frac {1-a x}{a x+1}}+15\right ) (a x+1)^3}{15 a^5}+\frac {\left (4 \sqrt {\frac {1-a x}{a x+1}}+9\right ) (a x+1)^2}{6 a^5}-\frac {x}{a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 261
Rule 1804
Rule 1814
Rule 6337
Rubi steps
\begin {align*} \int e^{-\text {sech}^{-1}(a x)} x^4 \, dx &=\int \frac {x^4}{\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}} \, dx\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {(-1+x)^5 x (1+x)^3}{\left (1+x^2\right )^6} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}-\frac {2 \operatorname {Subst}\left (\int \frac {-16+10 x+140 x^2-30 x^3-80 x^4+30 x^5+20 x^6-10 x^7}{\left (1+x^2\right )^5} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{5 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}+\frac {\operatorname {Subst}\left (\int \frac {-128+560 x+800 x^2-320 x^3-160 x^4+80 x^5}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}-\frac {\operatorname {Subst}\left (\int \frac {-320+2400 x+960 x^2-480 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{120 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}+\frac {\operatorname {Subst}\left (\int \frac {1920 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{480 a^5}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}+\frac {4 \operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^5}\\ &=-\frac {x}{a^4}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^5}{5 a^5}+\frac {(1+a x)^2 \left (9+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^5}+\frac {(1+a x)^4 \left (5+16 \sqrt {\frac {1-a x}{1+a x}}\right )}{20 a^5}-\frac {(1+a x)^3 \left (15+17 \sqrt {\frac {1-a x}{1+a x}}\right )}{15 a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 65, normalized size = 0.44 \[ \frac {15 a^4 x^4-4 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2 \left (3 a^3 x^3-3 a^2 x^2+2 a x-2\right )}{60 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.97, size = 65, normalized size = 0.44 \[ \frac {15 \, a^{3} x^{4} - 4 \, {\left (3 \, a^{4} x^{5} - a^{2} x^{3} - 2 \, x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{60 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.36, size = 531, normalized size = 3.61 \[ -\frac {\left (a x -1\right ) \left (15 a^{10} x^{10} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {5}{2}}+30 a^{8} x^{8} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {5}{2}}+30 \left (-\frac {a x -1}{a x}\right )^{\frac {3}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} x^{8} a^{8}+30 x^{6} \ln \left (a^{2} x^{2}\right ) \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {5}{2}} a^{6}-30 a^{7} x^{7} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {3}{2}}+60 \left (-\frac {a x -1}{a x}\right )^{\frac {3}{2}} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \ln \left (a^{2} x^{2}\right ) x^{6} a^{6}+12 x^{11} a^{11}-60 x^{5} \ln \left (a^{2} x^{2}\right ) \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {3}{2}} a^{5}+30 \sqrt {-\frac {a x -1}{a x}}\, \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \ln \left (a^{2} x^{2}\right ) x^{6} a^{6}+12 x^{10} a^{10}-60 \sqrt {-\frac {a x -1}{a x}}\, \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \ln \left (a^{2} x^{2}\right ) x^{5} a^{5}-40 x^{9} a^{9}+30 x^{4} \ln \left (a^{2} x^{2}\right ) \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \sqrt {-\frac {a x -1}{a x}}\, a^{4}-40 x^{8} a^{8}+40 x^{7} a^{7}+40 x^{6} a^{6}-20 x^{3} a^{3}-20 a^{2} x^{2}+8 a x +8\right )}{60 x^{7} a^{12} \left (\frac {a x +1}{a x}\right )^{\frac {7}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.14, size = 73, normalized size = 0.50 \[ \frac {x^4}{4\,a}+\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,x}{15\,a^4}+\frac {2}{15\,a^5}-\frac {x^5}{5}-\frac {x^4}{5\,a}+\frac {x^3}{15\,a^2}+\frac {x^2}{15\,a^3}\right )}{\sqrt {\frac {1}{a\,x}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {x^{5}}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________