Optimal. Leaf size=146 \[ \frac {8 a^5 \sqrt {1-a x}}{105 x \sqrt {\frac {1}{a x+1}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{a x+1}}}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{a x+1}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6335, 30, 103, 12, 95} \[ \frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{a x+1}}}+\frac {8 a^5 \sqrt {1-a x}}{105 x \sqrt {\frac {1}{a x+1}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{a x+1}}}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{a x+1}}}+\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 95
Rule 103
Rule 6335
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(a x)}}{x^7} \, dx &=-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}-\frac {\int \frac {1}{x^8} \, dx}{6 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^8 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{6 a}\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {6 a^2}{x^6 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{42 a}\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}-\frac {1}{7} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^6 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {1}{35} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {4 a^2}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}-\frac {1}{35} \left (4 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {1}{105} \left (4 a^3 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}-\frac {1}{105} \left (8 a^5 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{42 a x^7}-\frac {e^{\text {sech}^{-1}(a x)}}{6 x^6}+\frac {\sqrt {1-a x}}{42 a x^7 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{35 x^5 \sqrt {\frac {1}{1+a x}}}+\frac {4 a^3 \sqrt {1-a x}}{105 x^3 \sqrt {\frac {1}{1+a x}}}+\frac {8 a^5 \sqrt {1-a x}}{105 x \sqrt {\frac {1}{1+a x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 76, normalized size = 0.52 \[ \frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^2 \left (8 a^5 x^5-8 a^4 x^4+12 a^3 x^3-12 a^2 x^2+15 a x-15\right )-15}{105 a x^7} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 69, normalized size = 0.47 \[ \frac {{\left (8 \, a^{7} x^{7} + 4 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 15}{105 \, a x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 71, normalized size = 0.49 \[ \frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (8 x^{4} a^{4}+12 a^{2} x^{2}+15\right )}{105 x^{6}}-\frac {1}{7 a \,x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 60, normalized size = 0.41 \[ \frac {{\left (8 \, a^{6} x^{7} + 4 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - 15 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{105 \, a x^{8}} - \frac {1}{7 \, a x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.68, size = 95, normalized size = 0.65 \[ \frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{35}-\frac {\sqrt {\frac {1}{a\,x}+1}}{7}+\frac {4\,a^4\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{105}+\frac {8\,a^6\,x^6\,\sqrt {\frac {1}{a\,x}+1}}{105}\right )}{x^6}-\frac {1}{7\,a\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{x^{8}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{7}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________