Optimal. Leaf size=89 \[ -\frac {4 \sqrt {1-x^2}}{3 x}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {7}{8} \tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1807, 835, 807, 266, 63, 206} \begin {gather*} -\frac {4 \sqrt {1-x^2}}{3 x}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {2 \sqrt {1-x^2}}{3 x^3}-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {7}{8} \tanh ^{-1}\left (\sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 206
Rule 266
Rule 807
Rule 835
Rule 1807
Rubi steps
\begin {align*} \int \frac {(1+x)^2}{x^5 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {1}{4} \int \frac {-8-7 x}{x^4 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3}+\frac {1}{12} \int \frac {21+16 x}{x^3 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {1}{24} \int \frac {-32-21 x}{x^2 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {4 \sqrt {1-x^2}}{3 x}+\frac {7}{8} \int \frac {1}{x \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {4 \sqrt {1-x^2}}{3 x}+\frac {7}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {4 \sqrt {1-x^2}}{3 x}-\frac {7}{8} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {2 \sqrt {1-x^2}}{3 x^3}-\frac {7 \sqrt {1-x^2}}{8 x^2}-\frac {4 \sqrt {1-x^2}}{3 x}-\frac {7}{8} \tanh ^{-1}\left (\sqrt {1-x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 73, normalized size = 0.82 \begin {gather*} -\sqrt {1-x^2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-x^2\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sqrt {1-x^2} \left (8 x^2+3 x+4\right )}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.17, size = 58, normalized size = 0.65 \begin {gather*} \frac {7}{8} \log \left (\sqrt {1-x^2}-1\right )+\frac {\sqrt {1-x^2} \left (-32 x^3-21 x^2-16 x-6\right )}{24 x^4}-\frac {7 \log (x)}{8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 53, normalized size = 0.60 \begin {gather*} \frac {21 \, x^{4} \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - {\left (32 \, x^{3} + 21 \, x^{2} + 16 \, x + 6\right )} \sqrt {-x^{2} + 1}}{24 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 163, normalized size = 1.83 \begin {gather*} \frac {x^{4} {\left (\frac {16 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {48 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac {144 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}} - 3\right )}}{192 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{4}} - \frac {3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{4 \, x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{12 \, x^{3}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{4}}{64 \, x^{4}} + \frac {7}{8} \, \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 70, normalized size = 0.79 \begin {gather*} -\frac {7 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{8}-\frac {4 \sqrt {-x^{2}+1}}{3 x}-\frac {7 \sqrt {-x^{2}+1}}{8 x^{2}}-\frac {2 \sqrt {-x^{2}+1}}{3 x^{3}}-\frac {\sqrt {-x^{2}+1}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 82, normalized size = 0.92 \begin {gather*} -\frac {4 \, \sqrt {-x^{2} + 1}}{3 \, x} - \frac {7 \, \sqrt {-x^{2} + 1}}{8 \, x^{2}} - \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, x^{3}} - \frac {\sqrt {-x^{2} + 1}}{4 \, x^{4}} - \frac {7}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 77, normalized size = 0.87 \begin {gather*} \frac {7\,\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )}{8}-\sqrt {1-x^2}\,\left (\frac {4}{3\,x}+\frac {2}{3\,x^3}\right )-\sqrt {1-x^2}\,\left (\frac {3}{8\,x^2}+\frac {1}{4\,x^4}\right )-\frac {\sqrt {1-x^2}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 11.06, size = 223, normalized size = 2.51 \begin {gather*} 2 \left (\begin {cases} - \frac {\sqrt {1 - x^{2}}}{x} - \frac {\left (1 - x^{2}\right )^{\frac {3}{2}}}{3 x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) + \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x} \right )}}{2} - \frac {\sqrt {-1 + \frac {1}{x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x} \right )}}{2} - \frac {i}{2 x \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{2 x^{3} \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} + \begin {cases} - \frac {3 \operatorname {acosh}{\left (\frac {1}{x} \right )}}{8} + \frac {3}{8 x \sqrt {-1 + \frac {1}{x^{2}}}} - \frac {1}{8 x^{3} \sqrt {-1 + \frac {1}{x^{2}}}} - \frac {1}{4 x^{5} \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {3 i \operatorname {asin}{\left (\frac {1}{x} \right )}}{8} - \frac {3 i}{8 x \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{8 x^{3} \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{4 x^{5} \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________