Optimal. Leaf size=101 \[ -\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \left (-\sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1807, 835, 807, 266, 63, 208} \[ -\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}+a^3 \left (-\sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 1807
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^4} \, dx &=c \int \frac {(1+a x)^2}{x^4 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {1}{3} \int \frac {-6 a c-5 a^2 c x}{x^3 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}+\frac {\int \frac {10 a^2 c^2+6 a^3 c^2 x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{6 c}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+\left (a^3 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{2} \left (a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 x^3}-\frac {a \sqrt {c-a^2 c x^2}}{x^2}-\frac {5 a^2 \sqrt {c-a^2 c x^2}}{3 x}-a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 82, normalized size = 0.81 \[ a^3 \sqrt {c} \log (x)-\frac {\left (5 a^2 x^2+3 a x+1\right ) \sqrt {c-a^2 c x^2}}{3 x^3}-a^3 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 164, normalized size = 1.62 \[ \left [\frac {3 \, a^{3} \sqrt {c} x^{3} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )}}{6 \, x^{3}}, -\frac {3 \, a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a^{2} x^{2} + 3 \, a x + 1\right )}}{3 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.83, size = 250, normalized size = 2.48 \[ \frac {2 \, a^{3} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, {\left (3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{3} c - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{2} \sqrt {-c} c {\left | a \right |} + 12 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{2} \sqrt {-c} c^{2} {\left | a \right |} - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{3} c^{3} - 5 \, a^{2} \sqrt {-c} c^{3} {\left | a \right |}\right )}}{3 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 262, normalized size = 2.59 \[ -\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right ) a^{3}+\sqrt {-a^{2} c \,x^{2}+c}\, a^{3}-\frac {2 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{4} x \sqrt {-a^{2} c \,x^{2}+c}-\frac {2 a^{4} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}-2 a^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}+\frac {2 a^{4} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}}-\frac {a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 140, normalized size = 1.39 \[ -\frac {\sqrt {a x + 1} \sqrt {-a x + 1} a^{2} \sqrt {c}}{x} + \frac {a^{4} c^{\frac {3}{2}} \log \left (\frac {\sqrt {-a^{2} c x^{2} + c} - \sqrt {c}}{\sqrt {-a^{2} c x^{2} + c} + \sqrt {c}}\right ) - \frac {2 \, \sqrt {-a^{2} c x^{2} + c} a^{2} c}{x^{2}}}{2 \, a c} - \frac {{\left (2 \, a^{2} \sqrt {c} x^{2} + \sqrt {c}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{5} - x^{4}}\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{5} - x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________