Optimal. Leaf size=214 \[ -\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{1-a x}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x)}+\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ \frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{1-a x}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x)}-\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 92
Rule 97
Rule 149
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{3/2} (1+a x)^{3/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {\sqrt {1-a x} (1+a x)^{5/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {(1+a x)^{3/2} \left (2 a-3 a^2 x\right )}{x^2 \sqrt {1-a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {\sqrt {1+a x} \left (a^2-5 a^3 x\right )}{x \sqrt {1-a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {-a^3+4 a^4 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 a (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {\left (a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}-\frac {\left (2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac {\left (a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}-\frac {\left (2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac {2 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \sin ^{-1}(a x)}{(1-a x)^{3/2} (1+a x)^{3/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 115, normalized size = 0.54 \[ -\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1} \left (2 a^2 x^2-4 a x-1\right )+4 a^2 x^2 \log \left (\sqrt {a^2 x^2-1}+a x\right )+a^2 x^2 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{2 a^2 x \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.39, size = 316, normalized size = 1.48 \[ \left [\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + a \sqrt {-c} c x \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, -\frac {a c^{\frac {3}{2}} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 2 \, a c^{\frac {3}{2}} x \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.45, size = 265, normalized size = 1.24 \[ {\left (\frac {c^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} + \frac {2 \, c^{\frac {3}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {\sqrt {a^{2} c x^{2} - c} c \mathrm {sgn}\relax (x)}{a^{2}} - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{2} {\left | a \right |} \mathrm {sgn}\relax (x) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {5}{2}} \mathrm {sgn}\relax (x) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\relax (x) - 4 \, a c^{\frac {7}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 455, normalized size = 2.13 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} x \left (12 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{5} c -12 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x \,a^{5}+4 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{4} c -\sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{4} c +6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{3} c^{2}-3 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}-18 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{3} a^{3} c^{2}+18 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{2} a -6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{2} a +3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{2} c^{2}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{2} c^{3}\right )}{6 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________