Optimal. Leaf size=102 \[ -\frac {a^2 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} a^4 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6128, 835, 807, 266, 47, 63, 208} \[ -\frac {a^2 c^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {1}{8} a^4 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^5} \, dx &=c \int \frac {(c-a c x) \sqrt {1-a^2 x^2}}{x^5} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}-\frac {1}{4} c \int \frac {\left (4 a c-a^2 c x\right ) \sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (a^2 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {a^2 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}-\frac {1}{16} \left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a^2 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a^2 c^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{4 x^4}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} a^4 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 99, normalized size = 0.97 \[ \frac {c^2 \left (8 a^5 x^5-3 a^4 x^4-16 a^3 x^3+9 a^2 x^2+3 a^4 x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+8 a x-6\right )}{24 x^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 84, normalized size = 0.82 \[ -\frac {3 \, a^{4} c^{2} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (8 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 8 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.30, size = 240, normalized size = 2.35 \[ \frac {{\left (3 \, a^{5} c^{2} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c^{2}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {a^{5} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{2} {\left | a \right |}}{x} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{2} {\left | a \right |}}{x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{2} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 125, normalized size = 1.23 \[ c^{2} \left (-\frac {a^{3} \sqrt {-a^{2} x^{2}+1}}{x}-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )-\frac {a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 122, normalized size = 1.20 \[ \frac {1}{8} \, a^{4} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}}{3 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}}{8 \, x^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{3 \, x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.78, size = 113, normalized size = 1.11 \[ \frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^2\,c^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {a^3\,c^2\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {a^4\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 7.98, size = 415, normalized size = 4.07 \[ a^{3} c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________