Optimal. Leaf size=94 \[ -\frac {11 a^2 c \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {5}{2} a^3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6148, 1807, 807, 266, 63, 208} \[ -\frac {11 a^2 c \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {5}{2} a^3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^4} \, dx &=c \int \frac {(1+a x)^3}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {1}{3} c \int \frac {-9 a-11 a^2 x-3 a^3 x^2}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{6} c \int \frac {22 a^2+15 a^3 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {11 a^2 c \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{2} \left (5 a^3 c\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {11 a^2 c \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{4} \left (5 a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {11 a^2 c \sqrt {1-a^2 x^2}}{3 x}-\frac {1}{2} (5 a c) \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 {c \sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a c \sqrt {1-a^2 x^2}}{2 x^2}-\frac {11 a^2 c \sqrt {1-a^2 x^2}}{3 x}-\frac {5}{2} a^3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 60, normalized size = 0.64 \[ -\frac {c \sqrt {1-a^2 x^2} \left (22 a^2 x^2+9 a x+2\right )}{6 x^3}-\frac {5}{2} a^3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 66, normalized size = 0.70 \[ \frac {15 \, a^{3} c x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (22 \, a^{2} c x^{2} + 9 \, a c x + 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.23, size = 218, normalized size = 2.32 \[ \frac {{\left (a^{4} c + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c}{x} + \frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {5 \, a^{4} c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\frac {45 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 184, normalized size = 1.96 \[ -c \left (\frac {a^{3}}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 x \,a^{4}}{\sqrt {-a^{2} x^{2}+1}}+2 a^{3} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-\frac {10 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}+\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}-3 a \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 128, normalized size = 1.36 \[ \frac {11 \, a^{4} c x}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {5}{2} \, a^{3} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {3 \, a^{3} c}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {10 \, a^{2} c}{3 \, \sqrt {-a^{2} x^{2} + 1} x} - \frac {3 \, a c}{2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {c}{3 \, \sqrt {-a^{2} x^{2} + 1} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.89, size = 82, normalized size = 0.87 \[ -\frac {c\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {11\,a^2\,c\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {3\,a\,c\,\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {a^3\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 28.08, size = 267, normalized size = 2.84 \[ a^{3} c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________