Optimal. Leaf size=115 \[ -\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15}{8} a^4 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x} \]
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Rubi [A] time = 0.22, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6148, 1807, 835, 807, 266, 63, 208} \[ -\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {15}{8} a^4 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
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^5} \, dx &=c \int \frac {(1+a x)^3}{x^5 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {1}{4} c \int \frac {-12 a-15 a^2 x-4 a^3 x^2}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}+\frac {1}{12} c \int \frac {45 a^2+36 a^3 x}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {1}{24} c \int \frac {-72 a^3-45 a^4 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x}+\frac {1}{8} \left (15 a^4 c\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x}+\frac {1}{16} \left (15 a^4 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}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x}-\frac {1}{8} \left (15 a^2 c\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 {c \sqrt {1-a^2 x^2}}{4 x^4}-\frac {a c \sqrt {1-a^2 x^2}}{x^3}-\frac {15 a^2 c \sqrt {1-a^2 x^2}}{8 x^2}-\frac {3 a^3 c \sqrt {1-a^2 x^2}}{x}-\frac {15}{8} a^4 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.12, size = 97, normalized size = 0.84 \[ \frac {1}{2} a c \left (-\frac {\sqrt {1-a^2 x^2} \left (6 a^2 x^2+3 a x+2\right )}{x^3}-2 a^3 \sqrt {1-a^2 x^2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-a^2 x^2\right )-3 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 75, normalized size = 0.65 \[ \frac {15 \, a^{4} c x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (24 \, a^{3} c x^{3} + 15 \, a^{2} c x^{2} + 8 \, a c x + 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 280, normalized size = 2.43 \[ \frac {{\left (a^{5} c + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a c}{x^{2}} + \frac {104 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{a x^{3}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} - \frac {15 \, a^{5} c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {104 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c {\left | a \right |}}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} c {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c {\left | a \right |}}{x^{3}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 231, normalized size = 2.01 \[ -c \left (\frac {x \,a^{5}}{\sqrt {-a^{2} x^{2}+1}}+3 a^{4} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+2 a^{3} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )-\frac {13 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}+\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 149, normalized size = 1.30 \[ \frac {3 \, a^{5} c x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {15}{8} \, a^{4} c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {15 \, a^{4} c}{8 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {2 \, a^{3} c}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {13 \, a^{2} c}{8 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {a c}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {c}{4 \, \sqrt {-a^{2} x^{2} + 1} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 103, normalized size = 0.90 \[ -\frac {c\,\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {15\,a^2\,c\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {3\,a^3\,c\,\sqrt {1-a^2\,x^2}}{x}-\frac {a\,c\,\sqrt {1-a^2\,x^2}}{x^3}+\frac {a^4\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.00, size = 411, normalized size = 3.57 \[ a^{3} c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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