Optimal. Leaf size=203 \[ -\frac {(a x+1)^2}{5 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {2 (a x+1)^2 (43 a x+28) (1-a x)^3}{15 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {2 (a x+1)^{5/2} (1-a x)^{5/2} \sin ^{-1}(a x)}{a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {58 (a x+1)^2 (1-a x)^2}{15 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {2 (a x+1)^2 (1-a x)}{3 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.46, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6167, 6159, 6129, 98, 150, 143, 41, 216} \[ -\frac {2 (a x+1)^2 (43 a x+28) (1-a x)^3}{15 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {58 (a x+1)^2 (1-a x)^2}{15 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {2 (a x+1)^2 (1-a x)}{3 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}-\frac {(a x+1)^2}{5 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}+\frac {2 (a x+1)^{5/2} (1-a x)^{5/2} \sin ^{-1}(a x)}{a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 98
Rule 143
Rule 150
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx\\ &=-\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} x^5}{(1-a x)^{5/2} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x^5}{(1-a x)^{7/2} (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {(1+a x)^2}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x^3 (4+6 a x)}{(1-a x)^{5/2} (1+a x)^{3/2}} \, dx}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {(1+a x)^2}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {2 (1-a x) (1+a x)^2}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}+\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x^2 \left (-30 a-28 a^2 x\right )}{(1-a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {(1+a x)^2}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {2 (1-a x) (1+a x)^2}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}-\frac {58 (1-a x)^2 (1+a x)^2}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}+\frac {\left ((1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {x \left (116 a^2+86 a^3 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {(1+a x)^2}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {2 (1-a x) (1+a x)^2}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}-\frac {58 (1-a x)^2 (1+a x)^2}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+43 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {\left (2 (1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {(1+a x)^2}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {2 (1-a x) (1+a x)^2}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}-\frac {58 (1-a x)^2 (1+a x)^2}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+43 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {\left (2 (1-a x)^{5/2} (1+a x)^{5/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ &=-\frac {(1+a x)^2}{5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x}+\frac {2 (1-a x) (1+a x)^2}{3 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2}-\frac {58 (1-a x)^2 (1+a x)^2}{15 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}-\frac {2 (1-a x)^3 (1+a x)^2 (28+43 a x)}{15 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}+\frac {2 (1-a x)^{5/2} (1+a x)^{5/2} \sin ^{-1}(a x)}{a^6 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 105, normalized size = 0.52 \[ \frac {15 a^4 x^4-76 a^3 x^3+32 a^2 x^2+30 (a x-1)^2 \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )+82 a x-56}{15 a^2 c^2 x (a x-1)^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 352, normalized size = 1.73 \[ \left [\frac {15 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (15 \, a^{5} x^{5} - 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}}, -\frac {30 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (15 \, a^{5} x^{5} - 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 462, normalized size = 2.28 \[ \frac {\left (15 c^{\frac {5}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{5} a^{5}-45 x^{4} c^{\frac {5}{2}} a^{4} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}}-16 c^{\frac {5}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{4} a^{4}-60 c^{\frac {5}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{3}+16 c^{\frac {5}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{3}+30 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x \,a^{4} c +90 c^{\frac {5}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{2}+24 c^{\frac {5}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{2}-30 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{3} c +50 c^{\frac {5}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x a -24 c^{\frac {5}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x a -50 c^{\frac {5}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}}-6 c^{\frac {5}{2}} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}}\right ) \left (a x +1\right )}{15 \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{5} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} a^{6} c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a\,x+1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,\left (a\,x-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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