Optimal. Leaf size=283 \[ -\frac {(a x+1)^2}{7 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^3 (107 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {142 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {782 (a x+1)^2 (1-a x)^3}{105 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {124 (a x+1)^2 (1-a x)^2}{105 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^2 (1-a x)}{5 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.50, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \frac {142 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^3 (107 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {782 (a x+1)^2 (1-a x)^3}{105 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {124 (a x+1)^2 (1-a x)^2}{105 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {2 (a x+1)^2 (1-a x)}{5 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {(a x+1)^2}{7 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
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 )^{7/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx\\ &=-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} x^7}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^7}{(1-a x)^{9/2} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^5 (6+8 a x)}{(1-a x)^{7/2} (1+a x)^{5/2}} \, dx}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^4 \left (-70 a-54 a^2 x\right )}{(1-a x)^{5/2} (1+a x)^{5/2}} \, dx}{35 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^3 \left (496 a^2+286 a^3 x\right )}{(1-a x)^{3/2} (1+a x)^{5/2}} \, dx}{105 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x^2 \left (-2346 a^3-1068 a^4 x\right )}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{105 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {x \left (-2556 a^4-1926 a^5 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {(1+a x)^2}{7 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x}+\frac {2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2}-\frac {124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3}+\frac {782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}+\frac {142 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}+\frac {2 (1-a x)^4 (1+a x)^3 (72+107 a x)}{35 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {2 (1-a x)^{7/2} (1+a x)^{7/2} \sin ^{-1}(a x)}{a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 133, normalized size = 0.47 \[ \frac {105 a^6 x^6-562 a^5 x^5+74 a^4 x^4+1226 a^3 x^3-636 a^2 x^2+210 (a x-1)^3 (a x+1) \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )-654 a x+432}{105 a^2 c^3 x (a x-1)^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 496, normalized size = 1.75 \[ \left [\frac {105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac {210 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (105 \, a^{7} x^{7} - 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} + 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} - 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \]
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 \[ \int \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 572, normalized size = 2.02 \[ \frac {\left (105 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} a^{7}+96 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{6} a^{6}-553 x^{6} c^{\frac {7}{2}} a^{6} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}-96 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{5} a^{5}-392 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{5} a^{5}-240 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{4} a^{4}+1540 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{4} a^{4}+210 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \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 {5}{2}} x \,a^{6} c +240 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{3} a^{3}+350 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{3} a^{3}-210 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \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 {5}{2}} a^{5} c +180 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x^{2} a^{2}-1470 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{2} a^{2}-180 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}} x a -42 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x a -30 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c^{\frac {7}{2}}+462 c^{\frac {7}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}}\right ) \left (a x +1\right )}{105 \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} a^{8} c^{\frac {7}{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 \[ \int \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {a\,x+1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a\,x-1\right )} \,d x \]
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 \[ \int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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