Optimal. Leaf size=107 \[ -\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}+\frac {(a c-d)^2 \tan ^{-1}\left (\frac {d+a^2 c x}{\sqrt {a^2 c^2-d^2} \sqrt {1-a^2 x^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {867, 1668, 858,
222, 739, 210} \begin {gather*} \frac {(a c-d)^2 \text {ArcTan}\left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}}-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {\text {ArcSin}(a x) (a c-2 d)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 222
Rule 739
Rule 858
Rule 867
Rule 1668
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^2 (c+d x)} \, dx &=\int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {\int \frac {-a^2 d^2+a^3 (a c-2 d) d x}{(c+d x) \sqrt {1-a^2 x^2}} \, dx}{a^2 d^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a (a c-2 d)) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{d^2}+\frac {(a c-d)^2 \int \frac {1}{(c+d x) \sqrt {1-a^2 x^2}} \, dx}{d^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}-\frac {(a c-d)^2 \text {Subst}\left (\int \frac {1}{-a^2 c^2+d^2-x^2} \, dx,x,\frac {d+a^2 c x}{\sqrt {1-a^2 x^2}}\right )}{d^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}+\frac {(a c-d)^2 \tan ^{-1}\left (\frac {d+a^2 c x}{\sqrt {a^2 c^2-d^2} \sqrt {1-a^2 x^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(107)=214\).
time = 1.79, size = 501, normalized size = 4.68 \begin {gather*} -\frac {a d^3 (a c+d) \sqrt {1-a^2 x^2}-a (a c-d) \sqrt {2 a^2 c^2-d^2-2 a c \sqrt {a^2 c^2-d^2}} \left (a^2 c^2-d^2+a c \sqrt {a^2 c^2-d^2}\right ) \tan ^{-1}\left (\frac {d \left (\sqrt {-a^2} x-\sqrt {1-a^2 x^2}\right )}{\sqrt {2 a^2 c^2-d^2-2 a c \sqrt {a^2 c^2-d^2}}}\right )+a (a c-d) \left (-a^2 c^2+d^2+a c \sqrt {a^2 c^2-d^2}\right ) \sqrt {2 a^2 c^2-d^2+2 a c \sqrt {a^2 c^2-d^2}} \tan ^{-1}\left (\frac {d \left (\sqrt {-a^2} x-\sqrt {1-a^2 x^2}\right )}{\sqrt {2 a^2 c^2-d^2+2 a c \sqrt {a^2 c^2-d^2}}}\right )+\sqrt {-a^2} (a c-d) d^2 \sqrt {-a^2 c^2+d^2} \tan ^{-1}\left (\frac {-\sqrt {-a^2} d^2 x \sqrt {1-a^2 x^2}+a^2 \left (c^2-d^2 x^2\right )}{a c \sqrt {-a^2 c^2+d^2}}\right )+\sqrt {-a^2} (a c-2 d) d^2 (a c+d) \log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{a d^4 (a c+d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(865\) vs.
\(2(99)=198\).
time = 0.10, size = 866, normalized size = 8.09
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{d \sqrt {-a^{2} x^{2}+1}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{d^{2} \sqrt {a^{2}}}+\frac {2 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{d \sqrt {a^{2}}}-\frac {\ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right ) a^{2} c^{2}}{d^{3} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}+\frac {2 \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right ) a c}{d^{2} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}-\frac {\ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\) | \(532\) |
default | \(\frac {-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}-3 a \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a \left (a c +d \right )}-\frac {d \left (\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{\left (a c +d \right )^{2}}+\frac {d \left (\frac {\left (-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}\right )^{\frac {3}{2}}}{3}+\frac {a^{2} c \left (-\frac {\left (-2 a^{2} \left (x +\frac {c}{d}\right )+\frac {2 a^{2} c}{d}\right ) \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{4 a^{2}}-\frac {\left (\frac {4 a^{2} \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}-\frac {4 a^{4} c^{2}}{d^{2}}\right ) \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{d}-\frac {\left (a^{2} c^{2}-d^{2}\right ) \left (\sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}+\frac {a^{2} c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\right )}{d \sqrt {a^{2}}}+\frac {\left (a^{2} c^{2}-d^{2}\right ) \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\right )}{d^{2}}\right )}{\left (a c +d \right )^{2}}\) | \(866\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.34, size = 318, normalized size = 2.97 \begin {gather*} \left [-\frac {{\left (a c - d\right )} \sqrt {-\frac {a c - d}{a c + d}} \log \left (\frac {a^{2} c d x + d^{2} - {\left (a^{2} c^{2} - d^{2}\right )} \sqrt {-a^{2} x^{2} + 1} - {\left (a c d + d^{2} + {\left (a^{3} c^{2} + a^{2} c d\right )} x + \sqrt {-a^{2} x^{2} + 1} {\left (a c d + d^{2}\right )}\right )} \sqrt {-\frac {a c - d}{a c + d}}}{d x + c}\right ) - 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}, \frac {2 \, {\left (a c - d\right )} \sqrt {\frac {a c - d}{a c + d}} \arctan \left (\frac {{\left (d x - \sqrt {-a^{2} x^{2} + 1} c + c\right )} \sqrt {\frac {a c - d}{a c + d}}}{{\left (a c - d\right )} x}\right ) + 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (c + d x\right ) \left (a x - 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs.
\(2 (99) = 198\).
time = 3.97, size = 208, normalized size = 1.94 \begin {gather*} -{\left (\frac {{\left (a x - 1\right )} \sqrt {-\frac {2}{a x - 1} - 1} \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\left (a\right )}{a d} - \frac {2 \, {\left (a c \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\left (a\right ) - 2 \, d \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\left (a\right )\right )} \arctan \left (\sqrt {-\frac {2}{a x - 1} - 1}\right )}{a d^{2}} + \frac {2 \, {\left (a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\left (a\right ) - 2 \, a c d \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\left (a\right ) + d^{2} \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\left (a\right )\right )} \arctan \left (\frac {a c \sqrt {-\frac {2}{a x - 1} - 1} + d \sqrt {-\frac {2}{a x - 1} - 1}}{\sqrt {a^{2} c^{2} - d^{2}}}\right )}{\sqrt {a^{2} c^{2} - d^{2}} a d^{2}}\right )} {\left | a \right |} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 148, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}}{d}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\left (2\,a\,\sqrt {-a^2}-\frac {a^2\,c\,\sqrt {-a^2}}{d}\right )}{a^2\,d}-\frac {\left (\ln \left (\sqrt {1-\frac {a^2\,c^2}{d^2}}\,\sqrt {1-a^2\,x^2}+\frac {a^2\,c\,x}{d}+1\right )-\ln \left (c+d\,x\right )\right )\,\left (a^2\,c^2-2\,a\,c\,d+d^2\right )}{d^3\,\sqrt {1-\frac {a^2\,c^2}{d^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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