Optimal. Leaf size=69 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}-\frac{1}{8 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{6 a b (a+b x)^3} \]
[Out]
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Rubi [A] time = 0.106382, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}-\frac{1}{8 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{6 a b (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^3*(a^2 - b^2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 24.081, size = 54, normalized size = 0.78 \[ - \frac{1}{6 a b \left (a + b x\right )^{3}} - \frac{1}{8 a^{2} b \left (a + b x\right )^{2}} - \frac{1}{8 a^{3} b \left (a + b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{8 a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**3/(-b**2*x**2+a**2),x)
[Out]
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Mathematica [A] time = 0.0396529, size = 71, normalized size = 1.03 \[ \frac{-2 a \left (10 a^2+9 a b x+3 b^2 x^2\right )-3 (a+b x)^3 \log (a-b x)+3 (a+b x)^3 \log (a+b x)}{48 a^4 b (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^3*(a^2 - b^2*x^2)),x]
[Out]
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Maple [A] time = 0.014, size = 77, normalized size = 1.1 \[ -{\frac{\ln \left ( bx-a \right ) }{16\,{a}^{4}b}}+{\frac{\ln \left ( bx+a \right ) }{16\,{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b \left ( bx+a \right ) }}-{\frac{1}{8\,{a}^{2}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{6\,ab \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^3/(-b^2*x^2+a^2),x)
[Out]
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Maxima [A] time = 0.70203, size = 122, normalized size = 1.77 \[ -\frac{3 \, b^{2} x^{2} + 9 \, a b x + 10 \, a^{2}}{24 \,{\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac{\log \left (b x + a\right )}{16 \, a^{4} b} - \frac{\log \left (b x - a\right )}{16 \, a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216162, size = 181, normalized size = 2.62 \[ -\frac{6 \, a b^{2} x^{2} + 18 \, a^{2} b x + 20 \, a^{3} - 3 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{48 \,{\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.29796, size = 83, normalized size = 1.2 \[ - \frac{10 a^{2} + 9 a b x + 3 b^{2} x^{2}}{24 a^{6} b + 72 a^{5} b^{2} x + 72 a^{4} b^{3} x^{2} + 24 a^{3} b^{4} x^{3}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{16} - \frac{\log{\left (\frac{a}{b} + x \right )}}{16}}{a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**3/(-b**2*x**2+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.217289, size = 95, normalized size = 1.38 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac{3 \, a b^{2} x^{2} + 9 \, a^{2} b x + 10 \, a^{3}}{24 \,{\left (b x + a\right )}^{3} a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^3),x, algorithm="giac")
[Out]