Optimal. Leaf size=86 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^5 b}-\frac{1}{16 a^4 b (a+b x)}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{8 a b (a+b x)^4} \]
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Rubi [A] time = 0.130349, antiderivative size = 86, 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 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^5 b}-\frac{1}{16 a^4 b (a+b x)}-\frac{1}{16 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{8 a b (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^4*(a^2 - b^2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 27.59, size = 70, normalized size = 0.81 \[ - \frac{1}{8 a b \left (a + b x\right )^{4}} - \frac{1}{12 a^{2} b \left (a + b x\right )^{3}} - \frac{1}{16 a^{3} b \left (a + b x\right )^{2}} - \frac{1}{16 a^{4} b \left (a + b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{16 a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**4/(-b**2*x**2+a**2),x)
[Out]
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Mathematica [A] time = 0.0470439, size = 82, normalized size = 0.95 \[ \frac{-2 a \left (16 a^3+19 a^2 b x+12 a b^2 x^2+3 b^3 x^3\right )-3 (a+b x)^4 \log (a-b x)+3 (a+b x)^4 \log (a+b x)}{96 a^5 b (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^4*(a^2 - b^2*x^2)),x]
[Out]
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Maple [A] time = 0.014, size = 92, normalized size = 1.1 \[ -{\frac{\ln \left ( bx-a \right ) }{32\,{a}^{5}b}}+{\frac{\ln \left ( bx+a \right ) }{32\,{a}^{5}b}}-{\frac{1}{16\,{a}^{4}b \left ( bx+a \right ) }}-{\frac{1}{16\,{a}^{3}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{12\,{a}^{2}b \left ( bx+a \right ) ^{3}}}-{\frac{1}{8\,ab \left ( bx+a \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^4/(-b^2*x^2+a^2),x)
[Out]
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Maxima [A] time = 0.700616, size = 151, normalized size = 1.76 \[ -\frac{3 \, b^{3} x^{3} + 12 \, a b^{2} x^{2} + 19 \, a^{2} b x + 16 \, a^{3}}{48 \,{\left (a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{3} + 6 \, a^{6} b^{3} x^{2} + 4 \, a^{7} b^{2} x + a^{8} b\right )}} + \frac{\log \left (b x + a\right )}{32 \, a^{5} b} - \frac{\log \left (b x - a\right )}{32 \, a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226115, size = 240, normalized size = 2.79 \[ -\frac{6 \, a b^{3} x^{3} + 24 \, a^{2} b^{2} x^{2} + 38 \, a^{3} b x + 32 \, a^{4} - 3 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x - a\right )}{96 \,{\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.65671, size = 107, normalized size = 1.24 \[ - \frac{16 a^{3} + 19 a^{2} b x + 12 a b^{2} x^{2} + 3 b^{3} x^{3}}{48 a^{8} b + 192 a^{7} b^{2} x + 288 a^{6} b^{3} x^{2} + 192 a^{5} b^{4} x^{3} + 48 a^{4} b^{5} x^{4}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{32} - \frac{\log{\left (\frac{a}{b} + x \right )}}{32}}{a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**4/(-b**2*x**2+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.216516, size = 109, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{32 \, a^{5} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{32 \, a^{5} b} - \frac{3 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} + 19 \, a^{3} b x + 16 \, a^{4}}{48 \,{\left (b x + a\right )}^{4} a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^4),x, algorithm="giac")
[Out]