Optimal. Leaf size=213 \[ -\frac {\left (2 a^2+2 a+1\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \left (1-a^2\right )^{5/2}}-\frac {(a+4) (2 a+1) b^2 \sqrt {-a-b x+1} \sqrt {a+b x+1}}{6 (1-a)^3 (a+1)^2 x}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3}-\frac {(2 a+3) b \sqrt {-a-b x+1} \sqrt {a+b x+1}}{6 (1-a)^2 (a+1) x^2} \]
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Rubi [A] time = 0.18, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 99, 151, 12, 93, 208} \[ -\frac {\left (2 a^2+2 a+1\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) \left (1-a^2\right )^{5/2}}-\frac {(a+4) (2 a+1) b^2 \sqrt {-a-b x+1} \sqrt {a+b x+1}}{6 (1-a)^3 (a+1)^2 x}-\frac {(2 a+3) b \sqrt {-a-b x+1} \sqrt {a+b x+1}}{6 (1-a)^2 (a+1) x^2}-\frac {\sqrt {-a-b x+1} \sqrt {a+b x+1}}{3 (1-a) x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 208
Rule 6163
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {\sqrt {1+a+b x}}{x^4 \sqrt {1-a-b x}} \, dx\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}+\frac {\int \frac {(3+2 a) b+2 b^2 x}{x^3 \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{3 (1-a)}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}-\frac {(3+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^2 (1+a) x^2}-\frac {\int \frac {-(4+a) (1+2 a) b^2-(3+2 a) b^3 x}{x^2 \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{6 (1-a)^2 (1+a)}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}-\frac {(3+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^2 (1+a) x^2}-\frac {(4+a) (1+2 a) b^2 \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^3 (1+a)^2 x}+\frac {\int \frac {3 \left (1+2 a+2 a^2\right ) b^3}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{6 (1-a)^3 (1+a)^2}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}-\frac {(3+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^2 (1+a) x^2}-\frac {(4+a) (1+2 a) b^2 \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^3 (1+a)^2 x}+\frac {\left (\left (1+2 a+2 a^2\right ) b^3\right ) \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 (1-a)^3 (1+a)^2}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}-\frac {(3+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^2 (1+a) x^2}-\frac {(4+a) (1+2 a) b^2 \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^3 (1+a)^2 x}+\frac {\left (\left (1+2 a+2 a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1-a-(-1+a) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}}\right )}{(1-a)^3 (1+a)^2}\\ &=-\frac {\sqrt {1-a-b x} \sqrt {1+a+b x}}{3 (1-a) x^3}-\frac {(3+2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^2 (1+a) x^2}-\frac {(4+a) (1+2 a) b^2 \sqrt {1-a-b x} \sqrt {1+a+b x}}{6 (1-a)^3 (1+a)^2 x}-\frac {\left (1+2 a+2 a^2\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a)^3 (1+a)^2 \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 193, normalized size = 0.91 \[ \frac {\frac {3 \left (2 a^2+2 a+1\right ) b^2 x^2 \left (\sqrt {-a-1} \sqrt {a-1} \sqrt {-((a+b x-1) (a+b x+1))}-2 b x \tanh ^{-1}\left (\frac {\sqrt {-a-1} \sqrt {-a-b x+1}}{\sqrt {a-1} \sqrt {a+b x+1}}\right )\right )}{\sqrt {-a-1} (a-1)^{3/2}}-(4 a+1) b x \sqrt {-a-b x+1} (a+b x+1)^{3/2}+2 (a-1) (a+1) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{6 \left (a^2-1\right )^2 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.76, size = 488, normalized size = 2.29 \[ \left [-\frac {3 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, {\left (2 \, a^{6} + {\left (2 \, a^{4} + 9 \, a^{3} + 2 \, a^{2} - 9 \, a - 4\right )} b^{2} x^{2} - 6 \, a^{4} - {\left (2 \, a^{5} + 3 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} + 2 \, a + 3\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left (a^{7} - a^{6} - 3 \, a^{5} + 3 \, a^{4} + 3 \, a^{3} - 3 \, a^{2} - a + 1\right )} x^{3}}, -\frac {3 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (2 \, a^{6} + {\left (2 \, a^{4} + 9 \, a^{3} + 2 \, a^{2} - 9 \, a - 4\right )} b^{2} x^{2} - 6 \, a^{4} - {\left (2 \, a^{5} + 3 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} + 2 \, a + 3\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (a^{7} - a^{6} - 3 \, a^{5} + 3 \, a^{4} + 3 \, a^{3} - 3 \, a^{2} - a + 1\right )} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.05, size = 1376, normalized size = 6.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 683, normalized size = 3.21 \[ -\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 \left (-a^{2}+1\right ) x^{3}}-\frac {5 a b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 \left (-a^{2}+1\right )^{2} x^{2}}-\frac {5 a^{2} b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right )^{3} x}-\frac {5 a^{3} b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {7}{2}}}-\frac {3 a \,b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 \left (-a^{2}+1\right )^{2} x}-\frac {a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 \left (-a^{2}+1\right ) x^{3}}-\frac {5 a^{2} b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 \left (-a^{2}+1\right )^{2} x^{2}}-\frac {5 a^{3} b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right )^{3} x}-\frac {5 a^{4} b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {7}{2}}}-\frac {3 a^{2} b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {5}{2}}}-\frac {13 a \,b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 \left (-a^{2}+1\right )^{2} x}-\frac {b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x^{2}}-\frac {b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,x+1}{x^4\,\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \]
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 \[ \int \frac {a + b x + 1}{x^{4} \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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