Optimal. Leaf size=154 \[ -\frac {\left (2 a^2+1\right ) b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{3 \left (1-a^2\right )^{5/2}}+\frac {a b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right )^2 x}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{6 \left (1-a^2\right ) x^2}-\frac {\cosh ^{-1}(a+b x)}{3 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5866, 5802, 103, 151, 12, 93, 205} \[ \frac {a b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right )^2 x}-\frac {\left (2 a^2+1\right ) b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{3 \left (1-a^2\right )^{5/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{6 \left (1-a^2\right ) x^2}-\frac {\cosh ^{-1}(a+b x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 151
Rule 205
Rule 5802
Rule 5866
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a+b x)}{x^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^4} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cosh ^{-1}(a+b x)}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{6 \left (1-a^2\right ) x^2}-\frac {\cosh ^{-1}(a+b x)}{3 x^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\frac {2 a}{b}+\frac {x}{b}}{\sqrt {-1+x} \sqrt {1+x} \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{6 \left (1-a^2\right )}\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{6 \left (1-a^2\right ) x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {\cosh ^{-1}(a+b x)}{3 x^3}+\frac {b^4 \operatorname {Subst}\left (\int \frac {1+2 a^2}{b^2 \sqrt {-1+x} \sqrt {1+x} \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x\right )}{6 \left (1-a^2\right )^2}\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{6 \left (1-a^2\right ) x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {\cosh ^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (1+2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x\right )}{6 \left (1-a^2\right )^2}\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{6 \left (1-a^2\right ) x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {\cosh ^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (1+2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{b}-\frac {a}{b}-\left (\frac {1}{b}-\frac {a}{b}\right ) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {-1+a+b x}}\right )}{3 \left (1-a^2\right )^2}\\ &=\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{6 \left (1-a^2\right ) x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {\cosh ^{-1}(a+b x)}{3 x^3}-\frac {\left (1+2 a^2\right ) b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{3 \left (1-a^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 162, normalized size = 1.05 \[ \frac {1}{6} \left (-\frac {i \left (2 a^2+1\right ) b^3 \log \left (\frac {12 \left (1-a^2\right )^{3/2} \left (\sqrt {1-a^2} \sqrt {a+b x-1} \sqrt {a+b x+1}+i a^2+i a b x-i\right )}{b^3 \left (2 a^2 x+x\right )}\right )}{\left (1-a^2\right )^{5/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1} \left (-a^2+3 a b x+1\right )}{\left (a^2-1\right )^2 x^2}-\frac {2 \cosh ^{-1}(a+b x)}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 566, normalized size = 3.68 \[ \left [\frac {{\left (2 \, a^{2} + 1\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) + 3 \, {\left (a^{3} - a\right )} b^{3} x^{3} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (a^{6} - 3 \, a^{4} - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (3 \, {\left (a^{3} - a\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, \frac {2 \, {\left (2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) + 3 \, {\left (a^{3} - a\right )} b^{3} x^{3} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (a^{6} - 3 \, a^{4} - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (3 \, {\left (a^{3} - a\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.95, size = 340, normalized size = 2.21 \[ \frac {1}{3} \, b {\left (\frac {{\left (2 \, a^{2} b^{2} + b^{2}\right )} \arctan \left (-\frac {x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - 2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1}} - \frac {2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{3} a^{2} b^{2} - 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a^{4} b^{2} - 4 \, a^{5} b {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{3} b^{2} + 7 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} a^{2} b^{2} + 8 \, a^{3} b {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} b^{2} - 4 \, a b {\left | b \right |}}{{\left (a^{4} - 2 \, a^{2} + 1\right )} {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}^{2} - a^{2} + 1\right )}^{2}}\right )} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 397, normalized size = 2.58 \[ -\frac {\mathrm {arccosh}\left (b x +a \right )}{3 x^{3}}-\frac {b^{3} \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right ) a^{2}}{3 \left (a^{2}-1\right )^{\frac {3}{2}} \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}-\frac {b^{3} \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {\left (b x +a \right )^{2}-1}+2 a \left (b x +a \right )-2}{b x}\right )}{6 \left (a^{2}-1\right )^{\frac {3}{2}} \left (1+a \right ) \left (a -1\right ) \sqrt {\left (b x +a \right )^{2}-1}}+\frac {b^{2} \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a^{3}}{2 x \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right )}-\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a^{4}}{6 x^{2} \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right )}-\frac {b^{2} \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a}{2 x \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right )}+\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}\, a^{2}}{3 x^{2} \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right )}-\frac {b \sqrt {b x +a -1}\, \sqrt {b x +a +1}}{6 x^{2} \left (a^{2}-1\right )^{2} \left (1+a \right ) \left (a -1\right )} \]
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 = -0.01 \[ \int \frac {\mathrm {acosh}\left (a+b\,x\right )}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (a + b x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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