Optimal. Leaf size=167 \[ -\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}+\frac {5 a b^2 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^2 x^2}+\frac {\left (4-11 a^2\right ) b^3 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^3 x}-\frac {\sinh ^{-1}(a+b x)}{4 x^4}-\frac {a \left (3-2 a^2\right ) b^4 \tanh ^{-1}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{8 \left (1+a^2\right )^{7/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 167, 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 = {5859, 5828,
759, 849, 821, 739, 212} \begin {gather*} -\frac {a \left (3-2 a^2\right ) b^4 \tanh ^{-1}\left (\frac {a (a+b x)+1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{8 \left (a^2+1\right )^{7/2}}+\frac {\left (4-11 a^2\right ) b^3 \sqrt {(a+b x)^2+1}}{24 \left (a^2+1\right )^3 x}+\frac {5 a b^2 \sqrt {(a+b x)^2+1}}{24 \left (a^2+1\right )^2 x^2}-\frac {b \sqrt {(a+b x)^2+1}}{12 \left (a^2+1\right ) x^3}-\frac {\sinh ^{-1}(a+b x)}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 759
Rule 821
Rule 849
Rule 5828
Rule 5859
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)}{x^5} \, dx &=\frac {\text {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^5} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sinh ^{-1}(a+b x)}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^4 \sqrt {1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}-\frac {\sinh ^{-1}(a+b x)}{4 x^4}-\frac {b^2 \text {Subst}\left (\int \frac {\frac {3 a}{b}+\frac {2 x}{b}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sqrt {1+x^2}} \, dx,x,a+b x\right )}{12 \left (1+a^2\right )}\\ &=-\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}+\frac {5 a b^2 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^2 x^2}-\frac {\sinh ^{-1}(a+b x)}{4 x^4}+\frac {b^4 \text {Subst}\left (\int \frac {-\frac {2 \left (2-3 a^2\right )}{b^2}+\frac {5 a x}{b^2}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1+x^2}} \, dx,x,a+b x\right )}{24 \left (1+a^2\right )^2}\\ &=-\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}+\frac {5 a b^2 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^2 x^2}+\frac {\left (4-11 a^2\right ) b^3 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^3 x}-\frac {\sinh ^{-1}(a+b x)}{4 x^4}+\frac {\left (a \left (3-2 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 \left (1+a^2\right )^3}\\ &=-\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}+\frac {5 a b^2 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^2 x^2}+\frac {\left (4-11 a^2\right ) b^3 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^3 x}-\frac {\sinh ^{-1}(a+b x)}{4 x^4}-\frac {\left (a \left (3-2 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{b^2}+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {\frac {1}{b}+\frac {a (a+b x)}{b}}{\sqrt {1+(a+b x)^2}}\right )}{8 \left (1+a^2\right )^3}\\ &=-\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}+\frac {5 a b^2 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^2 x^2}+\frac {\left (4-11 a^2\right ) b^3 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^3 x}-\frac {\sinh ^{-1}(a+b x)}{4 x^4}-\frac {a \left (3-2 a^2\right ) b^4 \tanh ^{-1}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{8 \left (1+a^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 179, normalized size = 1.07 \begin {gather*} \frac {1}{8} \left (-\frac {b \sqrt {1+a^2+2 a b x+b^2 x^2} \left (2+2 a^4-5 a b x-5 a^3 b x-4 b^2 x^2+a^2 \left (4+11 b^2 x^2\right )\right )}{3 \left (1+a^2\right )^3 x^3}-\frac {2 \sinh ^{-1}(a+b x)}{x^4}-\frac {a \left (-3+2 a^2\right ) b^4 \log (x)}{\left (1+a^2\right )^{7/2}}+\frac {a \left (-3+2 a^2\right ) b^4 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\left (1+a^2\right )^{7/2}}\right ) \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. \(358\) vs.
\(2(145)=290\).
time = 2.03, size = 359, normalized size = 2.15
method | result | size |
derivativedivides | \(b^{4} \left (-\frac {\arcsinh \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 \left (a^{2}+1\right ) b^{3} x^{3}}-\frac {5 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {3 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (a^{2}+1\right )}-\frac {-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}}{6 \left (a^{2}+1\right )}\right )\) | \(359\) |
default | \(b^{4} \left (-\frac {\arcsinh \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 \left (a^{2}+1\right ) b^{3} x^{3}}-\frac {5 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {3 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (a^{2}+1\right )}-\frac {-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}}{6 \left (a^{2}+1\right )}\right )\) | \(359\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs.
\(2 (145) = 290\).
time = 0.27, size = 357, normalized size = 2.14 \begin {gather*} \frac {1}{24} \, {\left (\frac {15 \, a^{3} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {7}{2}}} - \frac {9 \, a b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {5}{2}}} - \frac {15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} b^{2}}{{\left (a^{2} + 1\right )}^{3} x} + \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}}{{\left (a^{2} + 1\right )}^{2} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b}{{\left (a^{2} + 1\right )}^{2} x^{2}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a^{2} + 1\right )} x^{3}}\right )} b - \frac {\operatorname {arsinh}\left (b x + a\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs.
\(2 (145) = 290\).
time = 0.43, size = 343, normalized size = 2.05 \begin {gather*} \frac {3 \, {\left (2 \, a^{3} - 3 \, a\right )} \sqrt {a^{2} + 1} b^{4} x^{4} \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 ) - {\left (11 \, a^{4} + 7 \, a^{2} - 4\right )} b^{4} x^{4} + 6 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 6 \, {\left (a^{8} + 4 \, a^{6} - {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left ({\left (11 \, a^{4} + 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} + 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{24 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4}} \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 {\operatorname {asinh}{\left (a + b x \right )}}{x^{5}}\, 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. 709 vs.
\(2 (145) = 290\).
time = 0.45, size = 709, normalized size = 4.25 \begin {gather*} -\frac {1}{24} \, b {\left (\frac {3 \, {\left (2 \, a^{3} b^{3} - 3 \, a b^{3}\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {2 \, {\left (6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a^{3} b^{3} - 16 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{5} b^{3} + 42 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{7} b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{6} b^{2} {\left | b \right |} + 20 \, a^{8} b^{2} {\left | b \right |} - 9 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a b^{3} + 8 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{3} b^{3} + 93 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{5} b^{3} + 36 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{4} b^{2} {\left | b \right |} + 56 \, a^{6} b^{2} {\left | b \right |} + 24 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a b^{3} + 60 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{3} b^{3} + 36 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{2} b^{2} {\left | b \right |} + 48 \, a^{4} b^{2} {\left | b \right |} + 9 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} b^{2} {\left | b \right |} + 8 \, a^{2} b^{2} {\left | b \right |} - 4 \, b^{2} {\left | b \right |}\right )}}{{\left (a^{6} + 3 \, a^{4} + 3 \, 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 )}^{3}}\right )} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{4 \, x^{4}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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