Optimal. Leaf size=186 \[ -\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\text {ArcSin}(a+b x)}{4 x^4}-\frac {a \left (3+2 a^2\right ) b^4 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 186, 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 = {4889, 4827,
759, 849, 821, 739, 212} \begin {gather*} -\frac {a \left (2 a^2+3\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}}-\frac {\left (11 a^2+4\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {\text {ArcSin}(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 4827
Rule 4889
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)}{x^5} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^5} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sin ^{-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 {\sin ^{-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 {\sin ^{-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 {\sin ^{-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 {\sin ^{-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 {\sin ^{-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 = 0.16, size = 194, normalized size = 1.04 \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 \text {ArcSin}(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. \(407\) vs.
\(2(164)=328\).
time = 0.01, size = 408, normalized size = 2.19
method | result | size |
derivativedivides | \(b^{4} \left (-\frac {\arcsin \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{12 \left (-a^{2}+1\right ) b^{3} x^{3}}+\frac {5 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b^{2} x^{2}}+\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (-a^{2}+1\right )}+\frac {-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}}{-6 a^{2}+6}\right )\) | \(408\) |
default | \(b^{4} \left (-\frac {\arcsin \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{12 \left (-a^{2}+1\right ) b^{3} x^{3}}+\frac {5 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b^{2} x^{2}}+\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (-a^{2}+1\right )}+\frac {-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}}{-6 a^{2}+6}\right )\) | \(408\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.61, size = 484, normalized size = 2.60 \begin {gather*} \left [-\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {-a^{2} + 1} b^{4} x^{4} \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 ) + 12 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - 2 \, {\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}}{48 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {a^{2} - 1} b^{4} x^{4} \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 ) + 6 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\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}}\right ] \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 {asin}{\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. 1112 vs.
\(2 (158) = 316\).
time = 0.45, size = 1112, normalized size = 5.98 \begin {gather*} -\frac {1}{12} \, b {\left (\frac {3 \, {\left (2 \, a^{3} b^{4} + 3 \, a b^{4}\right )} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{6} {\left | b \right |} - 3 \, a^{4} {\left | b \right |} + 3 \, a^{2} {\left | b \right |} - {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {\frac {36 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{7} b^{4}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {18 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{4} a^{7} b^{4}}{{\left (b^{2} x + a b\right )}^{4}} + 18 \, a^{7} b^{4} - \frac {81 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{6} b^{4}}{b^{2} x + a b} - \frac {108 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{6} b^{4}}{{\left (b^{2} x + a b\right )}^{3}} - \frac {27 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{5} a^{6} b^{4}}{{\left (b^{2} x + a b\right )}^{5}} + \frac {120 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{5} b^{4}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {81 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{4} a^{5} b^{4}}{{\left (b^{2} x + a b\right )}^{4}} - 5 \, a^{5} b^{4} + \frac {12 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{4} b^{4}}{b^{2} x + a b} - \frac {42 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{4} b^{4}}{{\left (b^{2} x + a b\right )}^{3}} + \frac {18 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{5} a^{4} b^{4}}{{\left (b^{2} x + a b\right )}^{5}} - \frac {18 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{3} b^{4}}{{\left (b^{2} x + a b\right )}^{2}} - \frac {36 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{4} a^{3} b^{4}}{{\left (b^{2} x + a b\right )}^{4}} + 2 \, a^{3} b^{4} - \frac {6 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{4}}{b^{2} x + a b} + \frac {8 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{2} b^{4}}{{\left (b^{2} x + a b\right )}^{3}} - \frac {6 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{5} a^{2} b^{4}}{{\left (b^{2} x + a b\right )}^{5}} + \frac {12 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a b^{4}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {12 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{4} a b^{4}}{{\left (b^{2} x + a b\right )}^{4}} - \frac {8 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} b^{4}}{{\left (b^{2} x + a b\right )}^{3}}}{{\left (a^{9} {\left | b \right |} - 3 \, a^{7} {\left | b \right |} + 3 \, a^{5} {\left | b \right |} - a^{3} {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}^{3}}\right )} - \frac {\arcsin \left (b x + a\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 {asin}\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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