Optimal. Leaf size=90 \[ -\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-\frac {b d \tanh ^{-1}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{2 \sqrt {1-c^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4842, 12, 1114, 724, 206} \[ -\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-\frac {b d \tanh ^{-1}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{2 \sqrt {1-c^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 1114
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x^3} \, dx &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}+\frac {1}{2} b \int \frac {2 d}{x \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}+(b d) \int \frac {1}{x \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}+\frac {1}{2} (b d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-(b d) \operatorname {Subst}\left (\int \frac {1}{4 \left (1-c^2\right )-x^2} \, dx,x,\frac {2 \left (1-c^2-c d x^2\right )}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-\frac {b d \tanh ^{-1}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{2 \sqrt {1-c^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 81, normalized size = 0.90 \[ \frac {1}{2} \left (-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x^2}-\frac {b d \tanh ^{-1}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {1-\left (c+d x^2\right )^2}}\right )}{\sqrt {1-c^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 280, normalized size = 3.11 \[ \left [-\frac {\sqrt {-c^{2} + 1} b d x^{2} \log \left (\frac {{\left (2 \, c^{2} - 1\right )} d^{2} x^{4} + 2 \, c^{4} + 4 \, {\left (c^{3} - c\right )} d x^{2} + 2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {-c^{2} + 1} - 4 \, c^{2} + 2}{x^{4}}\right ) + 2 \, a c^{2} + 2 \, {\left (b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) - 2 \, a}{4 \, {\left (c^{2} - 1\right )} x^{2}}, \frac {\sqrt {c^{2} - 1} b d x^{2} \arctan \left (\frac {\sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {c^{2} - 1}}{{\left (c^{2} - 1\right )} d^{2} x^{4} + c^{4} + 2 \, {\left (c^{3} - c\right )} d x^{2} - 2 \, c^{2} + 1}\right ) - a c^{2} - {\left (b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) + a}{2 \, {\left (c^{2} - 1\right )} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 89, normalized size = 0.99 \[ -\frac {a}{2 x^{2}}-\frac {b \arcsin \left (d \,x^{2}+c \right )}{2 x^{2}}-\frac {b d \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{2 \sqrt {-c^{2}+1}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^3} \,d x \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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