Optimal. Leaf size=90 \[ -\frac {a+b \text {ArcSin}\left (c+d x^2\right )}{2 x^2}-\frac {b d \tanh ^{-1}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{2 \sqrt {1-c^2}} \]
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Rubi [A]
time = 0.07, 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 = {4926, 12, 1128,
738, 212} \begin {gather*} -\frac {a+b \text {ArcSin}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 738
Rule 1128
Rule 4926
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) \text {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) \text {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 = 0.06, size = 81, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (-\frac {a+b \text {ArcSin}\left (c+d x^2\right )}{x^2}-\frac {b d \tanh ^{-1}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-\left (c+d x^2\right )^2}}\right )}{\sqrt {1-c^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 89, normalized size = 0.99
method | result | size |
default | \(-\frac {a}{2 x^{2}}-\frac {b \arcsin \left (d \,x^{2}+c \right )}{2 x^{2}}-\frac {b d \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{2 \sqrt {-c^{2}+1}}\) | \(89\) |
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 = 3.23, size = 280, normalized size = 3.11 \begin {gather*} \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 ] \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 {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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