Optimal. Leaf size=41 \[ -\frac {\sqrt {1-(a+b x)^2}}{b \text {ArcSin}(a+b x)}-\frac {\text {Si}(\text {ArcSin}(a+b x))}{b} \]
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Rubi [A]
time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4887, 4717,
4809, 3380} \begin {gather*} -\frac {\text {Si}(\text {ArcSin}(a+b x))}{b}-\frac {\sqrt {1-(a+b x)^2}}{b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4717
Rule 4809
Rule 4887
Rubi steps
\begin {align*} \int \frac {1}{\sin ^{-1}(a+b x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}-\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 37, normalized size = 0.90 \begin {gather*} -\frac {\frac {\sqrt {1-(a+b x)^2}}{\text {ArcSin}(a+b x)}+\text {Si}(\text {ArcSin}(a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 38, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{\arcsin \left (b x +a \right )}-\sinIntegral \left (\arcsin \left (b x +a \right )\right )}{b}\) | \(38\) |
default | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{\arcsin \left (b x +a \right )}-\sinIntegral \left (\arcsin \left (b x +a \right )\right )}{b}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 39, normalized size = 0.95 \begin {gather*} -\frac {\operatorname {Si}\left (\arcsin \left (b x + a\right )\right )}{b} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{b \arcsin \left (b x + a\right )} \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 = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {asin}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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