Optimal. Leaf size=65 \[ -\frac {\sqrt {1-(a+b x)^2}}{2 b \text {ArcSin}(a+b x)^2}+\frac {a+b x}{2 b \text {ArcSin}(a+b x)}-\frac {\text {CosIntegral}(\text {ArcSin}(a+b x))}{2 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4887, 4717,
4807, 4719, 3383} \begin {gather*} -\frac {\text {CosIntegral}(\text {ArcSin}(a+b x))}{2 b}+\frac {a+b x}{2 b \text {ArcSin}(a+b x)}-\frac {\sqrt {1-(a+b x)^2}}{2 b \text {ArcSin}(a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 4717
Rule 4719
Rule 4807
Rule 4887
Rubi steps
\begin {align*} \int \frac {1}{\sin ^{-1}(a+b x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}+\frac {a+b x}{2 b \sin ^{-1}(a+b x)}-\frac {\text {Subst}\left (\int \frac {1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}+\frac {a+b x}{2 b \sin ^{-1}(a+b x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}+\frac {a+b x}{2 b \sin ^{-1}(a+b x)}-\frac {\text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 65, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-(a+b x)^2}}{2 b \text {ArcSin}(a+b x)^2}+\frac {a+b x}{2 b \text {ArcSin}(a+b x)}-\frac {\text {CosIntegral}(\text {ArcSin}(a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 53, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{2 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{2 \arcsin \left (b x +a \right )}-\frac {\cosineIntegral \left (\arcsin \left (b x +a \right )\right )}{2}}{b}\) | \(53\) |
default | \(\frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{2 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{2 \arcsin \left (b x +a \right )}-\frac {\cosineIntegral \left (\arcsin \left (b x +a \right )\right )}{2}}{b}\) | \(53\) |
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}^{3}{\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 = 0.40, size = 57, normalized size = 0.88 \begin {gather*} -\frac {\operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b} + \frac {b x + a}{2 \, b \arcsin \left (b x + a\right )} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b \arcsin \left (b x + a\right )^{2}} \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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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