Optimal. Leaf size=319 \[ -\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.28, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {4789, 4771,
4721, 3798, 2221, 2317, 2438, 4723, 270} \begin {gather*} -\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 d x}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 d x^3}-\frac {2 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 \sqrt {d-c^2 d x^2}}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4723
Rule 4771
Rule 4789
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \sqrt {d-c^2 d x^2}} \, dx &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac {1}{3} \left (2 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^3} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}-\frac {\left (8 i b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x}+\frac {4 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 269, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {1-c^2 x^2} \left (a b c x+a^2 \sqrt {1-c^2 x^2}+2 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 c^2 x^2 \sqrt {1-c^2 x^2}+b^2 \left (2 i c^3 x^3+\sqrt {1-c^2 x^2}+2 c^2 x^2 \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2-b \text {ArcSin}(c x) \left (-b c x-2 a \sqrt {1-c^2 x^2} \left (1+2 c^2 x^2\right )+4 b c^3 x^3 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )-4 a b c^3 x^3 \log (c x)+2 i b^2 c^3 x^3 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )}{3 x^3 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2318 vs. \(2 (301 ) = 602\).
time = 0.56, size = 2319, normalized size = 7.27
method | result | size |
default | \(\text {Expression too large to display}\) | \(2319\) |
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 {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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