Optimal. Leaf size=507 \[ \frac {g \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b c^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b c^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.50, antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {4861, 4857,
3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \frac {g \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}-\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i g e^{i \text {ArcSin}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i g e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b c^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3404
Rule 3405
Rule 4857
Rule 4861
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \sin ^{-1}(c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{(c f+g \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (b c g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (2 c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i c^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i c^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (i b c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (i b c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {\left (b c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (b c^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {b c^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b c^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 295, normalized size = 0.58 \begin {gather*} \frac {c \sqrt {1-c^2 x^2} \left (\frac {g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c f+c g x}-b \log (f+g x)+\frac {c f \left (-i (a+b \text {ArcSin}(c x)) \left (\log \left (1+\frac {i e^{i \text {ArcSin}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-b \text {PolyLog}\left (2,-\frac {i e^{i \text {ArcSin}(c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )+b \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right ) \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. 1677 vs. \(2 (495 ) = 990\).
time = 0.76, size = 1678, normalized size = 3.31
method | result | size |
default | \(\text {Expression too large to display}\) | \(1678\) |
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 {a + b \operatorname {asin}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, 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 = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\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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