Optimal. Leaf size=380 \[ -\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}+\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}-\frac {b \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}+\frac {b \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.61, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4777, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac {b \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}+\frac {b \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}-\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}+\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3323
Rule 4773
Rule 4777
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(f+g x) \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) \sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {a+b x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (2 i g \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i g \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=-\frac {i \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=-\frac {i \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=-\frac {i \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \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 )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 232, normalized size = 0.61 \[ \frac {\sqrt {1-c^2 x^2} \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )\right )-b \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(c x)} g}{\sqrt {c^2 f^2-g^2}-c f}\right )+b \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{2} d g x^{3} + c^{2} d f x^{2} - d g x - d f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.39, size = 502, normalized size = 1.32 \[ -\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-c^{2} d \left (x +\frac {f}{g}\right )^{2}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} f^{2}+g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \ln \left (\frac {-i c f -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{-i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right ) \arcsin \left (c x \right )-i \ln \left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right ) \arcsin \left (c x \right )+\dilog \left (\frac {-i c f -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{-i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right )-\dilog \left (\frac {i c f +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) g +\sqrt {-c^{2} f^{2}+g^{2}}}{i c f +\sqrt {-c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{2} x^{2}-1\right ) \left (c^{2} f^{2}-g^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\left (f+g\,x\right )\,\sqrt {d-c^2\,d\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________