Optimal. Leaf size=1073 \[ -\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}+\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}-\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b g^3 \sqrt {1-c^2 x^2}}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b c g^4 (f+g x) \sqrt {1-c^2 x^2}}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b c g^2 (f+g x)}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) \log \left (1-\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.55, antiderivative size = 1073, normalized size of antiderivative = 1.00, number of steps
used = 29, number of rules used = 23, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.742, Rules used = {4861, 4851,
4741, 4737, 30, 4767, 4849, 697, 4841, 6874, 739, 210, 1668, 12, 4883, 4881, 8, 4857, 3404, 2296,
2221, 2317, 2438} \begin {gather*} \frac {b d x^3 \sqrt {d-c^2 d x^2} c^3}{9 g \sqrt {1-c^2 x^2}}-\frac {b d f x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {d f x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) c^2}{2 g^2}-\frac {d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 c}{2 b g^3 \sqrt {1-c^2 x^2}}+\frac {d f \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2 c}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {b d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} c}{g^3 \sqrt {1-c^2 x^2}}-\frac {b d x \sqrt {d-c^2 d x^2} c}{3 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g^3}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 g}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \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 )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \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 )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d 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 )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d 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 )}{g^4 \sqrt {1-c^2 x^2}}-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{2 b g^4 (f+g x) \sqrt {1-c^2 x^2} c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 210
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rule 4737
Rule 4741
Rule 4767
Rule 4841
Rule 4849
Rule 4851
Rule 4857
Rule 4861
Rule 4881
Rule 4883
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^2}-\frac {c^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g}+\frac {\left (-c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{g^2 (f+g x)}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (c^2 d \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{g \sqrt {1-c^2 x^2}}\\ &=\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1-c^2 x^2}}+\frac {\left (c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\left (d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (b c^2 d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (b c d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1-c^2 x^2}}+\frac {\left (a d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} 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 )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} 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 )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d 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 )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (i b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d 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 )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d 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 )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (b d \left (1-\frac {c^2 f^2}{g^2}\right ) (c f-g) (c f+g) \sqrt {d-c^2 d 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 )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 g}+\frac {c d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d \left (1-\frac {c^2 f^2}{g^2}\right ) x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {d \left (1-\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}+\frac {d \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d 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 )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d 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 )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.99, size = 507, normalized size = 0.47 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (-9 b c^3 f x^2+4 b c g x \left (-3+c^2 x^2\right )+18 c^2 f x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))+12 g \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {9 c f (a+b \text {ArcSin}(c x))^2}{b}+\frac {18 \left (c^2 f^2-g^2\right ) \left (-1+c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{b c (f+g x)}-\frac {18 \left (c^2 f^2-g^2\right ) \left (\left (c^2 f^2-g^2\right ) (a+b \text {ArcSin}(c x))^2+c^2 g x (f+g x) (a+b \text {ArcSin}(c x))^2-2 b c (f+g x) \left (b c g x-g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-i \sqrt {c^2 f^2-g^2} \left ((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 )-i b \text {PolyLog}\left (2,\frac {i e^{i \text {ArcSin}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+i 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 )\right )\right )}{b c g^2 (f+g x)}\right )}{36 g^2 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 2741 vs. \(2 (999 ) = 1998\).
time = 0.51, size = 2742, normalized size = 2.56
method | result | size |
default | \(\text {Expression too large to display}\) | \(2742\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{f + g x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________