Optimal. Leaf size=654 \[ \frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {\sqrt {1-c^2 x^2} \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f+g)}-\frac {\sqrt {1-c^2 x^2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {b g^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b g^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{d \sqrt {d-c^2 d x^2} (c f+g)} \]
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Rubi [A] time = 1.16, antiderivative size = 654, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {4777, 4775, 4773, 3318, 4184, 3475, 3323, 2264, 2190, 2279, 2391} \[ \frac {b g^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b g^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {i g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {\sqrt {1-c^2 x^2} \tan \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f+g)}-\frac {\sqrt {1-c^2 x^2} \cot \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)+\frac {\pi }{4}\right )\right )}{d \sqrt {d-c^2 d x^2} (c f+g)} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3318
Rule 3323
Rule 3475
Rule 4184
Rule 4773
Rule 4775
Rule 4777
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \sin ^{-1}(c x)}{(f+g x) \left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (-\frac {c \left (a+b \sin ^{-1}(c x)\right )}{2 (c f+g) (-1+c x) \sqrt {1-c^2 x^2}}+\frac {c \left (a+b \sin ^{-1}(c x)\right )}{2 (c f-g) (1+c x) \sqrt {1-c^2 x^2}}+\frac {g^2 \left (a+b \sin ^{-1}(c x)\right )}{(-c f+g) (c f+g) (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{(-1+c x) \sqrt {1-c^2 x^2}} \, dx}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a+b x}{c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a+b x}{-c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a+b x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{4 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^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 )}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}}\\ &=-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i g^3 \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 )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i g^3 \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 )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (i b g^2 \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 )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (i b g^2 \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 )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b g^2 \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 )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (b g^2 \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 )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 \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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan \left (\frac {\pi }{4}+\frac {1}{2} \sin ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 2.04, size = 359, normalized size = 0.55 \[ \frac {\sqrt {1-c^2 x^2} \left (\frac {2 g^2 \left (i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac {i g e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )\right )+b \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(c x)} g}{\sqrt {c^2 f^2-g^2}-c f}\right )-b \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{(c f-g) (c f+g) \sqrt {c^2 f^2-g^2}}+\frac {2 b \log \left (\sin \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-\cot \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )}{c f-g}+\frac {\tan \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )+2 b \log \left (\cos \left (\frac {1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c f+g}\right )}{2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{4} d^{2} g x^{5} + c^{4} d^{2} f x^{4} - 2 \, c^{2} d^{2} g x^{3} - 2 \, c^{2} d^{2} f x^{2} + d^{2} g x + d^{2} f}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.82, size = 1519, normalized size = 2.32 \[ \text {result too large to display} \]
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 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\left (f+g\,x\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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