Optimal. Leaf size=496 \[ \frac {g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b c^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {1-c^2 x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]
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Rubi [A] time = 0.72, antiderivative size = 496, 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 = {4778, 4774, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac {b c^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-\frac {g e^{i \cos ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-\frac {g e^{i \cos ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}+\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3321
Rule 3324
Rule 4774
Rule 4778
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-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 \cos ^{-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 ) \operatorname {Subst}\left (\int \frac {a+b x}{(c f+g \cos (x))^2} \, dx,x,\cos ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-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 ) \operatorname {Subst}\left (\int \frac {a+b x}{c f+g \cos (x)} \, dx,x,\cos ^{-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 ) \operatorname {Subst}\left (\int \frac {\sin (x)}{c f+g \cos (x)} \, dx,x,\cos ^{-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 \cos ^{-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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-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 \cos ^{-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 c^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f+2 e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 c^2 f g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f+2 e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cos ^{-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 \cos ^{-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 \cos ^{-1}(c x)\right ) \log \left (1+\frac {e^{i \cos ^{-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 \cos ^{-1}(c x)\right ) \log \left (1+\frac {e^{i \cos ^{-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 ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-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 ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-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 \cos ^{-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 \cos ^{-1}(c x)\right ) \log \left (1+\frac {e^{i \cos ^{-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 \cos ^{-1}(c x)\right ) \log \left (1+\frac {e^{i \cos ^{-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 ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-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 ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-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 \cos ^{-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 \cos ^{-1}(c x)\right ) \log \left (1+\frac {e^{i \cos ^{-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 \cos ^{-1}(c x)\right ) \log \left (1+\frac {e^{i \cos ^{-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 {e^{i \cos ^{-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 {e^{i \cos ^{-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 [B] time = 5.50, size = 1108, normalized size = 2.23 \[ -\frac {a f \log (f+g x) c^2}{\sqrt {d} \left (g^2-c^2 f^2\right )^{3/2}}-\frac {a f \log \left (d \left (f x c^2+g\right )+\sqrt {d} \sqrt {g^2-c^2 f^2} \sqrt {d-c^2 d x^2}\right ) c^2}{\sqrt {d} (c f-g) (c f+g) \sqrt {g^2-c^2 f^2}}-\frac {b \sqrt {1-c^2 x^2} \left (-\frac {g \sqrt {1-c^2 x^2} \cos ^{-1}(c x)}{(c f-g) (c f+g) (c f+c g x)}-\frac {\log \left (\frac {g x}{f}+1\right )}{c^2 f^2-g^2}-\frac {c f \left (2 \cos ^{-1}(c x) \tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )-2 \cos ^{-1}\left (-\frac {c f}{g}\right ) \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )+\left (\cos ^{-1}\left (-\frac {c f}{g}\right )-2 i \tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )+2 i \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right ) \log \left (\frac {e^{-\frac {1}{2} i \cos ^{-1}(c x)} \sqrt {g^2-c^2 f^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\cos ^{-1}\left (-\frac {c f}{g}\right )+2 i \left (\tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )-\tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} i \cos ^{-1}(c x)} \sqrt {g^2-c^2 f^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\cos ^{-1}\left (-\frac {c f}{g}\right )-2 i \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-i c f+i g+\sqrt {g^2-c^2 f^2}\right ) \left (\tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )-i\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac {c f}{g}\right )+2 i \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right ) \log \left (\frac {(c f+g) \left (i c f-i g+\sqrt {g^2-c^2 f^2}\right ) \left (\tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )+i\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )+i \left (\text {Li}_2\left (\frac {\left (c f-i \sqrt {g^2-c^2 f^2}\right ) \left (c f+g-\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )-\text {Li}_2\left (\frac {\left (c f+i \sqrt {g^2-c^2 f^2}\right ) \left (c f+g-\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )\right )\right )}{\left (g^2-c^2 f^2\right )^{3/2}}\right ) c}{\sqrt {d-c^2 d x^2}}-\frac {a g \sqrt {d-c^2 d x^2}}{d \left (g^2-c^2 f^2\right ) (f+g x)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )}}{c^{2} d g^{2} x^{4} + 2 \, c^{2} d f g x^{3} - 2 \, d f g x - d f^{2} + {\left (c^{2} d f^{2} - d g^{2}\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.07, size = 1622, normalized size = 3.27 \[ \text {result too large to display} \]
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 \[ \int \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acos}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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