Optimal. Leaf size=127 \[ -\frac{f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{b g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.223334, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {4778, 4764, 4642, 4678, 8} \[ -\frac{f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{b g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4778
Rule 4764
Rule 4642
Rule 4678
Rule 8
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{f \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{g x \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\right ) \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\left (f \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}+\frac{\left (g \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{\left (b g \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{c \sqrt{d-c^2 d x^2}}\\ &=-\frac{b g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.413451, size = 172, normalized size = 1.35 \[ \frac{-2 \sqrt{d} g \left (-a c^2 x^2+a+b c x \sqrt{1-c^2 x^2}\right )-2 a c f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-b c \sqrt{d} f \sqrt{1-c^2 x^2} \cos ^{-1}(c x)^2+2 b \sqrt{d} g \left (c^2 x^2-1\right ) \cos ^{-1}(c x)}{2 c^2 \sqrt{d} \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.299, size = 235, normalized size = 1.9 \begin{align*} -{\frac{ag}{{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{af\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b \left ( \arccos \left ( cx \right ) \right ) ^{2}f}{2\,dc \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{bg\arccos \left ( cx \right ){x}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bgx}{dc \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{bg\arccos \left ( cx \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (a g x + a f +{\left (b g x + b f\right )} \arccos \left (c x\right )\right )}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}{\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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