Optimal. Leaf size=255 \[ \frac{1}{2} f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{g (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac{b c f x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.545669, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {5836, 5822, 5683, 5676, 30, 5718} \[ \frac{1}{2} f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{g (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac{b c f x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5836
Rule 5822
Rule 5683
Rule 5676
Rule 30
Rule 5718
Rubi steps
\begin{align*} \int (f+g x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \sqrt{-1+c x} \sqrt{1+c x} (f+g x) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{d-c^2 d x^2} \int \left (f \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )+g x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\left (f \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (g \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{2} f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{g (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac{\left (f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b g \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c f x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{2} f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{g (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac{f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.05511, size = 251, normalized size = 0.98 \[ \frac{12 a \sqrt{d-c^2 d x^2} \left (3 c^2 f x+2 g \left (c^2 x^2-1\right )\right )-36 a c \sqrt{d} f \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-\frac{9 b c f \sqrt{d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{2 b g \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}}{72 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.389, size = 461, normalized size = 1.8 \begin{align*} -{\frac{ag}{3\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{afx}{2}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{afd}{2}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{bg{c}^{2}{\rm arccosh} \left (cx\right ){x}^{4}}{ \left ( 3\,cx+3 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{2\,bg{\rm arccosh} \left (cx\right ){x}^{2}}{ \left ( 3\,cx+3 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bf{c}^{2}{\rm arccosh} \left (cx\right ){x}^{3}}{ \left ( 2\,cx+2 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bf{\rm arccosh} \left (cx\right )x}{ \left ( 2\,cx+2 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bf}{8\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{bcf{x}^{2}}{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{bgc{x}^{3}}{9}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bgx}{3\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{bf \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{4\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bg{\rm arccosh} \left (cx\right )}{ \left ( 3\,cx+3 \right ){c}^{2} \left ( cx-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 (\sqrt{-c^{2} d x^{2} + d}{\left (a g x + a f +{\left (b g x + b f\right )} \operatorname{arcosh}\left (c x\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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