Optimal. Leaf size=479 \[ \frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 f g (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}-\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 1.17579, antiderivative size = 479, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {5836, 5822, 5683, 5676, 30, 5718, 5743, 5759} \[ \frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 f g (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}-\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5836
Rule 5822
Rule 5683
Rule 5676
Rule 30
Rule 5718
Rule 5743
Rule 5759
Rubi steps
\begin{align*} \int (f+g x)^2 \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)^2 \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^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )+2 f g x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 f 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{\left (g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \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^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 f 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^2 \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^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b f 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{\left (g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 f 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^2 \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}}-\frac{\left (g^2 \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}{8 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{2 f 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^2 \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}}-\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.20764, size = 356, normalized size = 0.74 \[ \frac{48 a c \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (c^2 x^2-1\right )+3 g^2 x \left (2 c^2 x^2-1\right )\right )-144 a \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (4 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-144 b c^2 f^2 \sqrt{d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+64 b c f g \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )-9 b g^2 \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{1152 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.425, size = 855, normalized size = 1.8 \begin{align*} \text{result too large to display} \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^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\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 )^{2}\, 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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