Optimal. Leaf size=478 \[ -\frac{3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{3 f g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b f^2 g x \sqrt{c x-1} \sqrt{c x+1}}{c \sqrt{d-c^2 d x^2}}-\frac{3 b f g^2 x^2 \sqrt{c x-1} \sqrt{c x+1}}{4 c \sqrt{d-c^2 d x^2}}-\frac{b g^3 x^3 \sqrt{c x-1} \sqrt{c x+1}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 b g^3 x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 1.2883, antiderivative size = 478, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {5836, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac{3 f^2 g (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{3 f g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b f^2 g x \sqrt{c x-1} \sqrt{c x+1}}{c \sqrt{d-c^2 d x^2}}-\frac{3 b f g^2 x^2 \sqrt{c x-1} \sqrt{c x+1}}{4 c \sqrt{d-c^2 d x^2}}-\frac{b g^3 x^3 \sqrt{c x-1} \sqrt{c x+1}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 b g^3 x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5836
Rule 5822
Rule 5676
Rule 5718
Rule 8
Rule 5759
Rule 30
Rubi steps
\begin{align*} \int \frac{(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (\frac{f^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 f^2 g x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 f g^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{g^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\left (f^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}+\frac{\left (3 f^2 g \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}+\frac{\left (3 f g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}+\frac{\left (g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{\left (3 b f^2 g \sqrt{-1+c x} \sqrt{1+c x}\right ) \int 1 \, dx}{c \sqrt{d-c^2 d x^2}}+\frac{\left (3 f g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b f g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x \, dx}{2 c \sqrt{d-c^2 d x^2}}+\frac{\left (2 g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^2 \, dx}{3 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{3 b f^2 g x \sqrt{-1+c x} \sqrt{1+c x}}{c \sqrt{d-c^2 d x^2}}-\frac{3 b f g^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{4 c \sqrt{d-c^2 d x^2}}-\frac{b g^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int 1 \, dx}{3 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{3 b f^2 g x \sqrt{-1+c x} \sqrt{1+c x}}{c \sqrt{d-c^2 d x^2}}-\frac{2 b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{3 b f g^2 x^2 \sqrt{-1+c x} \sqrt{1+c x}}{4 c \sqrt{d-c^2 d x^2}}-\frac{b g^3 x^3 \sqrt{-1+c x} \sqrt{1+c x}}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 2.05386, size = 405, normalized size = 0.85 \[ \frac{-\frac{12 a \sqrt{d-c^2 d x^2} \left (c^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+4 g^3\right )}{d}-\frac{36 a c f \left (2 c^2 f^2+3 g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{\sqrt{d}}+\frac{216 b c^2 f^2 g \sqrt{d-c^2 d x^2} \left (c x-\sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)\right )}{d \sqrt{c x-1} \sqrt{c x+1}}+\frac{36 b c^3 f^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{\sqrt{d-c^2 d x^2}}+\frac{27 b c f g^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+\sinh \left (2 \cosh ^{-1}(c x)\right )\right )-\cosh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt{d-c^2 d x^2}}+\frac{8 b g^3 \sqrt{d-c^2 d x^2} \left (c x \left (c^2 x^2+6\right )-3 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2+2\right ) \cosh ^{-1}(c x)\right )}{d \sqrt{c x-1} \sqrt{c x+1}}}{72 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.323, size = 859, 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 (-\frac{{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} +{\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \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 )}^{3}{\left (b \operatorname{arcosh}\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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