Optimal. Leaf size=212 \[ -\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d}+\frac{3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{c x-1} \sqrt{c x+1}}{16 c^3 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.647655, antiderivative size = 228, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {5798, 5759, 5676, 30} \[ -\frac{x^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{c x-1} \sqrt{c x+1}}{16 c^3 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5759
Rule 5676
Rule 30
Rubi steps
\begin{align*} \int \frac{x^4 \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{x^4 \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{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 \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}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^3 \, dx}{4 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (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}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x \, dx}{8 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{3 b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{16 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b c^5 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.843036, size = 171, normalized size = 0.81 \[ \frac{-\frac{16 a c x \left (2 c^2 x^2+3\right ) \sqrt{d-c^2 d x^2}}{d}-\frac{48 a \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{\sqrt{d}}+\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (-16 \cosh \left (2 \cosh ^{-1}(c x)\right )-\cosh \left (4 \cosh ^{-1}(c x)\right )+4 \cosh ^{-1}(c x) \left (6 \cosh ^{-1}(c x)+8 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt{d-c^2 d x^2}}}{128 c^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.355, size = 408, normalized size = 1.9 \begin{align*} -{\frac{{x}^{3}a}{4\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{3\,ax}{8\,d{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{3\,a}{8\,{c}^{4}}\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{x}^{4}}{16\,cd \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}+{\frac{3\,b{x}^{2}}{16\,d{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}-{\frac{3\,b \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{16\,{c}^{5}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}}-{\frac{b{\rm arccosh} \left (cx\right ){x}^{5}}{4\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{\rm arccosh} \left (cx\right ){x}^{3}}{8\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{3\,b{\rm arccosh} \left (cx\right )x}{8\,d{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{15\,b}{128\,{c}^{5}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx+1}\sqrt{cx-1}} \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 (b x^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}\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{x^{4} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\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 (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}}{\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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