Optimal. Leaf size=281 \[ \frac{1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.840716, antiderivative size = 293, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {5798, 5745, 5743, 5759, 5676, 30, 14} \[ \frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d x^3 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5745
Rule 5743
Rule 5759
Rule 5676
Rule 30
Rule 14
Rubi steps
\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (d \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}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d \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}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (-x^3+c^2 x^5\right ) \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (d \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}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b d x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.84984, size = 270, normalized size = 0.96 \[ \frac{d \left (-48 a c x \left (8 c^4 x^4-14 c^2 x^2+3\right ) \sqrt{d-c^2 d x^2}-144 a \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-\frac{18 b \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b \sqrt{d-c^2 d x^2} \left (72 \cosh ^{-1}(c x)^2-18 \cosh \left (2 \cosh ^{-1}(c x)\right )+9 \cosh \left (4 \cosh ^{-1}(c x)\right )+2 \cosh \left (6 \cosh ^{-1}(c x)\right )-12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right )}{2304 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.289, size = 456, normalized size = 1.6 \begin{align*} -{\frac{ax}{6\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{ax}{24\,{c}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{adx}{16\,{c}^{2}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{a{d}^{2}}{16\,{c}^{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{bd{c}^{4}{\rm arccosh} \left (cx\right ){x}^{7}}{ \left ( 6\,cx+6 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{11\,b{c}^{2}d{\rm arccosh} \left (cx\right ){x}^{5}}{ \left ( 24\,cx+24 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{17\,bd{\rm arccosh} \left (cx\right ){x}^{3}}{ \left ( 48\,cx+48 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bd{\rm arccosh} \left (cx\right )x}{ \left ( 16\,cx+16 \right ){c}^{2} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{7\,bd}{2304\,{c}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{db \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{32\,{c}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bd{c}^{3}{x}^{6}}{36}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{7\,bdc{x}^{4}}{96}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bd{x}^{2}}{32\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{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 (-{\left (a c^{2} d x^{4} - a d x^{2} +{\left (b c^{2} d x^{4} - b d x^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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