Optimal. Leaf size=145 \[ -\frac{192 b^3 \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )}{x \sqrt{d x^2} \sqrt{d x^2+2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2-\frac{8 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^4+384 b^4 x \]
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Rubi [A] time = 0.0360027, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5880, 8} \[ -\frac{192 b^3 \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )}{x \sqrt{d x^2} \sqrt{d x^2+2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2-\frac{8 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^4+384 b^4 x \]
Antiderivative was successfully verified.
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Rule 5880
Rule 8
Rubi steps
\begin{align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4 \, dx &=-\frac{8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2 \, dx\\ &=-\frac{192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )}{x \sqrt{d x^2} \sqrt{2+d x^2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2-\frac{8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx\\ &=384 b^4 x-\frac{192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )}{x \sqrt{d x^2} \sqrt{2+d x^2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2-\frac{8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4\\ \end{align*}
Mathematica [A] time = 0.220184, size = 264, normalized size = 1.82 \[ \frac{d x^2 \left (48 a^2 b^2+a^4+384 b^4\right )-8 a b \left (a^2+24 b^2\right ) \sqrt{d x^2} \sqrt{d x^2+2}+6 b^2 \cosh ^{-1}\left (d x^2+1\right )^2 \left (a^2 d x^2-4 a b \sqrt{d x^2} \sqrt{d x^2+2}+8 b^2 d x^2\right )+4 b \cosh ^{-1}\left (d x^2+1\right ) \left (-6 a^2 b \sqrt{d x^2} \sqrt{d x^2+2}+a^3 d x^2+24 a b^2 d x^2-48 b^3 \sqrt{d x^2} \sqrt{d x^2+2}\right )+4 b^3 \cosh ^{-1}\left (d x^2+1\right )^3 \left (a d x^2-2 b \sqrt{d x^2} \sqrt{d x^2+2}\right )+b^4 d x^2 \cosh ^{-1}\left (d x^2+1\right )^4}{d x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}+1\right ) \right ) ^{4}\, dx \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 [B] time = 2.08347, size = 626, normalized size = 4.32 \begin{align*} \frac{b^{4} d x^{2} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{4} +{\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 4 \,{\left (a b^{3} d x^{2} - 2 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}} b^{4}\right )} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{3} - 6 \,{\left (4 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}} a b^{3} -{\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x^{2}\right )} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{2} + 4 \,{\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x^{2} - 6 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}}{\left (a^{2} b^{2} + 8 \, b^{4}\right )}\right )} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right ) - 8 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}}{\left (a^{3} b + 24 \, a b^{3}\right )}}{d x} \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 \left (a + b \operatorname{acosh}{\left (d x^{2} + 1 \right )}\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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