Optimal. Leaf size=398 \[ -\frac{3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac{4 b d e^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{32 c^4}-\frac{3 b e^3 x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac{d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{3 b d^2 e x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{2 b d^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{2 b d e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{b e^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}+\frac{4 b^2 d e^2 x}{3 c^2}+\frac{3 b^2 e^3 x^2}{32 c^2}+\frac{3}{4} b^2 d^2 e x^2+2 b^2 d^3 x+\frac{2}{9} b^2 d e^2 x^3+\frac{1}{32} b^2 e^3 x^4 \]
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Rubi [A] time = 1.68906, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac{3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac{4 b d e^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{32 c^4}-\frac{3 b e^3 x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac{d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{3 b d^2 e x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{2 b d^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{2 b d e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{b e^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}+\frac{4 b^2 d e^2 x}{3 c^2}+\frac{3 b^2 e^3 x^2}{32 c^2}+\frac{3}{4} b^2 d^2 e x^2+2 b^2 d^3 x+\frac{2}{9} b^2 d e^2 x^3+\frac{1}{32} b^2 e^3 x^4 \]
Antiderivative was successfully verified.
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Rule 5802
Rule 5822
Rule 5676
Rule 5718
Rule 8
Rule 5759
Rule 30
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{(b c) \int \frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 e}\\ &=\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{(b c) \int \left (\frac{d^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 d^3 e x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{6 d^2 e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 d e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^4 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{2 e}\\ &=\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\left (2 b c d^3\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{\left (b c d^4\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\left (2 b c d e^2\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{2} \left (b c e^3\right ) \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 b d^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{2 b d e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac{b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac{d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac{1}{2} \left (3 b^2 d^2 e\right ) \int x \, dx-\frac{\left (3 b d^2 e\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c}+\frac{1}{3} \left (2 b^2 d e^2\right ) \int x^2 \, dx-\frac{\left (4 b d e^2\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c}+\frac{1}{8} \left (b^2 e^3\right ) \int x^3 \, dx-\frac{\left (3 b e^3\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c}\\ &=2 b^2 d^3 x+\frac{3}{4} b^2 d^2 e x^2+\frac{2}{9} b^2 d e^2 x^3+\frac{1}{32} b^2 e^3 x^4-\frac{2 b d^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{4 b d e^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{3 b e^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac{2 b d e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac{b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac{d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}+\frac{\left (4 b^2 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac{\left (3 b e^3\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^3}+\frac{\left (3 b^2 e^3\right ) \int x \, dx}{16 c^2}\\ &=2 b^2 d^3 x+\frac{4 b^2 d e^2 x}{3 c^2}+\frac{3}{4} b^2 d^2 e x^2+\frac{3 b^2 e^3 x^2}{32 c^2}+\frac{2}{9} b^2 d e^2 x^3+\frac{1}{32} b^2 e^3 x^4-\frac{2 b d^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{4 b d e^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac{3 b d^2 e x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac{3 b e^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{16 c^3}-\frac{2 b d e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac{b e^3 x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c}-\frac{d^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}-\frac{3 d^2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 e^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{32 c^4}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{4 e}\\ \end{align*}
Mathematica [A] time = 0.823838, size = 386, normalized size = 0.97 \[ \frac{c \left (72 a^2 c^3 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )+b^2 c x \left (c^2 \left (216 d^2 e x+576 d^3+64 d e^2 x^2+9 e^3 x^3\right )+3 e^2 (128 d+9 e x)\right )\right )-6 b c \cosh ^{-1}(c x) \left (b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )-24 a c^3 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right )-54 a b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )+9 b^2 \cosh ^{-1}(c x)^2 \left (8 c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 791, normalized size = 2. \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \operatorname{arcosh}\left (c x\right )^{2} + \frac{3}{2} \, a^{2} d^{2} e x^{2} + \frac{3}{2} \,{\left (2 \, x^{2} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} + \frac{\log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d^{2} e + \frac{2}{3} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b d e^{2} + \frac{1}{16} \,{\left (8 \, x^{4} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} x}{c^{4}} + \frac{3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b e^{3} + 2 \, b^{2} d^{3}{\left (x - \frac{\sqrt{c^{2} x^{2} - 1} \operatorname{arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac{2 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} a b d^{3}}{c} + \frac{1}{4} \,{\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} - \int \frac{{\left (b^{2} c^{3} e^{3} x^{6} + 4 \, b^{2} c^{3} d e^{2} x^{5} - 4 \, b^{2} c d e^{2} x^{3} - 6 \, b^{2} c d^{2} e x^{2} +{\left (6 \, c^{3} d^{2} e - c e^{3}\right )} b^{2} x^{4} +{\left (b^{2} c^{2} e^{3} x^{5} + 4 \, b^{2} c^{2} d e^{2} x^{4} + 6 \, b^{2} c^{2} d^{2} e x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{2 \,{\left (c^{3} x^{3} +{\left (c^{2} x^{2} - 1\right )} \sqrt{c x + 1} \sqrt{c x - 1} - c x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52465, size = 1010, normalized size = 2.54 \begin{align*} \frac{9 \,{\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \,{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \,{\left (8 \,{\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e + b^{2} c^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 96 \,{\left (3 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} + 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \,{\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x - 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} -{\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \,{\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 6 \,{\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \,{\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{288 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.14033, size = 750, normalized size = 1.88 \begin{align*} \text{result too large to display} \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 (e x + d\right )}^{3}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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