Optimal. Leaf size=609 \[ -\frac {32 b e^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^7}-\frac {16 b d e^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^5}-\frac {16 b e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^5}-\frac {4 b d^2 e \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {8 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^3}-\frac {12 b e^3 x^4 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^3}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b d^2 e x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {6 b d e^2 x^4 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b e^3 x^6 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {12 b^2 e^3 x^5}{1225 c^2}+2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7 \]
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Rubi [A] time = 2.10, antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5707, 5654, 5718, 8, 5662, 5759, 30} \[ -\frac {4 b d^2 e \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {8 b d e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b d e^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^5}-\frac {12 b e^3 x^4 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^3}-\frac {16 b e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^5}-\frac {32 b e^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^7}+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b d^2 e x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b d^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {6 b d e^2 x^4 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 b e^3 x^6 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3+2 b^2 d^3 x+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5654
Rule 5662
Rule 5707
Rule 5718
Rule 5759
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 \left (a+b \cosh ^{-1}(c x)\right )^2+3 d^2 e x^2 \left (a+b \cosh ^{-1}(c x)\right )^2+3 d e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )^2+e^3 x^6 \left (a+b \cosh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx+\left (3 d^2 e\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx+\left (3 d e^2\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx+e^3 \int x^6 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2-\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-\left (2 b c d^2 e\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{5} \left (6 b c d e^2\right ) \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{7} \left (2 b c e^3\right ) \int \frac {x^7 \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 {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac {1}{3} \left (2 b^2 d^2 e\right ) \int x^2 \, dx-\frac {\left (4 b d^2 e\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}{25} \left (6 b^2 d e^2\right ) \int x^4 \, dx-\frac {\left (24 b 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}{25 c}+\frac {1}{49} \left (2 b^2 e^3\right ) \int x^6 \, dx-\frac {\left (12 b e^3\right ) \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{49 c}\\ &=2 b^2 d^3 x+\frac {2}{9} b^2 d^2 e x^3+\frac {6}{125} b^2 d e^2 x^5+\frac {2}{343} b^2 e^3 x^7-\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^2 e \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^3}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (4 b^2 d^2 e\right ) \int 1 \, dx}{3 c^2}-\frac {\left (16 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}{25 c^3}+\frac {\left (8 b^2 d e^2\right ) \int x^2 \, dx}{25 c^2}-\frac {\left (48 b e^3\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{245 c^3}+\frac {\left (12 b^2 e^3\right ) \int x^4 \, dx}{245 c^2}\\ &=2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\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^2 e \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^5}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (16 b^2 d e^2\right ) \int 1 \, dx}{25 c^4}-\frac {\left (32 b e^3\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{245 c^5}+\frac {\left (16 b^2 e^3\right ) \int x^2 \, dx}{245 c^4}\\ &=2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\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^2 e \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^5}-\frac {32 b e^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (32 b^2 e^3\right ) \int 1 \, dx}{245 c^6}\\ &=2 b^2 d^3 x+\frac {4 b^2 d^2 e x}{3 c^2}+\frac {16 b^2 d e^2 x}{25 c^4}+\frac {32 b^2 e^3 x}{245 c^6}+\frac {2}{9} b^2 d^2 e x^3+\frac {8 b^2 d e^2 x^3}{75 c^2}+\frac {16 b^2 e^3 x^3}{735 c^4}+\frac {6}{125} b^2 d e^2 x^5+\frac {12 b^2 e^3 x^5}{1225 c^2}+\frac {2}{343} b^2 e^3 x^7-\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^2 e \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3}-\frac {16 b d e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^5}-\frac {32 b e^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c}-\frac {8 b d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}-\frac {12 b e^3 x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.85, size = 453, normalized size = 0.74 \[ \frac {11025 a^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-210 a b \sqrt {c x-1} \sqrt {c x+1} \left (c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )+240 e^3\right )-210 b \cosh ^{-1}(c x) \left (b \sqrt {c x-1} \sqrt {c x+1} \left (c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )+240 e^3\right )-105 a c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )\right )+11025 b^2 c^7 x \cosh ^{-1}(c x)^2 \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )+2 b^2 c x \left (c^6 \left (385875 d^3+42875 d^2 e x^2+9261 d e^2 x^4+1125 e^3 x^6\right )+210 c^4 e \left (1225 d^2+98 d e x^2+9 e^2 x^4\right )+840 c^2 e^2 \left (147 d+5 e x^2\right )+25200 e^3\right )}{385875 c^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 586, normalized size = 0.96 \[ \frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \, {\left (49 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d e^{2} + 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \, {\left (1225 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} e + 1176 \, b^{2} c^{5} d e^{2} + 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \, {\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 105 \, {\left (3675 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} + 4900 \, b^{2} c^{5} d^{2} e + 2352 \, b^{2} c^{3} d e^{2} + 480 \, b^{2} c e^{3}\right )} x + 210 \, {\left (525 \, a b c^{7} e^{3} x^{7} + 2205 \, a b c^{7} d e^{2} x^{5} + 3675 \, a b c^{7} d^{2} e x^{3} + 3675 \, a b c^{7} d^{3} x - {\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} + 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} + 240 \, b^{2} e^{3} + 9 \, {\left (49 \, b^{2} c^{6} d e^{2} + 10 \, b^{2} c^{4} e^{3}\right )} x^{4} + {\left (1225 \, b^{2} c^{6} d^{2} e + 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 210 \, {\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} + 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} + 240 \, a b e^{3} + 9 \, {\left (49 \, a b c^{6} d e^{2} + 10 \, a b c^{4} e^{3}\right )} x^{4} + {\left (1225 \, a b c^{6} d^{2} e + 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{385875 \, c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 632, normalized size = 1.04 \[ \frac {\frac {a^{2} \left (\frac {1}{7} e^{3} c^{7} x^{7}+\frac {3}{5} c^{7} d \,e^{2} x^{5}+c^{7} d^{2} e \,x^{3}+x \,c^{7} d^{3}\right )}{c^{6}}+\frac {b^{2} \left (\frac {e^{3} \left (3675 \mathrm {arccosh}\left (c x \right )^{2} c^{7} x^{7}-1050 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}-1260 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{4} x^{4}+150 c^{7} x^{7}-1680 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+252 c^{5} x^{5}-3360 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+560 c^{3} x^{3}+3360 c x \right )}{25725}+\frac {d \,e^{2} c^{2} \left (225 \mathrm {arccosh}\left (c x \right )^{2} c^{5} x^{5}-90 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{4} x^{4}-120 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+18 c^{5} x^{5}-240 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+40 c^{3} x^{3}+240 c x \right )}{375}+\frac {c^{4} d^{2} e \left (9 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{3}-6 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{9}+d^{3} c^{6} \left (\mathrm {arccosh}\left (c x \right )^{2} c x -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c^{6}}+\frac {2 a b \left (\frac {\mathrm {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) d \,e^{2} c^{7} x^{5}}{5}+\mathrm {arccosh}\left (c x \right ) c^{7} d^{2} e \,x^{3}+\mathrm {arccosh}\left (c x \right ) c^{7} x \,d^{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 d \,e^{2} c^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 684, normalized size = 1.12 \[ \frac {1}{7} \, b^{2} e^{3} x^{7} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} e^{3} x^{7} + \frac {3}{5} \, b^{2} d e^{2} x^{5} \operatorname {arcosh}\left (c x\right )^{2} + \frac {3}{5} \, a^{2} d e^{2} x^{5} + b^{2} d^{2} e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + a^{2} d^{2} e x^{3} + b^{2} d^{3} x \operatorname {arcosh}\left (c x\right )^{2} + \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^{2} e - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{2} e + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} a b d e^{2} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c \operatorname {arcosh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d e^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} a b e^{3} - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c \operatorname {arcosh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} 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} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.78, size = 996, normalized size = 1.64 \[ \begin {cases} a^{2} d^{3} x + a^{2} d^{2} e x^{3} + \frac {3 a^{2} d e^{2} x^{5}}{5} + \frac {a^{2} e^{3} x^{7}}{7} + 2 a b d^{3} x \operatorname {acosh}{\left (c x \right )} + 2 a b d^{2} e x^{3} \operatorname {acosh}{\left (c x \right )} + \frac {6 a b d e^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {2 a b e^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {2 a b d^{3} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {2 a b d^{2} e x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} - \frac {6 a b d e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {2 a b e^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {4 a b d^{2} e \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} - \frac {8 a b d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{3}} - \frac {12 a b e^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {16 a b d e^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{5}} - \frac {16 a b e^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {32 a b e^{3} \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} + b^{2} d^{3} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + b^{2} d^{2} e x^{3} \operatorname {acosh}^{2}{\left (c x \right )} + \frac {2 b^{2} d^{2} e x^{3}}{9} + \frac {3 b^{2} d e^{2} x^{5} \operatorname {acosh}^{2}{\left (c x \right )}}{5} + \frac {6 b^{2} d e^{2} x^{5}}{125} + \frac {b^{2} e^{3} x^{7} \operatorname {acosh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} e^{3} x^{7}}{343} - \frac {2 b^{2} d^{3} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {2 b^{2} d^{2} e x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{3 c} - \frac {6 b^{2} d e^{2} x^{4} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{25 c} - \frac {2 b^{2} e^{3} x^{6} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{49 c} + \frac {4 b^{2} d^{2} e x}{3 c^{2}} + \frac {8 b^{2} d e^{2} x^{3}}{75 c^{2}} + \frac {12 b^{2} e^{3} x^{5}}{1225 c^{2}} - \frac {4 b^{2} d^{2} e \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{3 c^{3}} - \frac {8 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{25 c^{3}} - \frac {12 b^{2} e^{3} x^{4} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{245 c^{3}} + \frac {16 b^{2} d e^{2} x}{25 c^{4}} + \frac {16 b^{2} e^{3} x^{3}}{735 c^{4}} - \frac {16 b^{2} d e^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{25 c^{5}} - \frac {16 b^{2} e^{3} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{245 c^{5}} + \frac {32 b^{2} e^{3} x}{245 c^{6}} - \frac {32 b^{2} e^{3} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right )^{2} \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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