Optimal. Leaf size=505 \[ -\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {16}{315} a c^3 x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {976 c^3 \sqrt {a x-1} \sqrt {a x+1}}{315 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {4322 c^3 x \cosh ^{-1}(a x)}{1225}+\frac {3 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)^2}{49 a}-\frac {18 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {8 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {48 c^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{35 a} \]
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Rubi [A] time = 1.41, antiderivative size = 505, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5681, 5718, 194, 5680, 12, 1610, 1799, 1850, 520, 1247, 698, 460, 74, 5654} \[ \frac {6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {16}{315} a c^3 x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {976 c^3 \sqrt {a x-1} \sqrt {a x+1}}{315 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {4322 c^3 x \cosh ^{-1}(a x)}{1225}+\frac {3 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)^2}{49 a}-\frac {18 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {8 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {48 c^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{35 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 194
Rule 460
Rule 520
Rule 698
Rule 1247
Rule 1610
Rule 1799
Rule 1850
Rule 5654
Rule 5680
Rule 5681
Rule 5718
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^3 \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^3 \, dx+\frac {1}{7} \left (3 a c^3\right ) \int x (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx-\frac {1}{49} \left (6 c^3\right ) \int \left (-1+a^2 x^2\right )^3 \cosh ^{-1}(a x) \, dx-\frac {1}{35} \left (18 a c^3\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac {6}{49} c^3 x \cosh ^{-1}(a x)-\frac {6}{49} a^2 c^3 x^3 \cosh ^{-1}(a x)+\frac {18}{245} a^4 c^3 x^5 \cosh ^{-1}(a x)-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {1}{175} \left (36 c^3\right ) \int \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x) \, dx+\frac {1}{35} \left (16 c^3\right ) \int \cosh ^{-1}(a x)^3 \, dx+\frac {1}{49} \left (6 a c^3\right ) \int \frac {x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right )}{35 \sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {1}{35} \left (24 a c^3\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac {402 c^3 x \cosh ^{-1}(a x)}{1225}-\frac {318 a^2 c^3 x^3 \cosh ^{-1}(a x)}{1225}+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3-\frac {1}{35} \left (16 c^3\right ) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx+\frac {\left (6 a c^3\right ) \int \frac {x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{1715}-\frac {1}{175} \left (36 a c^3\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {1}{35} \left (48 a c^3\right ) \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {962 c^3 x \cosh ^{-1}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {1}{35} \left (96 c^3\right ) \int \cosh ^{-1}(a x) \, dx-\frac {1}{875} \left (12 a c^3\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {1}{35} \left (16 a c^3\right ) \int \frac {x \left (-3+a^2 x^2\right )}{3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {\left (6 a c^3 \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right )}{\sqrt {-1+a^2 x^2}} \, dx}{1715 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac {1}{105} \left (16 a c^3\right ) \int \frac {x \left (-3+a^2 x^2\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {1}{35} \left (96 a c^3\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {\left (3 a c^3 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-35+35 a^2 x-21 a^4 x^2+5 a^6 x^3}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{1715 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (12 a c^3 \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {-1+a^2 x^2}} \, dx}{875 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {96 c^3 \sqrt {-1+a x} \sqrt {1+a x}}{35 a}+\frac {16}{315} a c^3 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3-\frac {1}{45} \left (16 a c^3\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {\left (3 a c^3 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {16}{\sqrt {-1+a^2 x}}+8 \sqrt {-1+a^2 x}-6 \left (-1+a^2 x\right )^{3/2}+5 \left (-1+a^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{1715 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (6 a c^3 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {15-10 a^2 x+3 a^4 x^2}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{875 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {976 c^3 \sqrt {-1+a x} \sqrt {1+a x}}{315 a}+\frac {16}{315} a c^3 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {96 c^3 \left (1-a^2 x^2\right )}{1715 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {16 c^3 \left (1-a^2 x^2\right )^2}{1715 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {36 c^3 \left (1-a^2 x^2\right )^3}{8575 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3-\frac {\left (6 a c^3 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {-1+a^2 x}}-4 \sqrt {-1+a^2 x}+3 \left (-1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{875 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {976 c^3 \sqrt {-1+a x} \sqrt {1+a x}}{315 a}+\frac {16}{315} a c^3 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.42, size = 179, normalized size = 0.35 \[ \frac {c^3 \left (2 \sqrt {a x-1} \sqrt {a x+1} \left (16875 a^6 x^6-134541 a^4 x^4+747937 a^2 x^2-22329151\right )-385875 a x \left (5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right ) \cosh ^{-1}(a x)^3+11025 \sqrt {a x-1} \sqrt {a x+1} \left (75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right ) \cosh ^{-1}(a x)^2-210 a x \left (1125 a^6 x^6-7371 a^4 x^4+26495 a^2 x^2-226905\right ) \cosh ^{-1}(a x)\right )}{13505625 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 248, normalized size = 0.49 \[ -\frac {385875 \, {\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 11025 \, {\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 210 \, {\left (1125 \, a^{7} c^{3} x^{7} - 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} - 226905 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (16875 \, a^{6} c^{3} x^{6} - 134541 \, a^{4} c^{3} x^{4} + 747937 \, a^{2} c^{3} x^{2} - 22329151 \, c^{3}\right )} \sqrt {a^{2} x^{2} - 1}}{13505625 \, a} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 294, normalized size = 0.58 \[ -\frac {c^{3} \left (1929375 \mathrm {arccosh}\left (a x \right )^{3} a^{7} x^{7}-826875 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-8103375 \mathrm {arccosh}\left (a x \right )^{3} a^{5} x^{5}+3869775 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+236250 \,\mathrm {arccosh}\left (a x \right ) a^{7} x^{7}-33750 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}+13505625 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{3}-8345925 \mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-1547910 \,\mathrm {arccosh}\left (a x \right ) a^{5} x^{5}+269082 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}-13505625 a x \mathrm {arccosh}\left (a x \right )^{3}+23825025 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+5563950 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )-1495874 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-47650050 a x \,\mathrm {arccosh}\left (a x \right )+44658302 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{13505625 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 276, normalized size = 0.55 \[ \frac {1}{1225} \, {\left (75 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 351 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 757 \, \sqrt {a^{2} x^{2} - 1} c^{3} x^{2} - \frac {2161 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} - \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{13505625} \, {\left (16875 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 134541 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt {a^{2} x^{2} - 1} c^{3} x^{2} - \frac {22329151 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}} - \frac {105 \, {\left (1125 \, a^{6} c^{3} x^{7} - 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} - 226905 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right )}{a}\right )} a \]
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 \[ \int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.36, size = 367, normalized size = 0.73 \[ \begin {cases} - \frac {a^{6} c^{3} x^{7} \operatorname {acosh}^{3}{\left (a x \right )}}{7} - \frac {6 a^{6} c^{3} x^{7} \operatorname {acosh}{\left (a x \right )}}{343} + \frac {3 a^{5} c^{3} x^{6} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{49} + \frac {6 a^{5} c^{3} x^{6} \sqrt {a^{2} x^{2} - 1}}{2401} + \frac {3 a^{4} c^{3} x^{5} \operatorname {acosh}^{3}{\left (a x \right )}}{5} + \frac {702 a^{4} c^{3} x^{5} \operatorname {acosh}{\left (a x \right )}}{6125} - \frac {351 a^{3} c^{3} x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{1225} - \frac {29898 a^{3} c^{3} x^{4} \sqrt {a^{2} x^{2} - 1}}{1500625} - a^{2} c^{3} x^{3} \operatorname {acosh}^{3}{\left (a x \right )} - \frac {1514 a^{2} c^{3} x^{3} \operatorname {acosh}{\left (a x \right )}}{3675} + \frac {757 a c^{3} x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{1225} + \frac {1495874 a c^{3} x^{2} \sqrt {a^{2} x^{2} - 1}}{13505625} + c^{3} x \operatorname {acosh}^{3}{\left (a x \right )} + \frac {4322 c^{3} x \operatorname {acosh}{\left (a x \right )}}{1225} - \frac {2161 c^{3} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{1225 a} - \frac {44658302 c^{3} \sqrt {a^{2} x^{2} - 1}}{13505625 a} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} c^{3} x}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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