Optimal. Leaf size=369 \[ -\frac {16 \sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (1-a^2 x^2\right )}{35 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
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Rubi [A] time = 1.01, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {192, 191, 5705, 12, 519, 6715, 1622, 949, 78, 63, 217, 206} \[ \frac {4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac {16 \sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {a \left (1-a^2 x^2\right )}{35 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 78
Rule 191
Rule 192
Rule 206
Rule 217
Rule 519
Rule 949
Rule 1622
Rule 5705
Rule 6715
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx &=\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{35 c^4 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{7/2}} \, dx\\ &=\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4}\\ &=\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\sqrt {-1+a^2 x^2} \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\sqrt {-1+a^2 x} (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {5 c^2 \left (17 a^2 c+15 d\right )+100 c d \left (a^2 c+d\right ) x+40 d^2 \left (a^2 c+d\right ) x^2}{\sqrt {-1+a^2 x} (c+d x)^{5/2}} \, dx,x,x^2\right )}{175 c^4 \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (2 a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {5 c \left (23 a^4 c^2+39 a^2 c d+18 d^2\right )+60 d \left (a^2 c+d\right )^2 x}{\sqrt {-1+a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{525 c^4 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (8 a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{35 c^4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (16 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d}{a^2}+\frac {d x^2}{a^2}}} \, dx,x,\sqrt {-1+a^2 x^2}\right )}{35 a c^4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (16 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{a^2}} \, dx,x,\frac {\sqrt {-1+a^2 x^2}}{\sqrt {c+d x^2}}\right )}{35 a c^4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {16 \sqrt {-1+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {-1+a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [C] time = 6.27, size = 844, normalized size = 2.29 \[ \frac {32 \sqrt {\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \left (1+\frac {2}{a x-1}\right )}{\sqrt {c} a+i \sqrt {d}}} \left (\left (\sqrt {c} a+i \sqrt {d}\right ) \sqrt {-\frac {i \left (\frac {c a^2}{a x-1}+i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}} \left (\frac {a \sqrt {c}}{a x-1}-i \sqrt {d} \left (1+\frac {1}{a x-1}\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}}}{\sqrt {2}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )+a \sqrt {c} \left (i \sqrt {d}-a \sqrt {c}\right ) \sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}} \sqrt {\frac {\left (c a^2+d\right ) \left (\frac {c a^2}{(a x-1)^2}+d \left (1+\frac {1}{a x-1}\right )^2\right )}{a^2 c d}} \Pi \left (\frac {2 a \sqrt {c}}{\sqrt {c} a+i \sqrt {d}};\sin ^{-1}\left (\frac {\sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}}}{\sqrt {2}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )\right ) (a x-1)^{3/2}}{35 a c^4 \left (c a^2+d\right ) \sqrt {a x+1} \sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}} \sqrt {\frac {d (a x-1)^2 \left (1+\frac {1}{a x-1}\right )^2}{a^2}+c}}+\sqrt {a x+1} \sqrt {d x^2+c} \left (-\frac {4 \left (11 c^2 a^4+15 c d a^2+6 d^2\right ) a}{105 c^3 \left (c a^2+d\right )^3 \left (d x^2+c\right )}-\frac {2 \left (5 c a^2+3 d\right ) a}{105 c^2 \left (c a^2+d\right )^2 \left (d x^2+c\right )^2}-\frac {a}{35 c \left (c a^2+d\right ) \left (d x^2+c\right )^3}\right ) \sqrt {a x-1}+\frac {x \left (16 d^3 x^6+56 c d^2 x^4+70 c^2 d x^2+35 c^3\right ) \cosh ^{-1}(a x)}{35 c^4 \left (d x^2+c\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 1752, normalized size = 4.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.19, size = 876, normalized size = 2.37 \[ \frac {8}{105} \, a {\left (\frac {3 \, {\left | d \right |} \log \left ({\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2}\right )}{c^{4} d^{\frac {3}{2}} {\left | a \right |}} - \frac {11 \, a^{10} c^{4} d^{\frac {9}{2}} {\left | d \right |} + 49 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{8} c^{3} d^{\frac {7}{2}} {\left | d \right |} + 37 \, a^{8} c^{3} d^{\frac {11}{2}} {\left | d \right |} + 77 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{6} c^{2} d^{\frac {5}{2}} {\left | d \right |} + 112 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{6} c^{2} d^{\frac {9}{2}} {\left | d \right |} + 47 \, a^{6} c^{2} d^{\frac {13}{2}} {\left | d \right |} + 33 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{6} a^{4} c d^{\frac {3}{2}} {\left | d \right |} + 93 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{4} c d^{\frac {7}{2}} {\left | d \right |} + 87 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{4} c d^{\frac {11}{2}} {\left | d \right |} + 27 \, a^{4} c d^{\frac {15}{2}} {\left | d \right |} + 6 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{8} a^{2} \sqrt {d} {\left | d \right |} + 24 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{6} a^{2} d^{\frac {5}{2}} {\left | d \right |} + 36 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{2} d^{\frac {9}{2}} {\left | d \right |} + 24 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{2} d^{\frac {13}{2}} {\left | d \right |} + 6 \, a^{2} d^{\frac {17}{2}} {\left | d \right |}}{{\left (a^{2} c d + {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} + d^{2}\right )}^{5} c^{3} d {\left | a \right |}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{35 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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