Optimal. Leaf size=479 \[ \frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {c x-1} \sqrt {c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {3 b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {c x-1} \sqrt {c x+1}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 1.14, antiderivative size = 509, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {5798, 5748, 5756, 5761, 4180, 2279, 2391, 207, 199, 290, 325} \[ -\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {c x-1} \sqrt {c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {c x-1} \sqrt {c x+1}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {3 b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 207
Rule 290
Rule 325
Rule 2279
Rule 2391
Rule 4180
Rule 5748
Rule 5756
Rule 5761
Rule 5798
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )} \, dx}{4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\left (-1+c^2 x^2\right )^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{12 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{6 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 7.29, size = 500, normalized size = 1.04 \[ -\frac {5 a c^2 \log \left (\sqrt {d} \sqrt {-d \left (c^2 x^2-1\right )}+d\right )}{2 d^{5/2}}+\frac {5 a c^2 \log (x)}{2 d^{5/2}}+\sqrt {-d \left (c^2 x^2-1\right )} \left (-\frac {2 a c^2}{d^3 \left (c^2 x^2-1\right )}+\frac {a c^2}{3 d^3 \left (c^2 x^2-1\right )^2}-\frac {a}{2 d^3 x^2}\right )+\frac {b c^2 \left (\frac {6 (c x-1) (c x+1) \cosh ^{-1}(c x)}{c^2 x^2}-30 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )+30 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )+\frac {6 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c x}+26 \cosh ^{-1}(c x) \cosh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-30 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+30 i \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-26 \cosh ^{-1}(c x) \sinh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \tanh ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \coth ^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-26 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{12 d^2 \sqrt {-d (c x-1) (c x+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 801, normalized size = 1.67 \[ -\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \mathrm {arccosh}\left (c x \right ) c^{4}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {10 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}+\frac {13 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {13 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, a {\left (\frac {15 \, c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right )}{d^{\frac {5}{2}}} - \frac {15 \, c^{2}}{\sqrt {-c^{2} d x^{2} + d} d^{2}} - \frac {5 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} + \frac {3}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{2}}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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