Optimal. Leaf size=321 \[ \frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.84, antiderivative size = 333, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 5740, 5738, 30, 5761, 4180, 2279, 2391, 14} \[ \frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \sqrt {d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2279
Rule 2391
Rule 4180
Rule 5738
Rule 5740
Rule 5761
Rule 5798
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+c^2 x^2}{x^4} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {1}{x^4}+\frac {c^2}{x^2}\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x^2} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {\left (3 c^4 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 i b c^4 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{8 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^4 d \sqrt {d-c^2 d x^2} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.28, size = 574, normalized size = 1.79 \[ \frac {-15 a c^4 d^2 x^4 \sqrt {\frac {c x-1}{c x+1}}+21 a c^2 d^2 x^2 \sqrt {\frac {c x-1}{c x+1}}+9 a c^4 d^{3/2} x^4 \sqrt {\frac {c x-1}{c x+1}} \log (x) \sqrt {d-c^2 d x^2}-9 a c^4 d^{3/2} x^4 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )-6 a d^2 \sqrt {\frac {c x-1}{c x+1}}-9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-9 i b c^4 d^2 x^4 (c x-1) \text {Li}_2\left (-i e^{-\cosh ^{-1}(c x)}\right )+9 i b c^4 d^2 x^4 (c x-1) \text {Li}_2\left (i e^{-\cosh ^{-1}(c x)}\right )-15 b c^4 d^2 x^4-15 b c^4 d^2 x^4 \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)+9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+15 b c^3 d^2 x^3+2 b c^2 d^2 x^2+21 b c^2 d^2 x^2 \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)-2 b c d^2 x-6 b d^2 \sqrt {\frac {c x-1}{c x+1}} \cosh ^{-1}(c x)}{24 x^4 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a c^{2} d x^{2} - a d + {\left (b c^{2} d x^{2} - b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{5}}, x\right ) \]
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 = 0.73, size = 570, normalized size = 1.78 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{4 d \,x^{4}}+\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 d \,x^{2}}+\frac {a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {3 a \,c^{4} d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}+\frac {3 a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}\, d}{8}+\frac {5 b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{4}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3}}{8 \sqrt {c x +1}\, x \sqrt {c x -1}}-\frac {7 b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{8 \left (c x +1\right ) x^{2} \left (c x -1\right )}-\frac {b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{12 \sqrt {c x +1}\, x^{3} \sqrt {c x -1}}+\frac {b d \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{4 \left (c x +1\right ) x^{4} \left (c x -1\right )}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d \,c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d \,c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d \,c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d \,c^{4}}{8 \sqrt {c x -1}\, \sqrt {c x +1}} \]
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 \[ -\frac {1}{8} \, {\left (3 \, c^{4} d^{\frac {3}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} - 3 \, \sqrt {-c^{2} d x^{2} + d} c^{4} d - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{2}} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{4}}\right )} a + b \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{5}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^5} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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