Optimal. Leaf size=246 \[ -\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.52, antiderivative size = 305, normalized size of antiderivative = 1.24, number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5713, 5697, 5780, 5448, 3303, 3298, 3301} \[ -\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c x+1)^{3/2} \sqrt {1-c^2 x^2} (1-c x)^2}{b c \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5697
Rule 5713
Rule 5780
Rubi steps
\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 c \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (-1+c^2 x^2\right )}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 232, normalized size = 0.94 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 \sinh \left (\frac {2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 b \cosh \left (\frac {2 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+b \cosh \left (\frac {4 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 b c^4 x^4+4 b c^2 x^2-2 b\right )}{2 b^2 c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 737, normalized size = 3.00 \[ -\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right )}{16 \left (c x +1\right ) \left (c x -1\right ) c b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+4 a}{b}}}{4 \left (c x +1\right ) \left (c x -1\right ) c \,b^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} b \,c^{3}+8 x^{4} b \,c^{4}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x b c -8 x^{2} b \,c^{2}+4 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} b +4 \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} a +b \right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c \,b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {3 \sqrt {-c^{2} x^{2}+1}}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right )}{4 \left (c x +1\right ) \left (c x -1\right ) c b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+2 a}{b}}}{2 \left (c x +1\right ) \left (c x -1\right ) c \,b^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (2 \sqrt {c x +1}\, \sqrt {c x -1}\, x b c +2 x^{2} b \,c^{2}+2 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} b +2 \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} a -b \right )}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c \,b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\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 \[ \frac {{\left ({\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{5} x^{5} - 2 \, c^{3} x^{3} + c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} - \int \frac {{\left ({\left (4 \, c^{4} x^{4} - 3 \, c^{2} x^{2} - 1\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + 4 \, {\left (2 \, c^{5} x^{5} - 3 \, c^{3} x^{3} + c x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (4 \, c^{6} x^{6} - 9 \, c^{4} x^{4} + 6 \, c^{2} x^{2} - 1\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{4} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{2} x^{2} - 2 \, a b c^{2} x^{2} + 2 \, {\left (a b c^{3} x^{3} - a b c x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + a b + {\left (b^{2} c^{4} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{2} x^{2} - 2 \, b^{2} c^{2} x^{2} + 2 \, {\left (b^{2} c^{3} x^{3} - b^{2} c x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + b^{2}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
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 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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