Optimal. Leaf size=337 \[ -\frac {5 \sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 c^6 \sqrt {1-c x}}-\frac {15 \sqrt {c x-1} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6 \sqrt {1-c x}}-\frac {5 \sqrt {c x-1} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6 \sqrt {1-c x}}+\frac {5 \sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 c^6 \sqrt {1-c x}}+\frac {15 \sqrt {c x-1} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6 \sqrt {1-c x}}+\frac {5 \sqrt {c x-1} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6 \sqrt {1-c x}}-\frac {x^5 \sqrt {c x-1}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.86, antiderivative size = 424, normalized size of antiderivative = 1.26, number of steps used = 14, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5798, 5775, 5670, 5448, 3303, 3298, 3301} \[ -\frac {5 \sqrt {c x-1} \sqrt {c x+1} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^6 \sqrt {1-c^2 x^2}}-\frac {15 \sqrt {c x-1} \sqrt {c x+1} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}-\frac {5 \sqrt {c x-1} \sqrt {c x+1} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}+\frac {5 \sqrt {c x-1} \sqrt {c x+1} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^6 \sqrt {1-c^2 x^2}}+\frac {15 \sqrt {c x-1} \sqrt {c x+1} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}+\frac {5 \sqrt {c x-1} \sqrt {c x+1} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}-\frac {x^5 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5670
Rule 5775
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {x^5 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {1-c^2 x^2}}\\ &=-\frac {x^5 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^6 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^5 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{8 (a+b x)}+\frac {3 \sinh (3 x)}{16 (a+b x)}+\frac {\sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^6 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^5 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^6 \sqrt {1-c^2 x^2}}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^6 \sqrt {1-c^2 x^2}}+\frac {\left (15 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^6 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^5 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^6 \sqrt {1-c^2 x^2}}+\frac {\left (15 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^6 \sqrt {1-c^2 x^2}}+\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^6 \sqrt {1-c^2 x^2}}-\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x} \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^6 \sqrt {1-c^2 x^2}}-\frac {\left (15 \sqrt {-1+c x} \sqrt {1+c x} \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^6 \sqrt {1-c^2 x^2}}-\frac {\left (5 \sqrt {-1+c x} \sqrt {1+c x} \sinh \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^6 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^5 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {5 \sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^6 \sqrt {1-c^2 x^2}}-\frac {15 \sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}-\frac {5 \sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}+\frac {5 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^6 \sqrt {1-c^2 x^2}}+\frac {15 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}+\frac {5 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^6 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 190, normalized size = 0.56 \[ \frac {\sqrt {1-c^2 x^2} \left (\frac {16 b c^5 x^5}{a+b \cosh ^{-1}(c x)}+5 \left (2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )\right )}{16 b^2 c^6 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} x^{5}}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]
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 [B] time = 0.84, size = 1046, normalized size = 3.10 \[ \text {result too large to display} \]
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 \[ -\frac {c^{3} x^{8} - c x^{6} + {\left (c^{2} x^{7} - x^{5}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int \frac {5 \, c^{5} x^{9} - 11 \, c^{3} x^{7} + 6 \, c x^{5} + {\left (5 \, c^{3} x^{7} - 4 \, c x^{5}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 5 \, {\left (2 \, c^{4} x^{8} - 3 \, c^{2} x^{6} + x^{4}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} b^{2} c^{3} x^{2} + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} a b c^{3} x^{2} + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]
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 \[ \int \frac {x^5}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{5}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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