Optimal. Leaf size=448 \[ \frac {5 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {27 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}+\frac {25 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {7 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}+\frac {27 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {25 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}+\frac {7 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.33, antiderivative size = 555, normalized size of antiderivative = 1.24, number of steps used = 29, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5798, 5778, 5700, 3312, 3303, 3298, 3301, 5780, 5448} \[ \frac {5 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {27 \sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {25 \sqrt {1-c^2 x^2} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 \sqrt {1-c^2 x^2} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {27 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {25 \sqrt {1-c^2 x^2} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {x (c x+1)^{5/2} \sqrt {1-c^2 x^2} (1-c x)^3}{b c \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 3312
Rule 5448
Rule 5700
Rule 5778
Rule 5780
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x (-1+c x)^{5/2} (1+c x)^{5/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (1-c x)^3 (1+c x)^{5/2} \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} \int \frac {\left (-1+c^2 x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 c \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (-1+c^2 x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (1-c x)^3 (1+c x)^{5/2} \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 ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (i \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 (a+b x)}-\frac {5 i \sinh (3 x)}{16 (a+b x)}+\frac {i \sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 (a+b x)}+\frac {\sinh (3 x)}{64 (a+b x)}-\frac {3 \sinh (5 x)}{64 (a+b x)}+\frac {\sinh (7 x)}{64 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (1-c x)^3 (1+c x)^{5/2} \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 (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (7 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (21 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (35 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (35 \sqrt {1-c^2 x^2} \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 )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2} \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^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2} \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 )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2} \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^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \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^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (21 \sqrt {1-c^2 x^2} \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 )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (7 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (35 \sqrt {1-c^2 x^2} \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 )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 \sqrt {1-c^2 x^2} \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^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (7 \sqrt {1-c^2 x^2} \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 )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 \sqrt {1-c^2 x^2} \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^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \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^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (21 \sqrt {1-c^2 x^2} \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 )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (7 \sqrt {1-c^2 x^2} \sinh \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x (1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {5 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {27 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {25 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {27 \sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 \sqrt {1-c^2 x^2} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{64 b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.43, size = 436, normalized size = 0.97 \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \left (-5 \sinh \left (\frac {a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+27 \sinh \left (\frac {3 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-25 a \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-25 b \sinh \left (\frac {5 a}{b}\right ) \cosh ^{-1}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+7 a \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+7 b \sinh \left (\frac {7 a}{b}\right ) \cosh ^{-1}(c x) \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+5 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+5 b \cosh \left (\frac {a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-27 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-27 b \cosh \left (\frac {3 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+25 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+25 b \cosh \left (\frac {5 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-7 b \cosh \left (\frac {7 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+64 b c^7 x^7-192 b c^5 x^5+192 b c^3 x^3-64 b c x\right )}{64 b^2 c^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{5} - 2 \, c^{2} x^{3} + x\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.56, size = 1499, normalized size = 3.35 \[ \text {result too large to display} \]
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 \[ -\frac {{\left ({\left (c^{6} x^{7} - 3 \, c^{4} x^{5} + 3 \, c^{2} x^{3} - x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{7} x^{8} - 3 \, c^{5} x^{6} + 3 \, c^{3} x^{4} - c x^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left (7 \, {\left (c^{7} x^{7} - 2 \, c^{5} x^{5} + c^{3} x^{3}\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + {\left (14 \, c^{8} x^{8} - 37 \, c^{6} x^{6} + 33 \, c^{4} x^{4} - 11 \, c^{2} x^{2} + 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (7 \, c^{9} x^{9} - 23 \, c^{7} x^{7} + 27 \, c^{5} x^{5} - 13 \, c^{3} x^{3} + 2 \, c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
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 \[ \int \frac {x\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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