Optimal. Leaf size=281 \[ \frac {\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.07, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5777, 5779, 5448, 3303, 3298, 3301} \[ \frac {\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b^2 c^3}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5777
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \int \frac {x \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac {(8 c) \int \frac {x^3 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh ^5(x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {8 \operatorname {Subst}\left (\int \frac {\cosh ^5(x) \sinh ^3(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}+\frac {\sinh (6 x)}{32 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {8 \operatorname {Subst}\left (\int \left (-\frac {3 \sinh (2 x)}{64 (a+b x)}-\frac {\sinh (4 x)}{64 (a+b x)}+\frac {\sinh (6 x)}{64 (a+b x)}+\frac {\sinh (8 x)}{128 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (8 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {5 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (5 \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,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\left (3 \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,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac {\cosh \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\left (5 \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,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\left (3 \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,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac {\sinh \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \text {Chi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}\\ \end {align*}
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Mathematica [A] time = 1.17, size = 413, normalized size = 1.47 \[ -\frac {-\sinh \left (\frac {2 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+2 \sinh \left (\frac {4 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+3 a \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+3 b \sinh \left (\frac {6 a}{b}\right ) \sinh ^{-1}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+a \sinh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+b \sinh \left (\frac {8 a}{b}\right ) \sinh ^{-1}(c x) \text {Chi}\left (8 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+b \cosh \left (\frac {2 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-2 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-2 b \cosh \left (\frac {4 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 b \cosh \left (\frac {6 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-a \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-b \cosh \left (\frac {8 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (8 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+16 b c^8 x^8+48 b c^6 x^6+48 b c^4 x^4+16 b c^2 x^2}{16 b^2 c^3 \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{6} + 2 \, c^{2} x^{4} + x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.89, size = 1044, normalized size = 3.72 \[ \text {result too large to display} \]
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 \[ -\frac {{\left (c^{6} x^{8} + 3 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{7} x^{9} + 3 \, c^{5} x^{7} + 3 \, c^{3} x^{5} + c x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x + a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (8 \, c^{7} x^{8} + 17 \, c^{5} x^{6} + 10 \, c^{3} x^{4} + c x^{2}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (8 \, c^{8} x^{9} + 22 \, c^{6} x^{7} + 21 \, c^{4} x^{5} + 8 \, c^{2} x^{3} + x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (8 \, c^{9} x^{10} + 27 \, c^{7} x^{8} + 33 \, c^{5} x^{6} + 17 \, c^{3} x^{4} + 3 \, c x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c + {\left (b^{2} c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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