Optimal. Leaf size=219 \[ \frac {\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.72, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 19, 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} \[ \frac {\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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 )^{3/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \int \frac {x \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac {(6 c) \int \frac {x^3 \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\cosh ^3(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 )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {6 \operatorname {Subst}\left (\int \left (-\frac {3 \sinh (2 x)}{32 (a+b x)}+\frac {\sinh (6 x)}{32 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \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 (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {2 a}{b}\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 )}{2 b c^3}-\frac {\left (9 \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 {\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 {\left (3 \cosh \left (\frac {6 a}{b}\right )\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 {\sinh \left (\frac {2 a}{b}\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 )}{2 b c^3}+\frac {\left (9 \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 {\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 {\left (3 \sinh \left (\frac {6 a}{b}\right )\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 {x^2 \left (1+c^2 x^2\right )^2}{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 )}{4 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 {\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 )}{4 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}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 306, normalized size = 1.40 \[ -\frac {-\sinh \left (\frac {2 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+4 \sinh \left (\frac {4 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+3 a \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+3 b \sinh \left (\frac {6 a}{b}\right ) \sinh ^{-1}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+b \cosh \left (\frac {2 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 b \cosh \left (\frac {4 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 b \cosh \left (\frac {6 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+16 b c^6 x^6+32 b c^4 x^4+16 b c^2 x^2}{16 b^2 c^3 \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 704, normalized size = 3.21 \[ \frac {1}{16 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}-\frac {32 c^{6} x^{6}-32 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+48 c^{4} x^{4}-32 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+18 c^{2} x^{2}-6 c x \sqrt {c^{2} x^{2}+1}+1}{64 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}+\frac {3 \,{\mathrm e}^{\frac {6 a}{b}} \Ei \left (1, 6 \arcsinh \left (c x \right )+\frac {6 a}{b}\right )}{32 c^{3} b^{2}}-\frac {8 c^{4} x^{4}-8 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}-4 c x \sqrt {c^{2} x^{2}+1}+1}{32 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}+\frac {{\mathrm e}^{\frac {4 a}{b}} \Ei \left (1, 4 \arcsinh \left (c x \right )+\frac {4 a}{b}\right )}{8 c^{3} b^{2}}+\frac {2 c^{2} x^{2}-2 c x \sqrt {c^{2} x^{2}+1}+1}{64 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}-\frac {{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{32 c^{3} b^{2}}+\frac {2 x^{2} b \,c^{2}+2 b c \sqrt {c^{2} x^{2}+1}\, x +2 \arcsinh \left (c x \right ) {\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right ) b +2 \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right ) a +b}{64 c^{3} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {8 x^{4} b \,c^{4}+8 \sqrt {c^{2} x^{2}+1}\, x^{3} b \,c^{3}+8 x^{2} b \,c^{2}+4 b c \sqrt {c^{2} x^{2}+1}\, x +4 \arcsinh \left (c x \right ) \Ei \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} b +4 \Ei \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} a +b}{32 c^{3} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {32 x^{6} b \,c^{6}+32 \sqrt {c^{2} x^{2}+1}\, x^{5} b \,c^{5}+48 x^{4} b \,c^{4}+32 \sqrt {c^{2} x^{2}+1}\, x^{3} b \,c^{3}+18 x^{2} b \,c^{2}+6 b c \sqrt {c^{2} x^{2}+1}\, x +6 \arcsinh \left (c x \right ) \Ei \left (1, -6 \arcsinh \left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {6 a}{b}} b +6 \Ei \left (1, -6 \arcsinh \left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {6 a}{b}} a +b}{64 c^{3} b^{2} \left (a +b \arcsinh \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 \[ -\frac {{\left (c^{4} x^{6} + 2 \, c^{2} x^{4} + x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{5} x^{7} + 2 \, c^{3} x^{5} + c x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x + a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (6 \, c^{5} x^{6} + 7 \, c^{3} x^{4} + c x^{2}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (6 \, c^{6} x^{7} + 11 \, c^{4} x^{5} + 6 \, c^{2} x^{3} + x\right )} {\left (c^{2} x^{2} + 1\right )} + 3 \, {\left (2 \, c^{7} x^{8} + 5 \, c^{5} x^{6} + 4 \, c^{3} x^{4} + c x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c + {\left (b^{2} c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{3/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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