Optimal. Leaf size=359 \[ \frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.69, antiderivative size = 351, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5803, 5655, 5779, 3303, 3298, 3301, 5665} \[ \frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac {d^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5655
Rule 5665
Rule 5779
Rule 5803
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac {d^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac {2 d e x}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac {e^2 x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d^2 \int \frac {1}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+(2 d e) \int \frac {x}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+e^2 \int \frac {x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (c d^2\right ) \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}+\frac {(2 d e) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {e^2 \operatorname {Subst}\left (\int \left (-\frac {\sinh (x)}{4 (a+b x)}+\frac {3 \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {d^2 \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac {e^2 \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\left (2 d e \cosh \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 )}{b c^2}-\frac {\left (2 d e \sinh \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 )}{b c^2}\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}+\frac {\left (d^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,\sinh ^{-1}(c x)\right )}{b c}-\frac {\left (e^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,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\left (3 e^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,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (d^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,\sinh ^{-1}(c x)\right )}{b c}+\frac {\left (e^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,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (3 e^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,\sinh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac {d^2 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {e^2 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^3}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {2 d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b^2 c^2}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 288, normalized size = 0.80 \[ -\frac {\sinh \left (\frac {a}{b}\right ) \left (4 c^2 d^2-e^2\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-4 c^2 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+\frac {4 b c^2 d^2 \sqrt {c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac {8 b c^2 d e x \sqrt {c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac {4 b c^2 e^2 x^2 \sqrt {c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}-8 c d e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+8 c d e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{4 b^2 c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{2} x^{2} + 2 \, d e x + d^{2}}{b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 616, normalized size = 1.72 \[ \frac {\frac {\left (4 c^{3} x^{3}-4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+3 c x -\sqrt {c^{2} x^{2}+1}\right ) e^{2}}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, 3 \arcsinh \left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e^{2} \left (4 c^{3} x^{3}+3 c x +4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{8 b \,c^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (c x -\sqrt {c^{2} x^{2}+1}\right ) d^{2}}{2 b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {d^{2} {\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {\left (c x -\sqrt {c^{2} x^{2}+1}\right ) e^{2}}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d^{2} \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {d^{2} {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {e^{2} \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}+\frac {\left (2 c^{2} x^{2}-2 c x \sqrt {c^{2} x^{2}+1}+1\right ) d e}{2 c \left (a +b \arcsinh \left (c x \right )\right ) b}-\frac {e d \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{c \,b^{2}}-\frac {e d \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{2 b c \left (a +b \arcsinh \left (c x \right )\right )}-\frac {e d \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right )}{b^{2} c}}{c} \]
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 \[ -\frac {c^{3} e^{2} x^{5} + 2 \, c^{3} d e x^{4} + 2 \, c d e x^{2} + c d^{2} x + {\left (c^{3} d^{2} + c e^{2}\right )} x^{3} + {\left (c^{2} e^{2} x^{4} + 2 \, c^{2} d e x^{3} + 2 \, d e x + {\left (c^{2} d^{2} + e^{2}\right )} x^{2} + d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x + a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {3 \, c^{5} e^{2} x^{6} + 4 \, c^{5} d e x^{5} + 8 \, c^{3} d e x^{3} + {\left (c^{5} d^{2} + 6 \, c^{3} e^{2}\right )} x^{4} + 4 \, c d e x + c d^{2} + {\left (2 \, c^{3} d^{2} + 3 \, c e^{2}\right )} x^{2} + {\left (3 \, c^{3} e^{2} x^{4} + 4 \, c^{3} d e x^{3} - c d^{2} + {\left (c^{3} d^{2} + c e^{2}\right )} x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (6 \, c^{4} e^{2} x^{5} + 8 \, c^{4} d e x^{4} + 8 \, c^{2} d e x^{2} + {\left (2 \, c^{4} d^{2} + 7 \, c^{2} e^{2}\right )} x^{3} + 2 \, d e + {\left (c^{2} d^{2} + 2 \, e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c + {\left (b^{2} c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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