Optimal. Leaf size=91 \[ -\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{b^2 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{b^2 d}-\frac {\sqrt {(c+d x)^2+1}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )} \]
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Rubi [A] time = 0.17, antiderivative size = 87, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5863, 5655, 5779, 3303, 3298, 3301} \[ -\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\sqrt {(c+d x)^2+1}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5655
Rule 5779
Rule 5863
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {\sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {\sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {\sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {\text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )}{b^2 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 77, normalized size = 0.85 \[ \frac {-\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-\frac {b \sqrt {(c+d x)^2+1}}{a+b \sinh ^{-1}(c+d x)}}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arsinh}\left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 128, normalized size = 1.41 \[ \frac {\frac {-\sqrt {1+\left (d x +c \right )^{2}}+d x +c}{2 b \left (a +b \arcsinh \left (d x +c \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (d x +c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d x +c +\sqrt {1+\left (d x +c \right )^{2}}}{2 b \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (d x +c \right )-\frac {a}{b}\right )}{2 b^{2}}}{d} \]
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 {d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (3 \, c^{2} d + d\right )} x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} + c}{a b d^{3} x^{2} + 2 \, a b c d^{2} x + {\left (c^{2} d + d\right )} a b + {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + {\left (c^{2} d + d\right )} b^{2} + {\left (b^{2} d^{2} x + b^{2} c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (a b d^{2} x + a b c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}} + \int \frac {d^{4} x^{4} + 4 \, c d^{3} x^{3} + c^{4} + 2 \, {\left (3 \, c^{2} d^{2} + d^{2}\right )} x^{2} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} + 2 \, c^{2} + 4 \, {\left (c^{3} d + c d\right )} x + {\left (2 \, d^{3} x^{3} + 6 \, c d^{2} x^{2} + 2 \, c^{3} + {\left (6 \, c^{2} d + d\right )} x + c\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1}{a b d^{4} x^{4} + 4 \, a b c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} d^{2} + d^{2}\right )} a b x^{2} + 4 \, {\left (c^{3} d + c d\right )} a b x + {\left (c^{4} + 2 \, c^{2} + 1\right )} a b + {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} d^{2} + d^{2}\right )} b^{2} x^{2} + 4 \, {\left (c^{3} d + c d\right )} b^{2} x + {\left (c^{4} + 2 \, c^{2} + 1\right )} b^{2} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + {\left (3 \, c^{2} d + d\right )} b^{2} x + {\left (c^{3} + c\right )} b^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + {\left (3 \, c^{2} d + d\right )} a b x + {\left (c^{3} + c\right )} a b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{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 = -0.01 \[ \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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