Optimal. Leaf size=100 \[ -\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}-\frac {4 b \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2}-\frac {2 b^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac {2 b^2 \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2} \]
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Rubi [A] time = 0.18, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5865, 12, 5661, 5760, 4182, 2279, 2391} \[ -\frac {2 b^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac {2 b^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}-\frac {4 b \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2279
Rule 2391
Rule 4182
Rule 5661
Rule 5760
Rule 5865
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{(c e+d e x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}+\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}-\frac {4 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}-\frac {4 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2 (c+d x)}-\frac {4 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac {2 b^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac {2 b^2 \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 154, normalized size = 1.54 \[ \frac {-\frac {a^2}{c+d x}+a b \left (2 \log \left (\frac {2 \sinh ^2\left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )}{c+d x}\right )-\frac {2 \sinh ^{-1}(c+d x)}{c+d x}\right )+b^2 \left (2 \text {Li}_2\left (-e^{-\sinh ^{-1}(c+d x)}\right )-2 \text {Li}_2\left (e^{-\sinh ^{-1}(c+d x)}\right )+\sinh ^{-1}(c+d x) \left (-\frac {\sinh ^{-1}(c+d x)}{c+d x}+2 \log \left (1-e^{-\sinh ^{-1}(c+d x)}\right )-2 \log \left (e^{-\sinh ^{-1}(c+d x)}+1\right )\right )\right )}{d e^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arsinh}\left (d x + c\right ) + a^{2}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 229, normalized size = 2.29 \[ -\frac {a^{2}}{d \,e^{2} \left (d x +c \right )}-\frac {b^{2} \arcsinh \left (d x +c \right )^{2}}{d \,e^{2} \left (d x +c \right )}-\frac {2 b^{2} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{d \,e^{2}}-\frac {2 b^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{d \,e^{2}}+\frac {2 b^{2} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{d \,e^{2}}+\frac {2 b^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{d \,e^{2}}-\frac {2 a b \arcsinh \left (d x +c \right )}{d \,e^{2} \left (d x +c \right )}-\frac {2 a b \arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{d \,e^{2}} \]
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 \[ -b^{2} {\left (\frac {\log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2}}{d^{2} e^{2} x + c d e^{2}} - \int \frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (d x + c\right )} + 1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}{d^{4} e^{2} x^{4} + 4 \, c d^{3} e^{2} x^{3} + c^{4} e^{2} + c^{2} e^{2} + {\left (6 \, c^{2} d^{2} e^{2} + d^{2} e^{2}\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e^{2} + c d e^{2}\right )} x + {\left (d^{3} e^{2} x^{3} + 3 \, c d^{2} e^{2} x^{2} + c^{3} e^{2} + c e^{2} + {\left (3 \, c^{2} d e^{2} + d e^{2}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}\,{d x}\right )} - 2 \, a b {\left (\frac {\operatorname {arsinh}\left (d x + c\right )}{d^{2} e^{2} x + c d e^{2}} + \frac {\operatorname {arsinh}\left (\frac {d e^{2}}{{\left | d^{2} e^{2} x + c d e^{2} \right |}}\right )}{d e^{2}}\right )} - \frac {a^{2}}{d^{2} e^{2} x + c d e^{2}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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