Optimal. Leaf size=478 \[ \frac {b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac {a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac {a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {a b^3 \log (x)}{\left (a^2+1\right )^2}+\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^2}{3 \left (a^2+1\right ) x}+\frac {a b^2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right )^2 x}-\frac {b \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 \left (a^2+1\right ) x^2}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3} \]
[Out]
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Rubi [A] time = 1.57, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 16, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {5865, 5801, 5831, 3325, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31, 6741, 12, 6742, 32} \[ \frac {b^3 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac {a^2 b^3 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^3 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac {a^2 b^3 \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^2}{3 \left (a^2+1\right ) x}-\frac {a b^3 \log (x)}{\left (a^2+1\right )^2}+\frac {a b^2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right )^2 x}+\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 \left (a^2+1\right ) x^2}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 32
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3322
Rule 3324
Rule 3325
Rule 5801
Rule 5831
Rule 5865
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)^2}{x^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)^2}{\left (-\frac {a}{b}+\frac {x}{b}\right )^4} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sqrt {1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {x}{\left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^3} \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac {b \operatorname {Subst}\left (\int \frac {\cosh (x)}{\left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac {b \operatorname {Subst}\left (\int \frac {x \sinh (x)}{\left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x}{\left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {2 a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {b \operatorname {Subst}\left (\int \frac {b^2 x \sinh (x)}{(a-\sinh (x))^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}+\frac {\left (2 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {x}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+x\right )^2} \, dx,x,\frac {a}{b}+x\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {2 a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac {\left (4 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x}{-\frac {1}{b}-\frac {2 a e^x}{b}+\frac {e^{2 x}}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}-\frac {\left (2 a b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+x} \, dx,x,\frac {a}{b}+x\right )}{3 \left (1+a^2\right )^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {x \sinh (x)}{(a-\sinh (x))^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {2 a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}+\frac {\left (4 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x}{-\frac {2 a}{b}-\frac {2 \sqrt {1+a^2}}{b}+\frac {2 e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {\left (4 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {e^x x}{-\frac {2 a}{b}+\frac {2 \sqrt {1+a^2}}{b}+\frac {2 e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {b^3 \operatorname {Subst}\left (\int \left (\frac {a x}{(a-\sinh (x))^2}-\frac {x}{a-\sinh (x)}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {2 a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}-\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {b^3 \operatorname {Subst}\left (\int \frac {x}{a-\sinh (x)} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac {\left (a b^3\right ) \operatorname {Subst}\left (\int \frac {x}{(a-\sinh (x))^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}-\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {\left (a b^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{a-\sinh (x)} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}-\frac {\left (a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {x}{a-\sinh (x)} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x x}{1+2 a e^x-e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}-\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {\left (a b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,-a-b x\right )}{3 \left (1+a^2\right )^2}-\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x x}{1+2 a e^x-e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}-\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x x}{2 a-2 \sqrt {1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x x}{2 a+2 \sqrt {1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac {a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x x}{2 a-2 \sqrt {1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x x}{2 a+2 \sqrt {1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {b^3 \operatorname {Subst}\left (\int \log \left (1-\frac {2 e^x}{2 a-2 \sqrt {1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {b^3 \operatorname {Subst}\left (\int \log \left (1-\frac {2 e^x}{2 a+2 \sqrt {1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac {a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {\left (a^2 b^3\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 e^x}{2 a-2 \sqrt {1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {\left (a^2 b^3\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 e^x}{2 a+2 \sqrt {1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 x}{2 a-2 \sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 x}{2 a+2 \sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{3/2}}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac {a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac {b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {2 a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {\left (a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 x}{2 a-2 \sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac {\left (a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 x}{2 a+2 \sqrt {1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}\\ &=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac {a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac {a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 10.52, size = 1830, normalized size = 3.83 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.89, size = 730, normalized size = 1.53 \[ -\frac {b^{3} \arcsinh \left (b x +a \right ) a}{\left (a^{2}+1\right )^{2}}+\frac {a \,b^{2} \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{\left (a^{2}+1\right )^{2} x}-\frac {\arcsinh \left (b x +a \right )^{2} a^{4}}{3 \left (a^{2}+1\right )^{2} x^{3}}-\frac {b \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{2}}{3 \left (a^{2}+1\right )^{2} x^{2}}-\frac {b^{2} a^{2}}{3 \left (a^{2}+1\right )^{2} x}-\frac {2 \arcsinh \left (b x +a \right )^{2} a^{2}}{3 \left (a^{2}+1\right )^{2} x^{3}}-\frac {b \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{3 \left (a^{2}+1\right )^{2} x^{2}}-\frac {b^{2}}{3 \left (a^{2}+1\right )^{2} x}-\frac {\arcsinh \left (b x +a \right )^{2}}{3 \left (a^{2}+1\right )^{2} x^{3}}+\frac {2 b^{3} a \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{\left (a^{2}+1\right )^{2}}-\frac {b^{3} a \ln \left (\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}-2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-1\right )}{\left (a^{2}+1\right )^{2}}-\frac {b^{3} \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {b^{3} \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {b^{3} \dilog \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {b^{3} \dilog \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 b^{3} a^{2} \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 b^{3} a^{2} \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 b^{3} a^{2} \dilog \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 b^{3} a^{2} \dilog \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{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 \[ -\frac {\log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{3 \, x^{3}} + \int \frac {2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b^{2} x + a b\right )} + b\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{3 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} + {\left (3 \, a^{2} b + b\right )} x^{4} + {\left (a^{3} + a\right )} x^{3} + {\left (b^{2} x^{5} + 2 \, a b x^{4} + {\left (a^{2} + 1\right )} x^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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