Integrand size = 30, antiderivative size = 444 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=-\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \]
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Time = 0.47 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5845, 5843, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=-\frac {g \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right ) (f+g x)}+\frac {c^2 f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {c^2 f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c^2 f \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c \sqrt {c^2 x^2+1} \log (f+g x)}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )} \]
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 5843
Rule 5845
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}} \\ & = \frac {\left (c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{(c f+g \sinh (x))^2} \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {d+c^2 d x^2}} \\ & = -\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {\left (c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c f+g \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {\left (b c g \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x)}{c f+g \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}} \\ & = -\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {\left (2 c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c e^x f-g+e^{2 x} g} \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}} \\ & = -\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {\left (2 c^2 f g \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g-2 \sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {\left (2 c^2 f g \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g+2 \sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \\ & = -\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}-\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \\ & = -\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}-\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \\ & = -\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\frac {c \sqrt {1+c^2 x^2} \left (-\frac {g \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c f+c g x}+b \log (f+g x)+\frac {c f \left ((a+b \text {arcsinh}(c x)) \left (\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1769\) vs. \(2(442)=884\).
Time = 0.85 (sec) , antiderivative size = 1770, normalized size of antiderivative = 3.99
method | result | size |
default | \(\text {Expression too large to display}\) | \(1770\) |
parts | \(\text {Expression too large to display}\) | \(1770\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (f + g x\right )^{2}}\, dx \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \]
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