Integrand size = 24, antiderivative size = 171 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {b}{2 c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {1+c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}+\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2} \]
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Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5810, 5812, 5789, 4265, 2317, 2438, 267, 272, 45} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2}-\frac {b \sqrt {c^2 x^2+1}}{c^5 d^2}+\frac {b}{2 c^5 d^2 \sqrt {c^2 x^2+1}} \]
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Rule 45
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4265
Rule 5789
Rule 5810
Rule 5812
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \int \frac {x^3}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}+\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx}{2 c^2 d} \\ & = \frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {(3 b) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{2 c^3 d^2}+\frac {b \text {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{2 c^4 d} \\ & = -\frac {3 b \sqrt {1+c^2 x^2}}{2 c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{2 c^5 d^2}+\frac {b \text {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^{3/2}}+\frac {1}{c^2 \sqrt {1+c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2} \\ & = \frac {b}{2 c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {1+c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}+\frac {(3 i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 c^5 d^2}-\frac {(3 i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 c^5 d^2} \\ & = \frac {b}{2 c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {1+c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2} \\ & = \frac {b}{2 c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {b \sqrt {1+c^2 x^2}}{c^5 d^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d^2}+\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2}-\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.57 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {3 a c x+2 a c^3 x^3-b \sqrt {1+c^2 x^2}-2 b c^2 x^2 \sqrt {1+c^2 x^2}+3 b c x \text {arcsinh}(c x)+2 b c^3 x^3 \text {arcsinh}(c x)-3 a \arctan (c x)-3 a c^2 x^2 \arctan (c x)-3 i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-3 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+3 i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+3 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+3 i b \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-3 i b \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^5 d^2 \left (1+c^2 x^2\right )} \]
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Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.31
method | result | size |
derivativedivides | \(\frac {\frac {a \left (c x +\frac {c x}{2 c^{2} x^{2}+2}-\frac {3 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c x +\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}-\frac {c^{2} x^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}}{c^{5}}\) | \(224\) |
default | \(\frac {\frac {a \left (c x +\frac {c x}{2 c^{2} x^{2}+2}-\frac {3 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c x +\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}-\frac {c^{2} x^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}}{c^{5}}\) | \(224\) |
parts | \(\frac {a \left (\frac {x}{c^{4}}-\frac {-\frac {x}{2 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{2 c}}{c^{4}}\right )}{d^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) c x +\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}-\frac {c^{2} x^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2} c^{5}}\) | \(233\) |
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{4}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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