Integrand size = 24, antiderivative size = 186 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3} \]
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Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5810, 5789, 4265, 2317, 2438, 267, 272, 45} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c^5 d^3}-\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (c^2 x^2+1\right )}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3}-\frac {5 b}{8 c^5 d^3 \sqrt {c^2 x^2+1}}+\frac {b}{12 c^5 d^3 \left (c^2 x^2+1\right )^{3/2}} \]
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Rule 45
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4265
Rule 5789
Rule 5810
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {x^3}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 c d^3}+\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx}{4 c^2 d} \\ & = -\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {(3 b) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 c^3 d^3}+\frac {b \text {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{8 c d^3}+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{8 c^4 d^2} \\ & = -\frac {3 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{8 c^5 d^3}+\frac {b \text {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^{5/2}}+\frac {1}{c^2 \left (1+c^2 x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{8 c d^3} \\ & = \frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}-\frac {(3 i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 c^5 d^3}+\frac {(3 i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 c^5 d^3} \\ & = \frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3} \\ & = \frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x (a+b \text {arcsinh}(c x))}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c^5 d^3} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.83 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {9 a c x+15 a c^3 x^3+13 b \sqrt {1+c^2 x^2}+15 b c^2 x^2 \sqrt {1+c^2 x^2}+9 b c x \text {arcsinh}(c x)+15 b c^3 x^3 \text {arcsinh}(c x)-9 a \arctan (c x)-18 a c^2 x^2 \arctan (c x)-9 a c^4 x^4 \arctan (c x)-9 i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-18 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-9 i b c^4 x^4 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+9 i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+18 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+9 i b c^4 x^4 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+9 i b \left (1+c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-9 i b \left (1+c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{24 c^5 d^3 \left (1+c^2 x^2\right )^2} \]
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Time = 0.17 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {-\frac {5}{8} c^{3} x^{3}-\frac {3}{8} c x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (-\frac {5 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {13}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {5 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{d^{3}}}{c^{5}}\) | \(247\) |
default | \(\frac {\frac {a \left (\frac {-\frac {5}{8} c^{3} x^{3}-\frac {3}{8} c x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (-\frac {5 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {13}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {5 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{d^{3}}}{c^{5}}\) | \(247\) |
parts | \(\frac {a \left (\frac {-\frac {5 x^{3}}{8 c^{2}}-\frac {3 x}{8 c^{4}}}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (c x \right )}{8 c^{5}}\right )}{d^{3}}+\frac {b \left (-\frac {5 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {13}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {5 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{d^{3} c^{5}}\) | \(251\) |
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a x^{4}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
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