Integrand size = 24, antiderivative size = 295 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {19 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^3}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 d^3} \]
[Out]
Time = 0.26 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5809, 5788, 5789, 4265, 2317, 2438, 267, 272, 53, 65, 214, 44} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\frac {35 c^3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 d^3}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (c^2 x^2+1\right )^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (c^2 x^2+1\right )}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (c^2 x^2+1\right )^2}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 d^3}+\frac {19 b c^3 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{6 d^3}-\frac {b c}{6 d^3 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {c^2 x^2+1}}-\frac {b c^3}{12 d^3 \left (c^2 x^2+1\right )^{3/2}} \]
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 214
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4265
Rule 5788
Rule 5789
Rule 5809
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}-\frac {1}{3} \left (7 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^3 \left (1+c^2 x^2\right )^{5/2}} \, dx}{3 d^3} \\ & = -\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {1}{3} \left (35 c^4\right ) \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx+\frac {(b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{6 d^3}-\frac {\left (7 b c^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2}} \, dx}{3 d^3} \\ & = -\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{12 d^3}-\frac {\left (7 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{6 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{12 d^3}+\frac {\left (35 c^4\right ) \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d} \\ & = -\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{12 d^3}-\frac {\left (7 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac {\left (35 c^4\right ) \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{8 d^2} \\ & = -\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {\left (35 c^3\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{8 d^3}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{12 d^3}-\frac {\left (7 b c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 d^3} \\ & = -\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}-\frac {(5 b c) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{6 d^3}-\frac {(7 b c) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 d^3}-\frac {\left (35 i b c^3\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 d^3}+\frac {\left (35 i b c^3\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 d^3} \\ & = -\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {19 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^3}-\frac {\left (35 i b c^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 d^3}+\frac {\left (35 i b c^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 d^3} \\ & = -\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {19 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^3}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.06 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.27 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\frac {3 (a+b \text {arcsinh}(c x))}{x^3 \left (1+c^2 x^2\right )^2}+\frac {21 (a+b \text {arcsinh}(c x))}{2 \left (x^3+c^2 x^5\right )}+\frac {b c^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},2,-\frac {1}{2},1+c^2 x^2\right )}{\left (1+c^2 x^2\right )^{3/2}}+\frac {21 b c^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1+c^2 x^2\right )}{2 \sqrt {1+c^2 x^2}}+\frac {35 \left (-2 a+6 a c^2 x^2-b c x \sqrt {1+c^2 x^2}-2 b \text {arcsinh}(c x)+6 b c^2 x^2 \text {arcsinh}(c x)+6 a c^3 x^3 \arctan (c x)+7 b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )-6 b \left (-c^2\right )^{3/2} x^3 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+6 b \left (-c^2\right )^{3/2} x^3 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+6 b \left (-c^2\right )^{3/2} x^3 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-6 b \left (-c^2\right )^{3/2} x^3 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )\right )}{4 x^3}}{12 d^3} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {\frac {11}{8} c^{3} x^{3}+\frac {13}{8} c x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {11 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {13 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {103}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{6 c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {19}{6 \sqrt {c^{2} x^{2}+1}}+\frac {19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {35 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {35 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}\right )\) | \(333\) |
default | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {\frac {11}{8} c^{3} x^{3}+\frac {13}{8} c x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {11 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {13 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {103}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{6 c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {19}{6 \sqrt {c^{2} x^{2}+1}}+\frac {19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {35 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {35 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}\right )\) | \(333\) |
parts | \(\frac {a \left (c^{4} \left (\frac {\frac {11}{8} x^{3} c^{2}+\frac {13}{8} x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (c x \right )}{8 c}\right )-\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}\right )}{d^{3}}+\frac {b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {11 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {13 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {103}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{6 c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {19}{6 \sqrt {c^{2} x^{2}+1}}+\frac {19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {35 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {35 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}\) | \(336\) |
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx}{d^{3}} \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
[In]
[Out]