Integrand size = 24, antiderivative size = 168 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5809, 5788, 5789, 4265, 2317, 2438, 267, 272, 53, 65, 214} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {3 c \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{d^2}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (c^2 x^2+1\right )}-\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (c^2 x^2+1\right )}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2}-\frac {b c \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{d^2}-\frac {b c}{2 d^2 \sqrt {c^2 x^2+1}} \]
[In]
[Out]
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)}{d^2 x \left (1+c^2 x^2\right )}-\left (3 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2} \\ & = -\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (3 b c^3\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}-\frac {\left (3 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(3 c) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{2 d^2}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 d^2} \\ & = -\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {b \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{c d^2}+\frac {(3 i b c) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 d^2}-\frac {(3 i b c) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 d^2} \\ & = -\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d^2}+\frac {(3 i b c) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 d^2}-\frac {(3 i b c) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 d^2} \\ & = -\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}-\frac {b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.38 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.51 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {\frac {3 a}{x}-\frac {a}{x+c^2 x^3}+\frac {3 b \text {arcsinh}(c x)}{x}-\frac {b \text {arcsinh}(c x)}{x+c^2 x^3}+3 a c \arctan (c x)+3 b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+c^2 x^2\right )}{\sqrt {1+c^2 x^2}}+3 b \sqrt {-c^2} \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-3 b \sqrt {-c^2} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-3 b \sqrt {-c^2} \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+3 b \sqrt {-c^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{2 d^2} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(c \left (\frac {a \left (-\frac {1}{c x}-\frac {c x}{2 \left (c^{2} x^{2}+1\right )}-\frac {3 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}-\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d^{2}}\right )\) | \(227\) |
default | \(c \left (\frac {a \left (-\frac {1}{c x}-\frac {c x}{2 \left (c^{2} x^{2}+1\right )}-\frac {3 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}-\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d^{2}}\right )\) | \(227\) |
parts | \(\frac {a \left (-c^{2} \left (\frac {x}{2 c^{2} x^{2}+2}+\frac {3 \arctan \left (c x \right )}{2 c}\right )-\frac {1}{x}\right )}{d^{2}}+\frac {b c \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}-\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {c^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d^{2}}\) | \(231\) |
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
[In]
[Out]