Integrand size = 26, antiderivative size = 318 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3} \]
[Out]
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5810, 5788, 5789, 4265, 2611, 2320, 6724, 5798, 209, 205} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{4 c^3 d^3}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c^3 d^3}+\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c^3 d^3}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (c^2 x^2+1\right )}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {c^2 x^2+1}}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}+\frac {b^2 x}{12 c^2 d^3 \left (c^2 x^2+1\right )} \]
[In]
[Out]
Rule 205
Rule 209
Rule 2320
Rule 2611
Rule 4265
Rule 5788
Rule 5789
Rule 5798
Rule 5810
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx}{4 c^2 d} \\ & = -\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b^2 \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{6 c^2 d^3}-\frac {b \int \frac {x (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx}{8 c^2 d^2} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{8 c^3 d^3}+\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{12 c^2 d^3}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{4 c^2 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {(i b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3}+\frac {(i b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{4 c^3 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3} \\ & = \frac {b^2 x}{12 c^2 d^3 \left (1+c^2 x^2\right )}-\frac {b (a+b \text {arcsinh}(c x))}{6 c^3 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b (a+b \text {arcsinh}(c x))}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {x (a+b \text {arcsinh}(c x))^2}{8 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {(a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {b^2 \arctan (c x)}{6 c^3 d^3}-\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}+\frac {i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3}-\frac {i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 c^3 d^3} \\ \end{align*}
Time = 2.03 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.73 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {-\frac {6 a^2 c x}{\left (1+c^2 x^2\right )^2}+\frac {3 a^2 c x}{1+c^2 x^2}+\frac {a b \left ((2+i c x) \sqrt {1+c^2 x^2}+3 i \text {arcsinh}(c x)\right )}{(-i+c x)^2}+\frac {3 a b \left (-i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{-i+c x}+\frac {3 a b \left (i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{i+c x}-\frac {i a b \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(i+c x)^2}+3 a^2 \arctan (c x)+\frac {3}{2} i a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {3}{2} i a b \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+b^2 \left (\frac {2 c x}{1+c^2 x^2}-\frac {4 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {6 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {6 c x \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {3 c x \text {arcsinh}(c x)^2}{1+c^2 x^2}-8 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-3 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+3 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+6 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-6 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+6 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )}{24 c^3 d^3} \]
[In]
[Out]
\[\int \frac {x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (c^{2} d \,x^{2}+d \right )^{3}}d x\]
[In]
[Out]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2} x^{2}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
[In]
[Out]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
[In]
[Out]