Integrand size = 29, antiderivative size = 634 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
[Out]
Time = 0.56 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {4789, 4793, 4803, 4268, 2611, 2320, 6724, 4749, 4266, 2317, 2438, 272, 65, 214} \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 i b c^2 \sqrt {1-c^2 x^2} \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}} \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4266
Rule 4268
Rule 4749
Rule 4789
Rule 4793
Rule 4803
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {1}{2} \left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\arcsin (c x)\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {\left (b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (3 b^2 c^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \arcsin (c x))^2}{2 d \sqrt {d-c^2 d x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {3 b^2 c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 7.96 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a^2}{2 d^2 x^2}-\frac {a^2 c^2}{d^2 \left (-1+c^2 x^2\right )}\right )+\frac {3 a^2 c^2 \log (x)}{2 d^{3/2}}-\frac {3 a^2 c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{3/2}}+\frac {a b c \left (6 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-6 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-\frac {-2 \arcsin (c x)+6 \arcsin (c x) \cos (2 \arcsin (c x))+3 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )-3 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )+2 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+\sqrt {1-c^2 x^2} \left (-3 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+3 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+2 \sin (2 \arcsin (c x))}{c x}\right )}{4 d x \sqrt {d \left (1-c^2 x^2\right )}}+\frac {b^2 c^2 \sqrt {1-c^2 x^2} \left (8 \arcsin (c x)^2-4 \arcsin (c x) \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x)^2 \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )+8 \log \left (\tan \left (\frac {1}{2} \arcsin (c x)\right )\right )-16 \left (\arcsin (c x) \left (\log \left (1-i e^{i \arcsin (c x)}\right )-\log \left (1+i e^{i \arcsin (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )\right )+12 \left (\arcsin (c x)^2 \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )+2 i \arcsin (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )+\arcsin (c x)^2 \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )+\frac {8 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )}-\frac {8 \arcsin (c x)^2 \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}-4 \arcsin (c x) \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{8 d \sqrt {d \left (1-c^2 x^2\right )}} \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.37
method | result | size |
default | \(a^{2} \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )-\frac {i a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\) | \(869\) |
parts | \(a^{2} \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+4 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-4 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )-\frac {i a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}-3 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+i x^{3} c^{3}-3 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-4 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-3 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )-i c x \right )}{d^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}\) | \(869\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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