Integrand size = 27, antiderivative size = 429 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {15 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^3}+\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^3}-\frac {15 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {15 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3} \]
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Time = 0.51 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 205, 4793, 4803, 4268, 2317, 2438} \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {15 i c \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{4 d^3}-\frac {4 b c \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {15 i b c \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{4 d^3}-\frac {15 i b c \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{4 d^3}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^3}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^3}-\frac {15 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {15 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}+\frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )} \]
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Rule 205
Rule 212
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4266
Rule 4268
Rule 4747
Rule 4749
Rule 4767
Rule 4789
Rule 4793
Rule 4803
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\left (5 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{d^3} \\ & = \frac {2 b c (a+b \arcsin (c x))}{3 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^3}-\frac {\left (2 b^2 c^2\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^3}-\frac {\left (5 b c^3\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac {\left (15 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d} \\ & = -\frac {b^2 c^2 x}{3 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c (a+b \arcsin (c x))}{d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d^3}-\frac {\left (b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}+\frac {\left (5 b^2 c^2\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}-\frac {\left (2 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^3}-\frac {\left (15 b c^3\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac {\left (15 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{8 d^2} \\ & = \frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {7 b^2 c \text {arctanh}(c x)}{3 d^3}+\frac {(15 c) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{8 d^3}+\frac {(2 b c) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{d^3}+\frac {\left (5 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{12 d^3}+\frac {\left (15 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{4 d^3} \\ & = \frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {15 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}-\frac {(15 b c) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3}+\frac {(15 b c) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3}-\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^3} \\ & = \frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {15 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}+\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^3}-\frac {\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^3}-\frac {\left (15 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3}+\frac {\left (15 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{4 d^3} \\ & = \frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {15 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^3}+\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^3}-\frac {\left (15 b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {\left (15 b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{4 d^3} \\ & = \frac {b^2 c^2 x}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \arcsin (c x))}{4 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x (a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {15 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^3}+\frac {11 b^2 c \text {arctanh}(c x)}{6 d^3}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^3}+\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {15 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^3}-\frac {15 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {15 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1400\) vs. \(2(429)=858\).
Time = 11.77 (sec) , antiderivative size = 1400, normalized size of antiderivative = 3.26 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {a^2}{d^3 x}+\frac {a^2 c^2 x}{4 d^3 \left (-1+c^2 x^2\right )^2}-\frac {7 a^2 c^2 x}{8 d^3 \left (-1+c^2 x^2\right )}-\frac {15 a^2 c \log (1-c x)}{16 d^3}+\frac {15 a^2 c \log (1+c x)}{16 d^3}-\frac {2 a b c \left (-\frac {7 \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{16 (-1+c x)}+\frac {\arcsin (c x)}{c x}+\frac {7 \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{16 (1+c x)}-\frac {(-2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)}{48 (-1+c x)^2}+\frac {(2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)}{48 (1+c x)^2}+\text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {15}{16} \left (\frac {3}{2} i \pi \arcsin (c x)-\frac {1}{2} i \arcsin (c x)^2+2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )-\pi \log \left (1+i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )-\frac {15}{16} \left (\frac {1}{2} i \pi \arcsin (c x)-\frac {1}{2} i \arcsin (c x)^2+2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )+\pi \log \left (1-i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )\right )}{d^3}-\frac {b^2 c \left (-2 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+\frac {1}{24} \left (44 \arcsin (c x)+15 \arcsin (c x)^3-45 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )-45 \pi \arcsin (c x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \arcsin (c x)} \left (1-i e^{i \arcsin (c x)}\right )\right )+45 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )+45 \arcsin (c x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \arcsin (c x)} \left (-i+e^{i \arcsin (c x)}\right )\right )-45 \pi \arcsin (c x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \arcsin (c x)} \left (-i+e^{i \arcsin (c x)}\right )\right )-45 \arcsin (c x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \arcsin (c x)} \left ((1+i)+(1-i) e^{i \arcsin (c x)}\right )\right )+45 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+44 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-45 \arcsin (c x)^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-44 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+45 \arcsin (c x)^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+45 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-90 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+90 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+90 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-90 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )-\frac {4+88 c x \arcsin (c x)-54 \arcsin (c x)^2+30 c x \arcsin (c x)^3-240 \arcsin (c x)^2 \cos (2 \arcsin (c x))-4 \cos (4 \arcsin (c x))-90 \arcsin (c x)^2 \cos (4 \arcsin (c x))+96 c x \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-96 c x \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-768 i c x \left (1-c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-200 \arcsin (c x) \sin (2 \arcsin (c x))+132 \arcsin (c x) \sin (3 \arcsin (c x))+45 \arcsin (c x)^3 \sin (3 \arcsin (c x))+144 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))-144 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))-84 \arcsin (c x) \sin (4 \arcsin (c x))+44 \arcsin (c x) \sin (5 \arcsin (c x))+15 \arcsin (c x)^3 \sin (5 \arcsin (c x))+48 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (5 \arcsin (c x))-48 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (5 \arcsin (c x))}{384 c x \left (1-c^2 x^2\right )^2}\right )}{d^3} \]
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Time = 0.57 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.70
method | result | size |
derivativedivides | \(c \left (-\frac {a^{2} \left (\frac {1}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {7}{16 \left (c x -1\right )}+\frac {15 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {7}{16 \left (c x +1\right )}-\frac {15 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {45 \arcsin \left (c x \right )^{2} x^{4} c^{4}-42 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-75 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}+46 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +24 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {15 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {11 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {45 c^{4} x^{4} \arcsin \left (c x \right )-21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-75 c^{2} x^{2} \arcsin \left (c x \right )+23 c x \sqrt {-c^{2} x^{2}+1}+24 \arcsin \left (c x \right )}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {15 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(730\) |
default | \(c \left (-\frac {a^{2} \left (\frac {1}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {7}{16 \left (c x -1\right )}+\frac {15 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {7}{16 \left (c x +1\right )}-\frac {15 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {45 \arcsin \left (c x \right )^{2} x^{4} c^{4}-42 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-75 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}+46 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +24 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {15 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {11 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b \left (\frac {45 c^{4} x^{4} \arcsin \left (c x \right )-21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-75 c^{2} x^{2} \arcsin \left (c x \right )+23 c x \sqrt {-c^{2} x^{2}+1}+24 \arcsin \left (c x \right )}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {15 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(730\) |
parts | \(-\frac {a^{2} \left (\frac {1}{x}-\frac {c}{16 \left (c x -1\right )^{2}}+\frac {7 c}{16 \left (c x -1\right )}+\frac {15 c \ln \left (c x -1\right )}{16}+\frac {c}{16 \left (c x +1\right )^{2}}+\frac {7 c}{16 \left (c x +1\right )}-\frac {15 c \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} c \left (\frac {45 \arcsin \left (c x \right )^{2} x^{4} c^{4}-42 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-75 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}+46 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +24 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {15 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {15 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {11 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}-\frac {2 a b c \left (\frac {45 c^{4} x^{4} \arcsin \left (c x \right )-21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-75 c^{2} x^{2} \arcsin \left (c x \right )+23 c x \sqrt {-c^{2} x^{2}+1}+24 \arcsin \left (c x \right )}{24 c x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {15 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {15 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {15 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\) | \(732\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx}{d^{3}} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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