Integrand size = 27, antiderivative size = 439 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \]
[Out]
Time = 0.64 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 32, 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, 4793, 4803, 4268, 2317, 2438, 331} \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {5 i c^3 \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^2}-\frac {26 b c^3 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}-\frac {b^2 c^2}{3 d^2 x} \]
[In]
[Out]
Rule 212
Rule 331
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}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac {1}{3} \left (5 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x^3 \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{3 d^2}+\frac {\left (b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (10 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}+\frac {13 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d^2}+\frac {\left (10 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^2}+\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^2}-\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2}-\frac {\left (10 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^2}-\frac {\left (5 b c^5\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (5 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {4 b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{2 d^2}+\frac {\left (b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{d^2}+\frac {\left (10 b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{3 d^2}+\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}-\frac {\left (5 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (5 b c^3\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}-\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}-\frac {\left (10 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^2}+\frac {\left (10 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (10 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {\left (10 i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {\left (5 i b^2 c^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (5 i b^2 c^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2} \\ & = -\frac {b^2 c^2}{3 d^2 x}-\frac {2 b c^3 (a+b \arcsin (c x))}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {b c (a+b \arcsin (c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac {5 c^2 (a+b \arcsin (c x))^2}{3 d^2 x \left (1-c^2 x^2\right )}+\frac {5 c^4 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {5 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {26 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {b^2 c^3 \text {arctanh}(c x)}{d^2}+\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^2}+\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {5 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {13 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^2}-\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {5 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1390\) vs. \(2(439)=878\).
Time = 12.42 (sec) , antiderivative size = 1390, normalized size of antiderivative = 3.17 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {a^2}{3 d^2 x^3}-\frac {2 a^2 c^2}{d^2 x}-\frac {a^2 c^4 x}{2 d^2 \left (-1+c^2 x^2\right )}-\frac {5 a^2 c^3 \log (1-c x)}{4 d^2}+\frac {5 a^2 c^3 \log (1+c x)}{4 d^2}+\frac {2 a b \left (\frac {c^3 \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{4 (-1+c x)}-\frac {c^4 \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{4 \left (c+c^2 x\right )}+2 c^2 \left (-\frac {\arcsin (c x)}{x}-c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )-\frac {c x \sqrt {1-c^2 x^2}+2 \arcsin (c x)+c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 x^3}-\frac {5}{4} c^4 \left (\frac {3 i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}-\frac {\pi \log \left (1+i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}+\frac {\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c}\right )+\frac {5}{4} c^4 \left (\frac {i \pi \arcsin (c x)}{2 c}-\frac {i \arcsin (c x)^2}{2 c}+\frac {2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )}{c}+\frac {\pi \log \left (1-i e^{i \arcsin (c x)}\right )}{c}+\frac {2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )}{c}-\frac {2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )}{c}-\frac {\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c}\right )\right )}{d^2}+\frac {b^2 c^3 \left (-24 \arcsin (c x)-\frac {6 \arcsin (c x)^2}{-1+c x}-4 \cot \left (\frac {1}{2} \arcsin (c x)\right )-26 \arcsin (c x)^2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-2 \arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-\frac {1}{2} c x \arcsin (c x)^2 \csc ^4\left (\frac {1}{2} \arcsin (c x)\right )+104 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+60 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )+60 \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 )-60 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )-60 \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 )+60 \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 )-104 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+60 \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 )-60 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-24 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+60 \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 )+24 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-60 \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 )-60 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+104 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+120 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-120 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-104 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-120 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )+120 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-\frac {24 \arcsin (c x) \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 ^4\left (\frac {1}{2} \arcsin (c x)\right )}{c^3 x^3}-\frac {6 \arcsin (c x)^2}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}+\frac {24 \arcsin (c x) \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 \tan \left (\frac {1}{2} \arcsin (c x)\right )-26 \arcsin (c x)^2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{24 d^2} \]
[In]
[Out]
Time = 0.55 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.61
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{2}}+\frac {2 a b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\) | \(707\) |
default | \(c^{3} \left (\frac {a^{2} \left (-\frac {1}{3 c^{3} x^{3}}-\frac {2}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {5 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {5 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{2}}+\frac {2 a b \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\) | \(707\) |
parts | \(\frac {a^{2} \left (-\frac {1}{3 x^{3}}-\frac {2 c^{2}}{x}-\frac {c^{3}}{4 \left (c x -1\right )}-\frac {5 c^{3} \ln \left (c x -1\right )}{4}-\frac {c^{3}}{4 \left (c x +1\right )}+\frac {5 c^{3} \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} c^{3} \left (-\frac {15 \arcsin \left (c x \right )^{2} x^{4} c^{4}-4 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-10 \arcsin \left (c x \right )^{2} x^{2} c^{2}+2 c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -2 \arcsin \left (c x \right )^{2}-2 c^{2} x^{2}}{6 c^{3} x^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-5 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {5 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-5 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+5 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {13 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {13 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {13 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{2}}+\frac {2 a b \,c^{3} \left (-\frac {15 c^{4} x^{4} \arcsin \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-10 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right ) c^{3} x^{3}}+\frac {13 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}-\frac {13 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {5 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {5 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\) | \(718\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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