Integrand size = 27, antiderivative size = 572 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3} \]
[Out]
Time = 0.87 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 205, 4793, 4803, 4268, 2317, 2438, 296, 331} \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {35 i c^3 \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{4 d^3}-\frac {38 b c^3 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{4 d^3}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{4 d^3}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^2}{2 d^3 x}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )} \]
[In]
[Out]
Rule 205
Rule 212
Rule 296
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^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (7 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x^3 \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3} \\ & = -\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (35 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )^2} \, dx}{3 d^3}+\frac {\left (5 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{3 d^3} \\ & = \frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}+\frac {19 b c^3 (a+b \arcsin (c x))}{9 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx}{2 d^3}+\frac {\left (5 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{9 d^3}-\frac {\left (14 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{9 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{6 d^3}+\frac {\left (35 c^4\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}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {19 b^2 c^4 x}{18 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}+\frac {19 b c^3 (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {\left (5 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^3}+\frac {\left (14 b c^3\right ) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{3 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{18 d^3}+\frac {\left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{2 d^3}-\frac {\left (7 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{9 d^3}-\frac {\left (5 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{\left (1-c^2 x^2\right )^2} \, dx}{18 d^3}-\frac {\left (14 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac {\left (35 c^4\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{8 d^2} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {62 b^2 c^3 \text {arctanh}(c x)}{9 d^3}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{8 d^3}+\frac {\left (5 b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{3 d^3}+\frac {\left (14 b c^3\right ) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{3 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{36 d^3}+\frac {\left (35 b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx}{4 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}-\frac {\left (35 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 )}{4 d^3}+\frac {\left (35 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 )}{4 d^3}-\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3}+\frac {\left (5 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3}-\frac {\left (14 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3}+\frac {\left (14 b^2 c^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{3 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {\left (5 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^3}-\frac {\left (5 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^3}+\frac {\left (14 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^3}-\frac {\left (14 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^3}-\frac {\left (35 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 )}{4 d^3}+\frac {\left (35 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 )}{4 d^3} \\ & = -\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {\left (35 b^2 c^3\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 (35 b^2 c^3\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}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1-c^2 x^2\right )}-\frac {b^2 c^4 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b c^3 (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (a+b \arcsin (c x))}{3 d^3 x^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {29 b c^3 (a+b \arcsin (c x))}{12 d^3 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 (a+b \arcsin (c x))^2}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \arcsin (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {38 b c^3 (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {17 b^2 c^3 \text {arctanh}(c x)}{6 d^3}+\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{3 d^3}+\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {35 i b c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 i b^2 c^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{3 d^3}-\frac {35 b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 d^3}+\frac {35 b^2 c^3 \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. \(1657\) vs. \(2(572)=1144\).
Time = 12.71 (sec) , antiderivative size = 1657, normalized size of antiderivative = 2.90 \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {a^2}{3 d^3 x^3}-\frac {3 a^2 c^2}{d^3 x}+\frac {a^2 c^4 x}{4 d^3 \left (-1+c^2 x^2\right )^2}-\frac {11 a^2 c^4 x}{8 d^3 \left (-1+c^2 x^2\right )}-\frac {35 a^2 c^3 \log (1-c x)}{16 d^3}+\frac {35 a^2 c^3 \log (1+c x)}{16 d^3}-\frac {2 a b \left (\frac {c^3 \left ((2-c x) \sqrt {1-c^2 x^2}-3 \arcsin (c x)\right )}{48 (-1+c x)^2}-\frac {11 c^3 \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{16 (-1+c x)}+\frac {11 c^4 \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{16 \left (c+c^2 x\right )}+\frac {c^3 \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{48 (1+c x)^2}-3 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 {35}{16} 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 {35}{16} 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^3}-\frac {b^2 c^3 \left (-\frac {19}{3} i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+\frac {19}{3} i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\frac {1}{24} \left (68 \arcsin (c x)+35 \arcsin (c x)^3-105 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )-105 \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 )+105 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )+105 \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 )-105 \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 )-105 \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 )+105 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+68 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-105 \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 )-68 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+105 \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 )+105 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-210 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+210 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+210 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-210 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )+\frac {24-204 c x \arcsin (c x)+204 \arcsin (c x)^2-105 c x \arcsin (c x)^3+\left (20+658 \arcsin (c x)^2\right ) \cos (2 \arcsin (c x))-4 \left (6+35 \arcsin (c x)^2\right ) \cos (4 \arcsin (c x))-20 \cos (6 \arcsin (c x))-210 \arcsin (c x)^2 \cos (6 \arcsin (c x))-456 c x \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+456 c x \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+540 \arcsin (c x) \sin (2 \arcsin (c x))-204 \arcsin (c x) \sin (3 \arcsin (c x))-105 \arcsin (c x)^3 \sin (3 \arcsin (c x))-456 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))+456 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))+32 \arcsin (c x) \sin (4 \arcsin (c x))+68 \arcsin (c x) \sin (5 \arcsin (c x))+35 \arcsin (c x)^3 \sin (5 \arcsin (c x))+152 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (5 \arcsin (c x))-152 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (5 \arcsin (c x))-116 \arcsin (c x) \sin (6 \arcsin (c x))+68 \arcsin (c x) \sin (7 \arcsin (c x))+35 \arcsin (c x)^3 \sin (7 \arcsin (c x))+152 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (7 \arcsin (c x))-152 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (7 \arcsin (c x))}{1536 c^3 x^3 \left (1-c^2 x^2\right )^2}\right )}{d^3} \]
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Time = 0.68 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a^{2} \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {105 \arcsin \left (c x \right )^{2} c^{6} x^{6}-58 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}-175 \arcsin \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+56 \arcsin \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +8 \arcsin \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 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 {35 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 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 {35 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {19 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {19 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {19 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(821\) |
default | \(c^{3} \left (-\frac {a^{2} \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {105 \arcsin \left (c x \right )^{2} c^{6} x^{6}-58 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}-175 \arcsin \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+56 \arcsin \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +8 \arcsin \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 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 {35 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 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 {35 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {19 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {19 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {19 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(821\) |
parts | \(-\frac {a^{2} \left (\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}-\frac {c^{3}}{16 \left (c x -1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x -1\right )}+\frac {35 c^{3} \ln \left (c x -1\right )}{16}+\frac {c^{3}}{16 \left (c x +1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x +1\right )}-\frac {35 c^{3} \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} c^{3} \left (\frac {105 \arcsin \left (c x \right )^{2} c^{6} x^{6}-58 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}-175 \arcsin \left (c x \right )^{2} x^{4} c^{4}+10 c^{6} x^{6}+54 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+56 \arcsin \left (c x \right )^{2} x^{2} c^{2}-18 c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +8 \arcsin \left (c x \right )^{2}+8 c^{2} x^{2}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}+\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 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 {35 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {35 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 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 {35 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {19 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {19 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {17 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {19 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \,c^{3} \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\) | \(838\) |
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx}{d^{3}} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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