Integrand size = 31, antiderivative size = 410 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}} \]
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Time = 0.39 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {4861, 4847, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 4749, 4266} \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 i b g \sqrt {1-c^2 x^2} \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c d \sqrt {d-c^2 d x^2}}+\frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4745
Rule 4749
Rule 4765
Rule 4767
Rule 4847
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (\frac {f (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {g x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (f \sqrt {1-c^2 x^2}\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c f \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b f \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tan (x) \, dx,x,\arcsin (c x))}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b g \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g \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 )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g \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 )}{c^2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 g \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 )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 g \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 )}{c^2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 f \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {g (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {f x (a+b \arcsin (c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {i f \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d \sqrt {d-c^2 d x^2}}+\frac {4 i b g \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b f \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 g \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.58 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left ((c f-g) \left (-(a+b \arcsin (c x))^2 \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+i \left ((a+b \arcsin (c x)) \left (a+b \arcsin (c x)-4 i b \log \left (1+i e^{-i \arcsin (c x)}\right )\right )+4 b^2 \operatorname {PolyLog}\left (2,-i e^{-i \arcsin (c x)}\right )\right )\right )-(c f+g) \left (i \left ((a+b \arcsin (c x)) \left (a+b \arcsin (c x)+4 i b \log \left (1+i e^{i \arcsin (c x)}\right )\right )+4 b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )-(a+b \arcsin (c x))^2 \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{2 c^2 d \sqrt {d-c^2 d x^2}} \]
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Time = 1.20 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.49
method | result | size |
default | \(a^{2} \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} \left (i \sqrt {-c^{2} x^{2}+1}\, c f +c^{2} f x +g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right )^{2} c f +i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c f -2 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c f +2 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g -2 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g -2 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g +2 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \arcsin \left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}\, c f +c^{2} f x +g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f -g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f +g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(612\) |
parts | \(a^{2} \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} \left (i \sqrt {-c^{2} x^{2}+1}\, c f +c^{2} f x +g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right )^{2} c f +i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c f -2 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) c f +2 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g -2 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g -2 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g +2 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \arcsin \left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}\, c f +c^{2} f x +g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f -g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f +g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(612\) |
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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