Integrand size = 23, antiderivative size = 190 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d} \]
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Time = 0.18 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 12, 4727, 3384, 3380, 3383} \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))} \]
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4727
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^3 x^3}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^3 \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{2 x}+\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}-\frac {e^3 \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^2 d}+\frac {e^3 \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^2 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {\left (e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^2 d}-\frac {\left (e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^2 d}+\frac {\left (e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^2 d}-\frac {\left (e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b^2 d} \\ & = -\frac {e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}+\frac {e^3 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d}-\frac {e^3 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 b^2 d} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.16 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=-\frac {e^3 \left (\frac {2 b (c+d x)^3 \sqrt {1-(c+d x)^2}}{a+b \arcsin (c+d x)}-4 \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+3 \log (a+b \arcsin (c+d x))-4 \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+3 \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )-\log (a+b \arcsin (c+d x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )+\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )}{2 b^2 d} \]
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Time = 0.30 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {e^{3} \left (4 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -4 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b -4 \arcsin \left (d x +c \right ) \cos \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b +4 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b +4 \,\operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -4 \sin \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a -4 \cos \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a +4 \sin \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a +\sin \left (4 \arcsin \left (d x +c \right )\right ) b -2 \sin \left (2 \arcsin \left (d x +c \right )\right ) b \right )}{8 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\) | \(280\) |
default | \(\frac {e^{3} \left (4 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -4 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b -4 \arcsin \left (d x +c \right ) \cos \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) b +4 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) b +4 \,\operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -4 \sin \left (\frac {4 a}{b}\right ) \operatorname {Si}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a -4 \cos \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (d x +c \right )+\frac {4 a}{b}\right ) a +4 \sin \left (\frac {2 a}{b}\right ) \operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) a +\sin \left (4 \arcsin \left (d x +c \right )\right ) b -2 \sin \left (2 \arcsin \left (d x +c \right )\right ) b \right )}{8 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\) | \(280\) |
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\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (180) = 360\).
Time = 0.41 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.88 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^2} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \]
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