Integrand size = 23, antiderivative size = 258 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 d}-\frac {9 e^4 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 d}+\frac {5 e^4 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d} \]
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Time = 0.24 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 12, 4727, 3384, 3380, 3383} \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^4 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}-\frac {9 e^4 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}+\frac {5 e^4 \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))} \]
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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^4 x^4}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{(a+b \arcsin (x))^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \text {Subst}\left (\int \left (\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^2 d}+\frac {\left (5 e^4\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}-\frac {\left (9 e^4\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}-\frac {\left (e^4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^4 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}-\frac {\left (5 e^4 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}+\frac {\left (e^4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^4 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d}+\frac {\left (5 e^4 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b^2 d} \\ & = -\frac {e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d (a+b \arcsin (c+d x))}+\frac {e^4 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 d}-\frac {9 e^4 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 d}+\frac {5 e^4 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 d}-\frac {e^4 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b^2 d}+\frac {9 e^4 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d}-\frac {5 e^4 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b^2 d} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.10 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\frac {e^4 \left (-\frac {16 b (c+d x)^4 \sqrt {1-(c+d x)^2}}{a+b \arcsin (c+d x)}+16 \left (-3 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )+\operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )+5 \left (10 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right ) \sin \left (\frac {a}{b}\right )-5 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+\operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-10 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )+5 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )-\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )\right )}{16 b^2 d} \]
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Time = 0.75 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(-\frac {e^{4} \left (5 \arcsin \left (d x +c \right ) \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +9 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -5 \arcsin \left (d x +c \right ) \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -9 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b +5 \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +9 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -5 \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +2 \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -2 \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a +\cos \left (5 \arcsin \left (d x +c \right )\right ) b -3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b +2 \sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\) | \(396\) |
default | \(-\frac {e^{4} \left (5 \arcsin \left (d x +c \right ) \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +9 \arcsin \left (d x +c \right ) \operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -5 \arcsin \left (d x +c \right ) \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -2 \arcsin \left (d x +c \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) b -9 \arcsin \left (d x +c \right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) b +5 \cos \left (\frac {5 a}{b}\right ) \operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +9 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -5 \sin \left (\frac {5 a}{b}\right ) \operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) a +2 \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -2 \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a -9 \cos \left (\frac {3 a}{b}\right ) \operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) a +\cos \left (5 \arcsin \left (d x +c \right )\right ) b -3 \cos \left (3 \arcsin \left (d x +c \right )\right ) b +2 \sqrt {1-\left (d x +c \right )^{2}}\, b \right )}{16 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\) | \(396\) |
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\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=e^{4} \left (\int \frac {c^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (244) = 488\).
Time = 0.42 (sec) , antiderivative size = 1401, normalized size of antiderivative = 5.43 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^2} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \]
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