Integrand size = 25, antiderivative size = 243 \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\frac {e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}-\frac {e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 \sqrt {b} d}-\frac {e^2 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{2 \sqrt {b} d} \]
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Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4889, 12, 4731, 4491, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e^2 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}-\frac {\sqrt {\frac {\pi }{6}} e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} e^2 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}-\frac {\sqrt {\frac {\pi }{6}} e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d} \]
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4731
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^2 x^2}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = \frac {e^2 \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = -\frac {e^2 \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b d}+\frac {e^2 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b d} \\ & = \frac {\left (e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b d}-\frac {\left (e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b d}+\frac {\left (e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b d}-\frac {\left (e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4 b d} \\ & = \frac {\left (e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{2 b d}-\frac {\left (e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{2 b d}+\frac {\left (e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{2 b d}-\frac {\left (e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{2 b d} \\ & = \frac {e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}-\frac {e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 \sqrt {b} d}-\frac {e^2 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{2 \sqrt {b} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.02 \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=-\frac {i e^2 e^{-\frac {3 i a}{b}} \left (3 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )-3 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )+\sqrt {3} \left (-\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{24 d \sqrt {a+b \arcsin (c+d x)}} \]
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Time = 1.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {e^{2} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \left (\cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {3}{b}}\, b -\sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {-\frac {3}{b}}\, b +\cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right )-\sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right )\right )}{12 d}\) | \(207\) |
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Exception generated. \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=e^{2} \left (\int \frac {c^{2}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {d^{2} x^{2}}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {2 c d x}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx\right ) \]
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\[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{\sqrt {b \arcsin \left (d x + c\right ) + a}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.42 \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\frac {\sqrt {\pi } e^{2} \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}} - \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (\frac {3 i \, a}{b}\right )}}{4 \, {\left (\sqrt {6} \sqrt {b} + \frac {i \, \sqrt {6} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } e^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{4 \, d {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } e^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{4 \, d {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} + \frac {\sqrt {\pi } e^{2} \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {b}} + \frac {i \, \sqrt {6} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{2 \, {\left | b \right |}}\right ) e^{\left (-\frac {3 i \, a}{b}\right )}}{4 \, {\left (\sqrt {6} \sqrt {b} - \frac {i \, \sqrt {6} b^{\frac {3}{2}}}{{\left | b \right |}}\right )} d} \]
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Timed out. \[ \int \frac {(c e+d e x)^2}{\sqrt {a+b \arcsin (c+d x)}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )}} \,d x \]
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