Integrand size = 21, antiderivative size = 69 \[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=-\frac {e \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b d}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b d} \]
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Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4889, 12, 4731, 4491, 3384, 3380, 3383} \[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b d}-\frac {e \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b d} \]
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4731
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e x}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = -\frac {e \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = -\frac {e \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{2 b d} \\ & = \frac {\left (e \cos \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 d}-\frac {\left (e \sin \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 d} \\ & = -\frac {e \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{2 b d}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 b d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=\frac {e \left (-\operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c+d x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c+d x)\right )\right )}{2 b d} \]
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Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {e \left (\operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )-\operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{2 d b}\) | \(60\) |
default | \(\frac {e \left (\operatorname {Si}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )-\operatorname {Ci}\left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{2 d b}\) | \(60\) |
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\[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=\int { \frac {d e x + c e}{b \arcsin \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=e \left (\int \frac {c}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=\int { \frac {d e x + c e}{b \arcsin \left (d x + c\right ) + a} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.38 \[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=-\frac {e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{b d} + \frac {e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b d} - \frac {e \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{2 \, b d} \]
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Timed out. \[ \int \frac {c e+d e x}{a+b \arcsin (c+d x)} \, dx=\int \frac {c\,e+d\,e\,x}{a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]
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