Integrand size = 23, antiderivative size = 213 \[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\frac {e^4 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b d} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4889, 12, 4731, 4491, 3384, 3380, 3383} \[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\frac {e^4 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b d} \]
[In]
[Out]
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^4 x^4}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{a+b \arcsin (x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = \frac {e^4 \text {Subst}\left (\int \left (\frac {\cos \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {3 \cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{b d} \\ & = \frac {e^4 \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b d}+\frac {e^4 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{8 b d}-\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c+d x)\right )}{16 b d} \\ & = \frac {\left (e^4 \cos \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 d}-\frac {\left (3 e^4 \cos \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 d}+\frac {\left (e^4 \cos \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 d}+\frac {\left (e^4 \sin \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 d}-\frac {\left (3 e^4 \sin \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 d}+\frac {\left (e^4 \sin \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 d} \\ & = \frac {e^4 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{8 b d}-\frac {3 e^4 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 b d}+\frac {e^4 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 b d} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.70 \[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\frac {e^4 \left (2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )-3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )-3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )+\sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c+d x)\right )\right )\right )}{16 b d} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {e^{4} \left (\operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )+\operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+2 \,\operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+2 \,\operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-3 \,\operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )\right )}{16 d b}\) | \(153\) |
default | \(\frac {e^{4} \left (\operatorname {Ci}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )+\operatorname {Si}\left (5 \arcsin \left (d x +c \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+2 \,\operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+2 \,\operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-3 \,\operatorname {Si}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \,\operatorname {Ci}\left (3 \arcsin \left (d x +c \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )\right )}{16 d b}\) | \(153\) |
[In]
[Out]
\[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{b \arcsin \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=e^{4} \left (\int \frac {c^{4}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a + b \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{b \arcsin \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (201) = 402\).
Time = 0.35 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.97 \[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\frac {e^{4} \cos \left (\frac {a}{b}\right )^{5} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{b d} + \frac {e^{4} \cos \left (\frac {a}{b}\right )^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{b d} - \frac {5 \, e^{4} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} - \frac {3 \, e^{4} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} - \frac {3 \, e^{4} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} - \frac {3 \, e^{4} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{4 \, b d} + \frac {5 \, e^{4} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {9 \, e^{4} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {e^{4} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{8 \, b d} + \frac {e^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {3 \, e^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (d x + c\right )\right )}{16 \, b d} + \frac {e^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (d x + c\right )\right )}{8 \, b d} \]
[In]
[Out]
Timed out. \[ \int \frac {(c e+d e x)^4}{a+b \arcsin (c+d x)} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]
[In]
[Out]