Integrand size = 18, antiderivative size = 362 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3} \]
[Out]
Time = 0.34 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4829, 4717, 4809, 3384, 3380, 3383, 4727} \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\frac {e^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))} \]
[In]
[Out]
Rule 3380
Rule 3383
Rule 3384
Rule 4717
Rule 4727
Rule 4809
Rule 4829
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2}{(a+b \arcsin (c x))^2}+\frac {2 d e x}{(a+b \arcsin (c x))^2}+\frac {e^2 x^2}{(a+b \arcsin (c x))^2}\right ) \, dx \\ & = d^2 \int \frac {1}{(a+b \arcsin (c x))^2} \, dx+(2 d e) \int \frac {x}{(a+b \arcsin (c x))^2} \, dx+e^2 \int \frac {x^2}{(a+b \arcsin (c x))^2} \, dx \\ & = -\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {\left (c d^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx}{b}+\frac {(2 d e) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^2}+\frac {e^2 \text {Subst}\left (\int \left (-\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^3} \\ & = -\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {d^2 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c}+\frac {e^2 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2 c^3}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2 c^3}+\frac {\left (2 d e \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 x)\right )}{b^2 c^2}+\frac {\left (2 d e \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 x)\right )}{b^2 c^2} \\ & = -\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {\left (d^2 \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 x)\right )}{b^2 c}-\frac {\left (e^2 \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 x)\right )}{4 b^2 c^3}+\frac {\left (3 e^2 \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 x)\right )}{4 b^2 c^3}+\frac {\left (d^2 \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 x)\right )}{b^2 c}+\frac {\left (e^2 \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 x)\right )}{4 b^2 c^3}-\frac {\left (3 e^2 \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 x)\right )}{4 b^2 c^3} \\ & = -\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {2 d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c^3}+\frac {2 d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c^3} \\ \end{align*}
Time = 1.63 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {\frac {4 b c^2 d^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {8 b c^2 d e x \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {4 b c^2 e^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-8 c d e \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-\left (4 c^2 d^2+e^2\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+3 e^2 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+4 c^2 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-8 c d e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-3 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{4 b^2 c^3} \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.45
method | result | size |
derivativedivides | \(-\frac {4 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-4 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b c d e -8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b c d e +4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d^{2}-4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d^{2}+4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2}+\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,e^{2}-\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,e^{2}-3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b \,e^{2}+3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b \,e^{2}-8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a c d e -8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a c d e +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,e^{2}-\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,e^{2}+4 \sin \left (2 \arcsin \left (c x \right )\right ) b c d e -3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,e^{2}+3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,e^{2}+\sqrt {-c^{2} x^{2}+1}\, b \,e^{2}-\cos \left (3 \arcsin \left (c x \right )\right ) b \,e^{2}}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(526\) |
default | \(-\frac {4 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-4 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d^{2}-8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b c d e -8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b c d e +4 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d^{2}-4 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d^{2}+4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d^{2}+\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,e^{2}-\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,e^{2}-3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b \,e^{2}+3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b \,e^{2}-8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a c d e -8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a c d e +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,e^{2}-\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,e^{2}+4 \sin \left (2 \arcsin \left (c x \right )\right ) b c d e -3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a \,e^{2}+3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a \,e^{2}+\sqrt {-c^{2} x^{2}+1}\, b \,e^{2}-\cos \left (3 \arcsin \left (c x \right )\right ) b \,e^{2}}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(526\) |
[In]
[Out]
\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (348) = 696\).
Time = 0.38 (sec) , antiderivative size = 1276, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
[In]
[Out]