Integrand size = 16, antiderivative size = 115 \[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}-\frac {e \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4831, 6874, 3384, 3380, 3383, 4491, 12} \[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=-\frac {e \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2}+\frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c} \]
[In]
[Out]
Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4831
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos (x) (c d+e \sin (x))}{a+b x} \, dx,x,\arcsin (c x)\right )}{c^2} \\ & = \frac {\text {Subst}\left (\int \left (\frac {c d \cos (x)}{a+b x}+\frac {e \cos (x) \sin (x)}{a+b x}\right ) \, dx,x,\arcsin (c x)\right )}{c^2} \\ & = \frac {d \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{c}+\frac {e \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{c^2} \\ & = \frac {e \text {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\arcsin (c x)\right )}{c^2}+\frac {\left (d \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{c}+\frac {\left (d \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{c} \\ & = \frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {e \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\arcsin (c x)\right )}{2 c^2} \\ & = \frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {\left (e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{2 c^2}-\frac {\left (e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\arcsin (c x)\right )}{2 c^2} \\ & = \frac {d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}-\frac {e \operatorname {CosIntegral}\left (\frac {2 a}{b}+2 \arcsin (c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b c^2}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b c}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \arcsin (c x)\right )}{2 b c^2} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\frac {2 c d \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-e \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+2 c d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b c^2} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {d \left (\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{b}+\frac {e \left (\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )-\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{2 c b}}{c}\) | \(103\) |
default | \(\frac {\frac {d \left (\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{b}+\frac {e \left (\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )-\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{2 c b}}{c}\) | \(103\) |
[In]
[Out]
\[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\int { \frac {e x + d}{b \arcsin \left (c x\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\int \frac {d + e x}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\int { \frac {e x + d}{b \arcsin \left (c x\right ) + a} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.21 \[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\frac {d \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} + \frac {e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {d \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {e \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {d+e x}{a+b \arcsin (c x)} \, dx=\int \frac {d+e\,x}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
[In]
[Out]