Integrand size = 21, antiderivative size = 344 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \arcsin (c x)^2}{2 e^2}+\frac {g x (a+b \arcsin (c x))}{e}+\frac {b (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \arcsin (c x) \log (d+e x)}{e^2}+\frac {(e f-d g) (a+b \arcsin (c x)) \log (d+e x)}{e^2}-\frac {i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \] Output:
b*g*(-c^2*x^2+1)^(1/2)/c/e-1/2*I*b*(-d*g+e*f)*arcsin(c*x)^2/e^2+g*x*(a+b*a rcsin(c*x))/e+b*(-d*g+e*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2)) /(c*d-(c^2*d^2-e^2)^(1/2)))/e^2+b*(-d*g+e*f)*arcsin(c*x)*ln(1-I*e*(I*c*x+( -c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2-b*(-d*g+e*f)*arcsin(c*x) *ln(e*x+d)/e^2+(-d*g+e*f)*(a+b*arcsin(c*x))*ln(e*x+d)/e^2-I*b*(-d*g+e*f)*p olylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2-I*b *(-d*g+e*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1 /2)))/e^2
Time = 0.32 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.94 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {\frac {b e g \sqrt {1-c^2 x^2}}{c}-\frac {1}{2} i b (e f-d g) \arcsin (c x)^2+e g x (a+b \arcsin (c x))+b (e f-d g) \arcsin (c x) \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+b (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b (e f-d g) \arcsin (c x) \log (d+e x)+(e f-d g) (a+b \arcsin (c x)) \log (d+e x)-i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-i b (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \] Input:
Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x),x]
Output:
((b*e*g*Sqrt[1 - c^2*x^2])/c - (I/2)*b*(e*f - d*g)*ArcSin[c*x]^2 + e*g*x*( a + b*ArcSin[c*x]) + b*(e*f - d*g)*ArcSin[c*x]*Log[1 + (I*e*E^(I*ArcSin[c* x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + b*(e*f - d*g)*ArcSin[c*x]*Log[1 - ( I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - b*(e*f - d*g)*ArcSin [c*x]*Log[d + e*x] + (e*f - d*g)*(a + b*ArcSin[c*x])*Log[d + e*x] - I*b*(e *f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - I*b*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^2
Time = 1.02 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5252, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx\) |
\(\Big \downarrow \) 5252 |
\(\displaystyle -b c \int \frac {e g x+(e f-d g) \log (d+e x)}{e^2 \sqrt {1-c^2 x^2}}dx+\frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b c \int \frac {e g x+(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}}dx}{e^2}+\frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {b c \int \left (\frac {e g x}{\sqrt {1-c^2 x^2}}+\frac {(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right )dx}{e^2}+\frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e f-d g) \log (d+e x) (a+b \arcsin (c x))}{e^2}+\frac {g x (a+b \arcsin (c x))}{e}-\frac {b c \left (\frac {i (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}+\frac {i (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {\arcsin (c x) (e f-d g) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{c}-\frac {\arcsin (c x) (e f-d g) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{c}+\frac {i \arcsin (c x)^2 (e f-d g)}{2 c}+\frac {\arcsin (c x) (e f-d g) \log (d+e x)}{c}-\frac {e g \sqrt {1-c^2 x^2}}{c^2}\right )}{e^2}\) |
Input:
Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x),x]
Output:
(g*x*(a + b*ArcSin[c*x]))/e + ((e*f - d*g)*(a + b*ArcSin[c*x])*Log[d + e*x ])/e^2 - (b*c*(-((e*g*Sqrt[1 - c^2*x^2])/c^2) + ((I/2)*(e*f - d*g)*ArcSin[ c*x]^2)/c - ((e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/c - ((e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*Ar cSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/c + ((e*f - d*g)*ArcSin[c*x]*Log [d + e*x])/c + (I*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sq rt[c^2*d^2 - e^2])])/c + (I*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x])) /(c*d + Sqrt[c^2*d^2 - e^2])])/c))/e^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, x]}, Simp[(a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (357 ) = 714\).
Time = 1.82 (sec) , antiderivative size = 1566, normalized size of antiderivative = 4.55
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1566\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1584\) |
default | \(\text {Expression too large to display}\) | \(1584\) |
Input:
int((g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*(g/e*x+(-d*g+e*f)/e^2*ln(e*x+d))+I*b*c^2/e^2*d^3*g/(c^2*d^2-e^2)*dilog(( I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+ e^2)^(1/2)))-I*b*c^2/e*f/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1 /2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d^2+I*b*e*f/(c^ 2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2)) /(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*d*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+( I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1 /2)))+b*d*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2)) *e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*e*f*arcsin(c*x)/( c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/ (I*d*c+(-c^2*d^2+e^2)^(1/2)))-b*e*f*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I *c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/ 2)))-b*c^2/e^2*d^3*g*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+ 1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+b*c^2/e*f* arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2 +e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*d^2+b*c^2/e*f*arcsin(c*x)/(c^2* d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d *c+(-c^2*d^2+e^2)^(1/2)))*d^2-b*c^2/e^2*d^3*g*arcsin(c*x)/(c^2*d^2-e^2)*ln ((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^ 2+e^2)^(1/2)))+I*b*c^2/e^2*d^3*g/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^...
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d} \,d x } \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x, algorithm="fricas")
Output:
integral((a*g*x + a*f + (b*g*x + b*f)*arcsin(c*x))/(e*x + d), x)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{d + e x}\, dx \] Input:
integrate((g*x+f)*(a+b*asin(c*x))/(e*x+d),x)
Output:
Integral((a + b*asin(c*x))*(f + g*x)/(d + e*x), x)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d} \,d x } \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x, algorithm="maxima")
Output:
a*g*(x/e - d*log(e*x + d)/e^2) + a*f*log(e*x + d)/e + integrate((b*g*x + b *f)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e*x + d), x)
Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{d+e\,x} \,d x \] Input:
int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x),x)
Output:
int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x), x)
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{d+e x} \, dx=\frac {\mathit {asin} \left (c x \right ) b c e g x +\sqrt {-c^{2} x^{2}+1}\, b e g -\left (\int \frac {\mathit {asin} \left (c x \right )}{e x +d}d x \right ) b c d e g +\left (\int \frac {\mathit {asin} \left (c x \right )}{e x +d}d x \right ) b c \,e^{2} f -\mathrm {log}\left (e x +d \right ) a c d g +\mathrm {log}\left (e x +d \right ) a c e f +a c e g x}{c \,e^{2}} \] Input:
int((g*x+f)*(a+b*asin(c*x))/(e*x+d),x)
Output:
(asin(c*x)*b*c*e*g*x + sqrt( - c**2*x**2 + 1)*b*e*g - int(asin(c*x)/(d + e *x),x)*b*c*d*e*g + int(asin(c*x)/(d + e*x),x)*b*c*e**2*f - log(d + e*x)*a* c*d*g + log(d + e*x)*a*c*e*f + a*c*e*g*x)/(c*e**2)