Integrand size = 35, antiderivative size = 396 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=-\frac {(a+b \arcsin (c x))^2}{d e x \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 c^2 x (a+b \arcsin (c x))^2}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {4 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:
-(a+b*arcsin(c*x))^2/d/e/x/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+2*c^2*x*(a+b*a rcsin(c*x))^2/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-2*I*c*(-c^2*x^2+1)^(1/2 )*(a+b*arcsin(c*x))^2/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-4*b*c*(-c^2*x^2 +1)^(1/2)*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d/e/(c*d *x+d)^(1/2)/(-c*e*x+e)^(1/2)+4*b*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*ln (1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-I*b^ 2*c*(-c^2*x^2+1)^(1/2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d/e/(c*d*x +d)^(1/2)/(-c*e*x+e)^(1/2)-I*b^2*c*(-c^2*x^2+1)^(1/2)*polylog(2,(I*c*x+(-c ^2*x^2+1)^(1/2))^2)/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)
Time = 3.79 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {c \csc \left (\frac {1}{2} \arcsin (c x)\right ) \sec \left (\frac {1}{2} \arcsin (c x)\right ) \left (-2 a^2+4 a^2 c^2 x^2-4 a b \arcsin (c x) \cos (2 \arcsin (c x))-2 b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))+2 i b^2 \pi \arcsin (c x) \sin (2 \arcsin (c x))-2 i b^2 \arcsin (c x)^2 \sin (2 \arcsin (c x))+4 b^2 \pi \log \left (1+e^{-i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))+b^2 \pi \log \left (1-i e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))+2 b^2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-b^2 \pi \log \left (1+i e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))+2 b^2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))+2 b^2 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))+2 a b \log (c x) \sin (2 \arcsin (c x))-4 b^2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right ) \sin (2 \arcsin (c x))+b^2 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right ) \sin (2 \arcsin (c x))+2 a b \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right ) \sin (2 \arcsin (c x))+2 a b \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right ) \sin (2 \arcsin (c x))-b^2 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right ) \sin (2 \arcsin (c x))-2 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))-i b^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \sin (2 \arcsin (c x))\right )}{4 d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)), x]
Output:
(c*Csc[ArcSin[c*x]/2]*Sec[ArcSin[c*x]/2]*(-2*a^2 + 4*a^2*c^2*x^2 - 4*a*b*A rcSin[c*x]*Cos[2*ArcSin[c*x]] - 2*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + ( 2*I)*b^2*Pi*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*ArcSin[c*x]^2*Sin[2 *ArcSin[c*x]] + 4*b^2*Pi*Log[1 + E^((-I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + b^2*Pi*Log[1 - I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c* x]*Log[1 - I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - b^2*Pi*Log[1 + I*E^(I *ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSi n[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c* x])]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[c*x]*Sin[2*ArcSin[c*x]] - 4*b^2*Pi*Log [Cos[ArcSin[c*x]/2]]*Sin[2*ArcSin[c*x]] + b^2*Pi*Log[-Cos[(Pi + 2*ArcSin[c *x])/4]]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c* x]/2]]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x] /2]]*Sin[2*ArcSin[c*x]] - b^2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]*Sin[2*Ar cSin[c*x]] - (2*I)*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x ]] - (2*I)*b^2*PolyLog[2, I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - I*b^2* PolyLog[2, E^((2*I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]]))/(4*d*e*Sqrt[d + c*d *x]*Sqrt[e - c*e*x])
Time = 2.37 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.62, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5238, 5204, 5160, 5180, 3042, 4202, 2620, 2715, 2838, 5184, 4919, 3042, 4671, 2715, 2838}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^2 (c d x+d)^{3/2} (e-c e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5238 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 c^2 \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}dx+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-2 b c \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)\right )}{c}\right )+2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b c \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b c \int \frac {a+b \arcsin (c x)}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (4 b c \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (4 b c \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (4 b c \left (-\frac {1}{2} b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+\frac {1}{2} b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (4 b c \left (\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (4 b c \left (-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )+2 c^2 \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )-\frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]
Output:
(Sqrt[1 - c^2*x^2]*(-((a + b*ArcSin[c*x])^2/(x*Sqrt[1 - c^2*x^2])) + 2*c^2 *((x*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - (2*b*(((I/2)*(a + b*ArcSin [c*x])^2)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[ c*x])] - (b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/4)))/c) + 4*b*c*(-((a + b* ArcSin[c*x])*ArcTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLog[2, -E^((2*I )*ArcSin[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])))/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar t[q]) Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n , x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & & EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Time = 4.26 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.83
| method | result | size |
| default | \(\frac {a^{2} \left (-2 c^{2} x^{2}+1\right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d^{2} e^{2} \left (c x +1\right ) \left (c x -1\right ) x}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \left (2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-1\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{\left (c^{2} x^{2}-1\right ) x \,e^{2} d^{2}}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right ) c}{d^{2} e^{2} \left (c^{2} x^{2}-1\right )}\right )+\frac {2 a b \left (2 i \arcsin \left (c x \right ) x^{3} c^{3}-\ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) x^{3} c^{3}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 i \arcsin \left (c x \right ) x c +\ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) x c -\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{\left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x \,e^{2} d^{2}}\) | \(723\) |
| parts | \(\frac {a^{2} \left (-2 c^{2} x^{2}+1\right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d^{2} e^{2} \left (c x +1\right ) \left (c x -1\right ) x}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \left (2 i \sqrt {-c^{2} x^{2}+1}\, c x +2 c^{2} x^{2}-1\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{\left (c^{2} x^{2}-1\right ) x \,e^{2} d^{2}}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -i \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right ) c}{d^{2} e^{2} \left (c^{2} x^{2}-1\right )}\right )+\frac {2 a b \left (2 i \arcsin \left (c x \right ) x^{3} c^{3}-\ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) x^{3} c^{3}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 i \arcsin \left (c x \right ) x c +\ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) x c -\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{\left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x \,e^{2} d^{2}}\) | \(723\) |
Input:
int((a+b*arcsin(c*x))^2/x^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x,method=_RET URNVERBOSE)
Output:
a^2*(-2*c^2*x^2+1)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/d^2/e^2/(c*x+1)/(c *x-1)/x+b^2*(-arcsin(c*x)^2*(2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-1)*(-e*( c*x-1))^(1/2)*(d*(c*x+1))^(1/2)/(c^2*x^2-1)/x/e^2/d^2+2*I*(-c^2*x^2+1)^(1/ 2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/d^2/e^2/(c^2*x^2-1)*(I*arcsin(c*x) *ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+I*arcsin(c*x)*ln(1+I*(I*c*x+(-c^ 2*x^2+1)^(1/2))^(1/2))+I*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2) )+I*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+I*arcsin(c*x)*ln(1+ I*(I*c*x+(-c^2*x^2+1)^(1/2)))+I*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/ 2)))+2*arcsin(c*x)^2+2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+2*pol ylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+2*polylog(2,(I*c*x+(-c^2*x^2+1 )^(1/2))^(1/2))+2*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+polylog(2,- I*(I*c*x+(-c^2*x^2+1)^(1/2)))+polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))))*c)+ 2*a*b*(2*I*arcsin(c*x)*x^3*c^3-ln((I*c*x+(-c^2*x^2+1)^(1/2))^4-1)*x^3*c^3+ 2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2-2*I*arcsin(c*x)*c*x+ln((I*c*x+(-c ^2*x^2+1)^(1/2))^4-1)*x*c-arcsin(c*x)*(-c^2*x^2+1)^(1/2))*(-e*(c*x-1))^(1/ 2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-2*c^2*x^2+1)/x/e^2/d^2
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algo rithm="fricas")
Output:
integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*sqrt(c*d*x + d)*sqr t(-c*e*x + e)/(c^4*d^2*e^2*x^6 - 2*c^2*d^2*e^2*x^4 + d^2*e^2*x^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2} \left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (- e \left (c x - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*asin(c*x))**2/x**2/(c*d*x+d)**(3/2)/(-c*e*x+e)**(3/2),x)
Output:
Integral((a + b*asin(c*x))**2/(x**2*(d*(c*x + 1))**(3/2)*(-e*(c*x - 1))**( 3/2)), x)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arcsin(c*x))^2/x^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algo rithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algo rithm="giac")
Output:
integrate((b*arcsin(c*x) + a)^2/((c*d*x + d)^(3/2)*(-c*e*x + e)^(3/2)*x^2) , x)
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*asin(c*x))^2/(x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x)
Output:
int((a + b*asin(c*x))^2/(x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{4}-\sqrt {c x +1}\, \sqrt {-c x +1}\, x^{2}}d x \right ) a b x -\sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{4}-\sqrt {c x +1}\, \sqrt {-c x +1}\, x^{2}}d x \right ) b^{2} x +2 a^{2} c^{2} x^{2}-a^{2}}{\sqrt {e}\, \sqrt {d}\, \sqrt {c x +1}\, \sqrt {-c x +1}\, d e x} \] Input:
int((a+b*asin(c*x))^2/x^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x)
Output:
( - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**4 - sqrt(c*x + 1)*sqrt( - c*x + 1)*x**2),x)*a*b*x - sqrt( c*x + 1)*sqrt( - c*x + 1)*int(asin(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1) *c**2*x**4 - sqrt(c*x + 1)*sqrt( - c*x + 1)*x**2),x)*b**2*x + 2*a**2*c**2* x**2 - a**2)/(sqrt(e)*sqrt(d)*sqrt(c*x + 1)*sqrt( - c*x + 1)*d*e*x)