Integrand size = 27, antiderivative size = 261 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \arcsin (c x))}{d^2 x \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2}+\frac {c^2 (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {4 c^2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {2 i b c^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^2}-\frac {2 i b c^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^2}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{d^2} \] Output:
-b*c*(a+b*arcsin(c*x))/d^2/x/(-c^2*x^2+1)^(1/2)-1/2*(a+b*arcsin(c*x))^2/d^ 2/x^2+1/2*c^2*(a+b*arcsin(c*x))^2/d^2/(-c^2*x^2+1)-4*c^2*(a+b*arcsin(c*x)) ^2*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^2+b^2*c^2*ln(x)/d^2-1/2*b^2*c^2 *ln(-c^2*x^2+1)/d^2+2*I*b*c^2*(a+b*arcsin(c*x))*polylog(2,-(I*c*x+(-c^2*x^ 2+1)^(1/2))^2)/d^2-2*I*b*c^2*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+ 1)^(1/2))^2)/d^2-b^2*c^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^2+b^2* c^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(759\) vs. \(2(261)=522\).
Time = 1.50 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.91 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {a^2}{x^2}+\frac {a^2 c^2}{1-c^2 x^2}-\frac {2 a b c \sqrt {1-c^2 x^2}}{x}+\frac {a b c^2 \sqrt {1-c^2 x^2}}{-1+c x}+\frac {a b c^2 \sqrt {1-c^2 x^2}}{1+c x}-8 i a b c^2 \pi \arcsin (c x)-\frac {2 a b \arcsin (c x)}{x^2}+\frac {a b c^2 \arcsin (c x)}{1-c x}+\frac {a b c^2 \arcsin (c x)}{1+c x}-\frac {2 b^2 c^3 x \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c \sqrt {1-c^2 x^2} \arcsin (c x)}{x}-\frac {b^2 \arcsin (c x)^2}{x^2}+\frac {b^2 c^2 \arcsin (c x)^2}{1-c^2 x^2}-16 a b c^2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )-4 a b c^2 \pi \log \left (1-i e^{i \arcsin (c x)}\right )-8 a b c^2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+4 a b c^2 \pi \log \left (1+i e^{i \arcsin (c x)}\right )-8 a b c^2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+8 a b c^2 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+4 b^2 c^2 \arcsin (c x)^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-4 b^2 c^2 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+4 a^2 c^2 \log (x)+2 b^2 c^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-2 a^2 c^2 \log \left (1-c^2 x^2\right )+16 a b c^2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-4 a b c^2 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+4 a b c^2 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+8 i a b c^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+8 i a b c^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+4 i b^2 c^2 \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-4 i a b c^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-4 i b^2 c^2 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-2 b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )+2 b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^2} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^2),x]
Output:
(-(a^2/x^2) + (a^2*c^2)/(1 - c^2*x^2) - (2*a*b*c*Sqrt[1 - c^2*x^2])/x + (a *b*c^2*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (a*b*c^2*Sqrt[1 - c^2*x^2])/(1 + c* x) - (8*I)*a*b*c^2*Pi*ArcSin[c*x] - (2*a*b*ArcSin[c*x])/x^2 + (a*b*c^2*Arc Sin[c*x])/(1 - c*x) + (a*b*c^2*ArcSin[c*x])/(1 + c*x) - (2*b^2*c^3*x*ArcSi n[c*x])/Sqrt[1 - c^2*x^2] - (2*b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/x - (b ^2*ArcSin[c*x]^2)/x^2 + (b^2*c^2*ArcSin[c*x]^2)/(1 - c^2*x^2) - 16*a*b*c^2 *Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 4*a*b*c^2*Pi*Log[1 - I*E^(I*ArcSin[c*x ])] - 8*a*b*c^2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 4*a*b*c^2*Pi*Lo g[1 + I*E^(I*ArcSin[c*x])] - 8*a*b*c^2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c *x])] + 8*a*b*c^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 4*b^2*c^2*A rcSin[c*x]^2*Log[1 - E^((2*I)*ArcSin[c*x])] - 4*b^2*c^2*ArcSin[c*x]^2*Log[ 1 + E^((2*I)*ArcSin[c*x])] + 4*a^2*c^2*Log[x] + 2*b^2*c^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 2*a^2*c^2*Log[1 - c^2*x^2] + 16*a*b*c^2*Pi*Log[Cos[ArcSin[c *x]/2]] - 4*a*b*c^2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 4*a*b*c^2*Pi*Lo g[Sin[(Pi + 2*ArcSin[c*x])/4]] + (8*I)*a*b*c^2*PolyLog[2, (-I)*E^(I*ArcSin [c*x])] + (8*I)*a*b*c^2*PolyLog[2, I*E^(I*ArcSin[c*x])] + (4*I)*b^2*c^2*Ar cSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (4*I)*a*b*c^2*PolyLog[2, E^ ((2*I)*ArcSin[c*x])] - (4*I)*b^2*c^2*ArcSin[c*x]*PolyLog[2, E^((2*I)*ArcSi n[c*x])] - 2*b^2*c^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])] + 2*b^2*c^2*PolyLo g[3, E^((2*I)*ArcSin[c*x])])/(2*d^2)
Time = 2.11 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.24, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {5204, 27, 5194, 25, 354, 86, 2009, 5208, 5160, 240, 5184, 4919, 3042, 4671, 3011, 2720, 7143}
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.
\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 5204 |
\(\displaystyle 2 c^2 \int \frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )^2}dx+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}+\frac {b c \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 5194 |
\(\displaystyle \frac {2 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}+\frac {b c \left (-b c \int -\frac {1-2 c^2 x^2}{x \left (1-c^2 x^2\right )}dx+\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}+\frac {b c \left (b c \int \frac {1-2 c^2 x^2}{x \left (1-c^2 x^2\right )}dx+\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {2 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}+\frac {b c \left (\frac {1}{2} b c \int \frac {1-2 c^2 x^2}{x^2 \left (1-c^2 x^2\right )}dx^2+\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {2 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}+\frac {b c \left (\frac {1}{2} b c \int \left (\frac {c^2}{c^2 x^2-1}+\frac {1}{x^2}\right )dx^2+\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 c^2 \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 5208 |
\(\displaystyle \frac {2 c^2 \left (-b c \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 5160 |
\(\displaystyle \frac {2 c^2 \left (-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {2 c^2 \left (\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 5184 |
\(\displaystyle \frac {2 c^2 \left (\int \frac {(a+b \arcsin (c x))^2}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 4919 |
\(\displaystyle \frac {2 c^2 \left (2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c^2 \left (2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle \frac {2 c^2 \left (2 \left (-b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {2 c^2 \left (2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {2 c^2 \left (2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {2 c^2 \left (2 \left (-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \arcsin (c x))^2}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2 c^2 x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}+\frac {1}{2} b c \left (\log \left (1-c^2 x^2\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^2),x]
Output:
-1/2*(a + b*ArcSin[c*x])^2/(d^2*x^2*(1 - c^2*x^2)) + (b*c*(-((a + b*ArcSin [c*x])/(x*Sqrt[1 - c^2*x^2])) + (2*c^2*x*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2 *x^2] + (b*c*(Log[x^2] + Log[1 - c^2*x^2]))/2))/d^2 + (2*c^2*((a + b*ArcSi n[c*x])^2/(2*(1 - c^2*x^2)) - b*c*((x*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^ 2] + (b*Log[1 - c^2*x^2])/(2*c)) + 2*(-((a + b*ArcSin[c*x])^2*ArcTanh[E^(( 2*I)*ArcSin[c*x])]) + b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*Ar cSin[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/4) - b*((I/2)*(a + b* ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, E^((2*I)*Ar cSin[c*x])])/4))))/d^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_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[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[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_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 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_.)*((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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c *(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (301 ) = 602\).
Time = 0.74 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.53
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a^{2} \left (\frac {1}{4 c x +4}-\ln \left (c x +1\right )-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right ) \left (2 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {2 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-2 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}\right )\) | \(661\) |
default | \(c^{2} \left (\frac {a^{2} \left (\frac {1}{4 c x +4}-\ln \left (c x +1\right )-\frac {1}{2 c^{2} x^{2}}+2 \ln \left (c x \right )-\frac {1}{4 \left (c x -1\right )}-\ln \left (c x -1\right )\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (c x \right ) \left (2 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {2 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-2 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}\right )\) | \(661\) |
parts | \(\frac {a^{2} \left (-\frac {c^{2}}{4 \left (c x -1\right )}-c^{2} \ln \left (c x -1\right )+\frac {c^{2}}{4 c x +4}-c^{2} \ln \left (c x +1\right )-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\arcsin \left (c x \right ) \left (2 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \,c^{2} \left (-\frac {2 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )}{2 c^{2} x^{2} \left (c^{2} x^{2}-1\right )}-2 \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}\) | \(673\) |
Input:
int((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
c^2*(a^2/d^2*(1/4/(c*x+1)-ln(c*x+1)-1/2/c^2/x^2+2*ln(c*x)-1/4/(c*x-1)-ln(c *x-1))+b^2/d^2*(-1/2/c^2/x^2/(c^2*x^2-1)*arcsin(c*x)*(2*c^2*x^2*arcsin(c*x )-2*c*x*(-c^2*x^2+1)^(1/2)-arcsin(c*x))-ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2) +ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)+ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*arcsin(c* x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-4*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2 *x^2+1)^(1/2))+4*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+2*arcsin(c*x)^2*ln(1 -I*c*x-(-c^2*x^2+1)^(1/2))-4*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1 /2))+4*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))-2*arcsin(c*x)^2*ln(1+(I*c*x+(-c ^2*x^2+1)^(1/2))^2)+2*I*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^ 2)-polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2))+2*a*b/d^2*(-1/2*(2*c^2*x^2*ar csin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-arcsin(c*x))/c^2/x^2/(c^2*x^2-1)-2*arcsin (c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^ 2+1)^(1/2))+2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*polylog(2,-I* c*x-(-c^2*x^2+1)^(1/2))-2*I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+I*polylog( 2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)))
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^4*d^2*x^7 - 2*c^ 2*d^2*x^5 + d^2*x^3), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \] Input:
integrate((a+b*asin(c*x))**2/x**3/(-c**2*d*x**2+d)**2,x)
Output:
(Integral(a**2/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(b**2*asin(c *x)**2/(c**4*x**7 - 2*c**2*x**5 + x**3), x) + Integral(2*a*b*asin(c*x)/(c* *4*x**7 - 2*c**2*x**5 + x**3), x))/d**2
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
Output:
-1/2*a^2*(2*c^2*log(c*x + 1)/d^2 + 2*c^2*log(c*x - 1)/d^2 - 4*c^2*log(x)/d ^2 + (2*c^2*x^2 - 1)/(c^2*d^2*x^4 - d^2*x^2)) + integrate((b^2*arctan2(c*x , sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt( -c*x + 1)))/(c^4*d^2*x^7 - 2*c^2*d^2*x^5 + d^2*x^3), x)
Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*x))^2/x^3/(-c^2*d*x^2+d)^2,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. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \] Input:
int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^2),x)
Output:
int((a + b*asin(c*x))^2/(x^3*(d - c^2*d*x^2)^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3 \left (d-c^2 d x^2\right )^2} \, dx=\frac {4 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) a b \,c^{2} x^{4}-4 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) a b \,x^{2}+2 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) b^{2} c^{2} x^{4}-2 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{c^{4} x^{7}-2 c^{2} x^{5}+x^{3}}d x \right ) b^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a^{2} c^{4} x^{4}-4 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}-2 a^{2} c^{4} x^{4}+a^{2}}{2 d^{2} x^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*asin(c*x))^2/x^3/(-c^2*d*x^2+d)^2,x)
Output:
(4*int(asin(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*a*b*c**2*x**4 - 4*int (asin(c*x)/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*a*b*x**2 + 2*int(asin(c*x)* *2/(c**4*x**7 - 2*c**2*x**5 + x**3),x)*b**2*c**2*x**4 - 2*int(asin(c*x)**2 /(c**4*x**7 - 2*c**2*x**5 + x**3),x)*b**2*x**2 - 2*log(c**2*x - c)*a**2*c* *4*x**4 + 2*log(c**2*x - c)*a**2*c**2*x**2 - 2*log(c**2*x + c)*a**2*c**4*x **4 + 2*log(c**2*x + c)*a**2*c**2*x**2 + 4*log(x)*a**2*c**4*x**4 - 4*log(x )*a**2*c**2*x**2 - 2*a**2*c**4*x**4 + a**2)/(2*d**2*x**2*(c**2*x**2 - 1))