Integrand size = 25, antiderivative size = 291 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b c^3}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {11 b c^3}{8 d^3 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {1-c^2 x^2}}{6 d^3 x^2}-\frac {a+b \arcsin (c x)}{3 d^3 x^3}-\frac {3 c^2 (a+b \arcsin (c x))}{d^3 x}+\frac {c^4 x (a+b \arcsin (c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {11 c^4 x (a+b \arcsin (c x))}{8 d^3 \left (1-c^2 x^2\right )}-\frac {35 i c^3 (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{4 d^3}-\frac {19 b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{8 d^3}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{8 d^3} \] Output:
-1/12*b*c^3/d^3/(-c^2*x^2+1)^(3/2)-11/8*b*c^3/d^3/(-c^2*x^2+1)^(1/2)-1/6*b *c*(-c^2*x^2+1)^(1/2)/d^3/x^2-1/3*(a+b*arcsin(c*x))/d^3/x^3-3*c^2*(a+b*arc sin(c*x))/d^3/x+1/4*c^4*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^2+11/8*c^4*x* (a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)-35/4*I*c^3*(a+b*arcsin(c*x))*arctan(I*c *x+(-c^2*x^2+1)^(1/2))/d^3-19/6*b*c^3*arctanh((-c^2*x^2+1)^(1/2))/d^3+35/8 *I*b*c^3*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3-35/8*I*b*c^3*polylog (2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(587\) vs. \(2(291)=582\).
Time = 1.52 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.02 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {\frac {16 a}{x^3}+\frac {144 a c^2}{x}+\frac {8 b c \sqrt {1-c^2 x^2}}{x^2}+\frac {2 b c^3 \sqrt {1-c^2 x^2}}{(-1+c x)^2}-\frac {b c^4 x \sqrt {1-c^2 x^2}}{(-1+c x)^2}-\frac {33 b c^3 \sqrt {1-c^2 x^2}}{-1+c x}+\frac {2 b c^3 \sqrt {1-c^2 x^2}}{(1+c x)^2}+\frac {b c^4 x \sqrt {1-c^2 x^2}}{(1+c x)^2}+\frac {33 b c^3 \sqrt {1-c^2 x^2}}{1+c x}-\frac {12 a c^4 x}{\left (-1+c^2 x^2\right )^2}+\frac {66 a c^4 x}{-1+c^2 x^2}+105 i b c^3 \pi \arcsin (c x)+\frac {16 b \arcsin (c x)}{x^3}+\frac {144 b c^2 \arcsin (c x)}{x}-\frac {3 b c^3 \arcsin (c x)}{(-1+c x)^2}+\frac {33 b c^3 \arcsin (c x)}{-1+c x}+\frac {3 b c^3 \arcsin (c x)}{(1+c x)^2}+\frac {33 b c^3 \arcsin (c x)}{1+c x}+152 b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-105 b c^3 \pi \log \left (1-i e^{i \arcsin (c x)}\right )-210 b c^3 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-105 b c^3 \pi \log \left (1+i e^{i \arcsin (c x)}\right )+210 b c^3 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+105 a c^3 \log (1-c x)-105 a c^3 \log (1+c x)+105 b c^3 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+105 b c^3 \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-210 i b c^3 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+210 i b c^3 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{48 d^3} \] Input:
Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^3),x]
Output:
-1/48*((16*a)/x^3 + (144*a*c^2)/x + (8*b*c*Sqrt[1 - c^2*x^2])/x^2 + (2*b*c ^3*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (b*c^4*x*Sqrt[1 - c^2*x^2])/(-1 + c*x )^2 - (33*b*c^3*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (2*b*c^3*Sqrt[1 - c^2*x^2] )/(1 + c*x)^2 + (b*c^4*x*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (33*b*c^3*Sqrt[1 - c^2*x^2])/(1 + c*x) - (12*a*c^4*x)/(-1 + c^2*x^2)^2 + (66*a*c^4*x)/(-1 + c^2*x^2) + (105*I)*b*c^3*Pi*ArcSin[c*x] + (16*b*ArcSin[c*x])/x^3 + (144* b*c^2*ArcSin[c*x])/x - (3*b*c^3*ArcSin[c*x])/(-1 + c*x)^2 + (33*b*c^3*ArcS in[c*x])/(-1 + c*x) + (3*b*c^3*ArcSin[c*x])/(1 + c*x)^2 + (33*b*c^3*ArcSin [c*x])/(1 + c*x) + 152*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 105*b*c^3*Pi*Log [1 - I*E^(I*ArcSin[c*x])] - 210*b*c^3*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c* x])] - 105*b*c^3*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 210*b*c^3*ArcSin[c*x]*L og[1 + I*E^(I*ArcSin[c*x])] + 105*a*c^3*Log[1 - c*x] - 105*a*c^3*Log[1 + c *x] + 105*b*c^3*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 105*b*c^3*Pi*Log[Si n[(Pi + 2*ArcSin[c*x])/4]] - (210*I)*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x ])] + (210*I)*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^3
Time = 1.54 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.34, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.920, Rules used = {5204, 27, 243, 52, 61, 61, 73, 221, 5204, 243, 61, 61, 73, 221, 5162, 241, 5162, 241, 5164, 3042, 4669, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {7}{3} c^2 \int \frac {a+b \arcsin (c x)}{d^3 x^2 \left (1-c^2 x^2\right )^3}dx+\frac {b c \int \frac {1}{x^3 \left (1-c^2 x^2\right )^{5/2}}dx}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \int \frac {1}{x^3 \left (1-c^2 x^2\right )^{5/2}}dx}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \int \frac {1}{x^4 \left (1-c^2 x^2\right )^{5/2}}dx^2}{6 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2}}dx^2-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx^2+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (\int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (-\frac {2 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2}+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \arcsin (c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^3}dx+b c \int \frac {1}{x \left (1-c^2 x^2\right )^{5/2}}dx-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^3}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{5/2}}dx^2-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^3}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}dx^2+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^3}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^3}dx+\frac {1}{2} b c \left (-\frac {2 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}}{c^2}+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^3}dx-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {1}{4} b c \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arcsin (c x))}{4 \left (1-c^2 x^2\right )^2}-\frac {b}{12 c \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \arcsin (c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+\frac {2}{\sqrt {1-c^2 x^2}}+\frac {2}{3 \left (1-c^2 x^2\right )^{3/2}}\right )-\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}}\right )}{6 d^3}\) |
Input:
Int[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^3),x]
Output:
-1/3*(a + b*ArcSin[c*x])/(d^3*x^3*(1 - c^2*x^2)^2) + (b*c*(-(1/(x^2*(1 - c ^2*x^2)^(3/2))) + (5*c^2*(2/(3*(1 - c^2*x^2)^(3/2)) + 2/Sqrt[1 - c^2*x^2] - 2*ArcTanh[Sqrt[1 - c^2*x^2]]))/2))/(6*d^3) + (7*c^2*(-((a + b*ArcSin[c*x ])/(x*(1 - c^2*x^2)^2)) + (b*c*(2/(3*(1 - c^2*x^2)^(3/2)) + 2/Sqrt[1 - c^2 *x^2] - 2*ArcTanh[Sqrt[1 - c^2*x^2]]))/2 + 5*c^2*(-1/12*b/(c*(1 - c^2*x^2) ^(3/2)) + (x*(a + b*ArcSin[c*x]))/(4*(1 - c^2*x^2)^2) + (3*(-1/2*b/(c*Sqrt [1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(2*(1 - c^2*x^2)) + ((-2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, (-I)*E^(I*ArcSin [c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*c)))/4)))/(3*d^3)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cSin[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 *x^2)^p] Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sec[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_.)*((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]
Time = 0.68 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}+\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(372\) |
| default | \(c^{3} \left (-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}+\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\right )\) | \(372\) |
| parts | \(-\frac {a \left (-\frac {c^{3}}{16 \left (c x -1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x -1\right )}+\frac {35 c^{3} \ln \left (c x -1\right )}{16}+\frac {c^{3}}{16 \left (c x +1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x +1\right )}-\frac {35 c^{3} \ln \left (c x +1\right )}{16}+\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}\right )}{d^{3}}-\frac {b \,c^{3} \left (\frac {105 \arcsin \left (c x \right ) c^{6} x^{6}-29 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-175 c^{4} x^{4} \arcsin \left (c x \right )+27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+56 c^{2} x^{2} \arcsin \left (c x \right )+4 c x \sqrt {-c^{2} x^{2}+1}+8 \arcsin \left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{6}+\frac {19 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{6}+\frac {35 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}\) | \(386\) |
Input:
int((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
c^3*(-a/d^3*(1/16/(c*x+1)^2+11/16/(c*x+1)-35/16*ln(c*x+1)+1/3/c^3/x^3+3/c/ x-1/16/(c*x-1)^2+11/16/(c*x-1)+35/16*ln(c*x-1))-b/d^3*(1/24*(105*arcsin(c* x)*c^6*x^6-29*c^5*x^5*(-c^2*x^2+1)^(1/2)-175*c^4*x^4*arcsin(c*x)+27*c^3*x^ 3*(-c^2*x^2+1)^(1/2)+56*c^2*x^2*arcsin(c*x)+4*c*x*(-c^2*x^2+1)^(1/2)+8*arc sin(c*x))/(c^4*x^4-2*c^2*x^2+1)/c^3/x^3-19/6*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1 )+19/6*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+35/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2 *x^2+1)^(1/2)))-35/8*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-35/8*I *dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+35/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1 )^(1/2)))))
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b*arcsin(c*x) + a)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x^ 6 - d^3*x^4), x)
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx}{d^{3}} \] Input:
integrate((a+b*asin(c*x))/x**4/(-c**2*d*x**2+d)**3,x)
Output:
-(Integral(a/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4), x) + Integra l(b*asin(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4), x))/d**3
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
1/48*a*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 - 2*(105*c^6*x ^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3) ) + 1/48*(105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*arctan2(c*x, sqrt(c*x + 1)*s qrt(-c*x + 1))*log(c*x + 1) - 105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*arctan2( c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(105*c^6*x^6 - 175*c^ 4*x^4 + 56*c^2*x^2 + 8)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 48*(c ^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)*integrate(-1/48*(210*c^7*x^6 - 350*c ^5*x^4 + 112*c^3*x^2 - 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(c*x + 1) + 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(-c*x + 1) + 16*c)*sqrt(c*x + 1)*sq rt(-c*x + 1)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x))* b/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3)
Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^3,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. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*asin(c*x))/(x^4*(d - c^2*d*x^2)^3),x)
Output:
int((a + b*asin(c*x))/(x^4*(d - c^2*d*x^2)^3), x)
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-48 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b \,c^{4} x^{7}+96 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b \,c^{2} x^{5}-48 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b \,x^{3}-105 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{7} x^{7}+210 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{5} x^{5}-105 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{3} x^{3}+105 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{7} x^{7}-210 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{5} x^{5}+105 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{3} x^{3}-210 a \,c^{6} x^{6}+350 a \,c^{4} x^{4}-112 a \,c^{2} x^{2}-16 a}{48 d^{3} x^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asin(c*x))/x^4/(-c^2*d*x^2+d)^3,x)
Output:
( - 48*int(asin(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4),x)*b* c**4*x**7 + 96*int(asin(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x** 4),x)*b*c**2*x**5 - 48*int(asin(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x* *6 - x**4),x)*b*x**3 - 105*log(c**2*x - c)*a*c**7*x**7 + 210*log(c**2*x - c)*a*c**5*x**5 - 105*log(c**2*x - c)*a*c**3*x**3 + 105*log(c**2*x + c)*a*c **7*x**7 - 210*log(c**2*x + c)*a*c**5*x**5 + 105*log(c**2*x + c)*a*c**3*x* *3 - 210*a*c**6*x**6 + 350*a*c**4*x**4 - 112*a*c**2*x**2 - 16*a)/(48*d**3* x**3*(c**4*x**4 - 2*c**2*x**2 + 1))