Integrand size = 16, antiderivative size = 313 \[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=-\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}-\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \]
1/3*x^3*(a+b*arccos(c*x))^(3/2)+1/144*b^(3/2)*cos(3*a/b)*FresnelS(6^(1/2)/ Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/c^3-1/144*b^(3/ 2)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*6 ^(1/2)*Pi^(1/2)/c^3+3/16*b^(3/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*a rccos(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^3-3/16*b^(3/2)*FresnelC(2^(1 /2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/c^ 3-1/3*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^(1/2)/c^3-1/6*b*x^2*(-c^2*x^2 +1)^(1/2)*(a+b*arccos(c*x))^(1/2)/c
Result contains complex when optimal does not.
Time = 7.08 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.77 \[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=-\frac {i a b e^{-\frac {3 i a}{b}} \left (-9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )+\sqrt {3} \left (-\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arccos (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arccos (c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \arccos (c x)}}-\frac {\sqrt {b} \left (18 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (3 \sqrt {1-c^2 x^2}-2 c x \arccos (c x)\right )-9 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )-9 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )-\sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (b \cos \left (\frac {3 a}{b}\right )+2 a \sin \left (\frac {3 a}{b}\right )\right )-\sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {3 a}{b}\right )-b \sin \left (\frac {3 a}{b}\right )\right )+6 \sqrt {b} \sqrt {a+b \arccos (c x)} (-2 \arccos (c x) \cos (3 \arccos (c x))+\sin (3 \arccos (c x)))\right )}{144 c^3} \]
((-1/72*I)*a*b*(-9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamm a[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*Ar cCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(-(Sqrt[((- I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b]) + E ^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*Ar cCos[c*x]))/b])))/(c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]]) - (Sqrt[b] *(18*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*ArcCos[c *x]) - 9*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] *(3*b*Cos[a/b] + 2*a*Sin[a/b]) - 9*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) - Sqrt[6*Pi]*Fres nelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(b*Cos[(3*a)/b] + 2*a*S in[(3*a)/b]) - Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sq rt[b]]*(2*a*Cos[(3*a)/b] - b*Sin[(3*a)/b]) + 6*Sqrt[b]*Sqrt[a + b*ArcCos[c *x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(144*c^3)
Time = 2.17 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.35, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.062, Rules used = {5141, 5211, 5147, 25, 4906, 2009, 5183, 5135, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}
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.
| \(\displaystyle \int x^2 (a+b \arccos (c x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 5141 |
| \(\displaystyle \frac {1}{2} b c \int \frac {x^3 \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \int \frac {x^2}{\sqrt {a+b \arccos (c x)}}dx}{6 c}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 5147 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {\int -\frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} b c \left (-\frac {\int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {1}{2} b c \left (-\frac {\int \left (\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{4 \sqrt {a+b \arccos (c x)}}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{4 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (-\frac {b \int \frac {1}{\sqrt {a+b \arccos (c x)}}dx}{2 c}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 5135 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {\int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arccos (c x)}{b}\right )d\sqrt {a+b \arccos (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}+\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {1}{2} b c \left (\frac {-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {2 \left (\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^{3/2}\) |
(x^3*(a + b*ArcCos[c*x])^(3/2))/3 + (b*c*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*Sqrt [a + b*ArcCos[c*x]])/c^2 + (2*(-((Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x] ])/c^2) + (Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*Arc Cos[c*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*A rcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c^2)))/(3*c^2) + ((Sqrt[b]*Sqrt[Pi/2]*C os[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/2 + (Sqrt[ b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/S qrt[b]])/2 - (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c* x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[ a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/(6*c^4)))/2
3.2.78.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(b*c)^(-1) Subst[Int[x^n*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcCos[c*x])^n/(m + 1)), x] + Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (b*c^(m + 1))^(-1) Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x , a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*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*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(547\) vs. \(2(241)=482\).
Time = 2.03 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.75
| method | result | size |
| default | \(\frac {-\sqrt {a +b \arccos \left (c x \right )}\, \sqrt {-\frac {3}{b}}\, \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sin \left (\frac {3 a}{b}\right ) \sqrt {\pi }\, \sqrt {2}\, b^{2}-27 \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-27 \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-\sqrt {a +b \arccos \left (c x \right )}\, \sqrt {-\frac {3}{b}}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}\, b^{2}+36 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+12 \arccos \left (c x \right )^{2} \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+72 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b +54 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+24 \arccos \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b +6 \arccos \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+36 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}+54 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b +12 \cos \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2}+6 \sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b}{144 c^{3} \sqrt {a +b \arccos \left (c x \right )}}\) | \(548\) |
1/144/c^3/(a+b*arccos(c*x))^(1/2)*(-(a+b*arccos(c*x))^(1/2)*(-3/b)^(1/2)*F resnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*sin(3*a /b)*Pi^(1/2)*2^(1/2)*b^2-27*(a+b*arccos(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1 /2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2) *2^(1/2)*b^2-27*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2) /(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*b^2 -(a+b*arccos(c*x))^(1/2)*(-3/b)^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/ 2)/(-3/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)*b^2+36*arccos( c*x)^2*cos(-(a+b*arccos(c*x))/b+a/b)*b^2+12*arccos(c*x)^2*cos(-3*(a+b*arcc os(c*x))/b+3*a/b)*b^2+72*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*a*b+54* arccos(c*x)*sin(-(a+b*arccos(c*x))/b+a/b)*b^2+24*arccos(c*x)*cos(-3*(a+b*a rccos(c*x))/b+3*a/b)*a*b+6*arccos(c*x)*sin(-3*(a+b*arccos(c*x))/b+3*a/b)*b ^2+36*cos(-(a+b*arccos(c*x))/b+a/b)*a^2+54*sin(-(a+b*arccos(c*x))/b+a/b)*a *b+12*cos(-3*(a+b*arccos(c*x))/b+3*a/b)*a^2+6*sin(-3*(a+b*arccos(c*x))/b+3 *a/b)*a*b)
Exception generated. \[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=\int x^{2} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 1967, normalized size of antiderivative = 6.28 \[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=\text {Too large to display} \]
-1/8*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a) /sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a /b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*sqrt(2)*sqrt(pi)*a *b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2) *sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)* sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a) *sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sq rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b )/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 1/4*I*sqrt(pi)*a^2*b^(3 /2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt( b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b ^3/abs(b))*c^3) + 1/12*sqrt(pi)*a*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arccos(c *x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e ^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/8*sqrt(2)*sqrt(p i)*a*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqr t(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b) ) + b*sqrt(abs(b)))*c^3) - 3/32*I*sqrt(2)*sqrt(pi)*b^3*erf(-1/2*I*sqrt(2)* sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) +...
Timed out. \[ \int x^2 (a+b \arccos (c x))^{3/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2} \,d x \]