Integrand size = 29, antiderivative size = 511 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=-\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}-\frac {5 d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {d^2 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {3 d^2 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {d^2 \sqrt {\frac {7 \pi }{2}} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {5 d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 b^{3/2} c^3}-\frac {d^2 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^{3/2} c^3}-\frac {3 d^2 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^{3/2} c^3}-\frac {d^2 \sqrt {\frac {7 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {7 a}{b}\right )}{16 b^{3/2} c^3} \] Output:
-2*d^2*x^2*(-c^2*x^2+1)^(5/2)/b/c/(a+b*arccos(c*x))^(1/2)-5/32*d^2*2^(1/2) *Pi^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/ 2))/b^(3/2)/c^3+1/32*d^2*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^( 1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+3/32*d^2*10^(1/2)*Pi^(1/ 2)*cos(5*a/b)*FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/ b^(3/2)/c^3+1/32*d^2*14^(1/2)*Pi^(1/2)*cos(7*a/b)*FresnelS(14^(1/2)/Pi^(1/ 2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^3+5/32*d^2*2^(1/2)*Pi^(1/2)* FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(3/2 )/c^3-1/32*d^2*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arccos(c*x) )^(1/2)/b^(1/2))*sin(3*a/b)/b^(3/2)/c^3-3/32*d^2*10^(1/2)*Pi^(1/2)*Fresnel C(10^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(5*a/b)/b^(3/2)/c^ 3-1/32*d^2*14^(1/2)*Pi^(1/2)*FresnelC(14^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^ (1/2)/b^(1/2))*sin(7*a/b)/b^(3/2)/c^3
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\frac {d^2 e^{-\frac {7 i a}{b}} \left (10 e^{\frac {7 i a}{b}} \sqrt {1-c^2 x^2}+5 i e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arccos (c x))}{b}\right )-5 i e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arccos (c x))}{b}\right )+i \sqrt {3} e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arccos (c x))}{b}\right )-i \sqrt {3} e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arccos (c x))}{b}\right )-3 i \sqrt {5} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {5 i (a+b \arccos (c x))}{b}\right )+3 i \sqrt {5} e^{\frac {12 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {5 i (a+b \arccos (c x))}{b}\right )+i \sqrt {7} \sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {7 i (a+b \arccos (c x))}{b}\right )-i \sqrt {7} e^{\frac {14 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {7 i (a+b \arccos (c x))}{b}\right )+2 e^{\frac {7 i a}{b}} \sin (3 \arccos (c x))-6 e^{\frac {7 i a}{b}} \sin (5 \arccos (c x))+2 e^{\frac {7 i a}{b}} \sin (7 \arccos (c x))\right )}{64 b c^3 \sqrt {a+b \arccos (c x)}} \] Input:
Integrate[(x^2*(d - c^2*d*x^2)^2)/(a + b*ArcCos[c*x])^(3/2),x]
Output:
(d^2*(10*E^(((7*I)*a)/b)*Sqrt[1 - c^2*x^2] + (5*I)*E^(((6*I)*a)/b)*Sqrt[(( -I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] - (5* I)*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*Ar cCos[c*x]))/b] + I*Sqrt[3]*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x])) /b]*Gamma[1/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] - I*Sqrt[3]*E^(((10*I)*a)/b )*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcCos[c*x]))/b ] - (3*I)*Sqrt[5]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma [1/2, ((-5*I)*(a + b*ArcCos[c*x]))/b] + (3*I)*Sqrt[5]*E^(((12*I)*a)/b)*Sqr t[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcCos[c*x]))/b] + I *Sqrt[7]*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-7*I)*(a + b*ArcC os[c*x]))/b] - I*Sqrt[7]*E^(((14*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]* Gamma[1/2, ((7*I)*(a + b*ArcCos[c*x]))/b] + 2*E^(((7*I)*a)/b)*Sin[3*ArcCos [c*x]] - 6*E^(((7*I)*a)/b)*Sin[5*ArcCos[c*x]] + 2*E^(((7*I)*a)/b)*Sin[7*Ar cCos[c*x]]))/(64*b*c^3*E^(((7*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]])
Time = 1.92 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.57, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {5215, 5225, 4906, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5215 |
| \(\displaystyle -\frac {4 d^2 \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b c}+\frac {14 c d^2 \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 5225 |
| \(\displaystyle -\frac {14 d^2 \int \frac {\cos ^3\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}+\frac {4 d^2 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^3}+\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle -\frac {14 d^2 \int \left (\frac {\cos \left (\frac {7 a}{b}-\frac {7 (a+b \arccos (c x))}{b}\right )}{64 \sqrt {a+b \arccos (c x)}}-\frac {\cos \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{64 \sqrt {a+b \arccos (c x)}}-\frac {3 \cos \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{64 \sqrt {a+b \arccos (c x)}}+\frac {3 \cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{64 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{b^2 c^3}+\frac {4 d^2 \int \left (\frac {\cos \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{16 \sqrt {a+b \arccos (c x)}}-\frac {3 \cos \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{16 \sqrt {a+b \arccos (c x)}}+\frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{8 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{b^2 c^3}+\frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d^2 x^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}+\frac {4 d^2 \left (\frac {1}{4} \sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \cos \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}{8} \sqrt {b} \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )-\frac {1}{8} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )+\frac {1}{8} \sqrt {b} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )\right )}{b^2 c^3}-\frac {14 d^2 \left (\frac {3}{32} \sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \cos \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}{32} \sqrt {b} \sqrt {\frac {\pi }{10}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{14}} \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{32} \sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )-\frac {1}{32} \sqrt {b} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )-\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )+\frac {1}{32} \sqrt {b} \sqrt {\frac {\pi }{14}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {14}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {7 a}{b}\right )\right )}{b^2 c^3}\) |
Input:
Int[(x^2*(d - c^2*d*x^2)^2)/(a + b*ArcCos[c*x])^(3/2),x]
Output:
(2*d^2*x^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcCos[c*x]]) + (4*d^2*((S qrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/S qrt[b]])/4 - (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqr t[a + b*ArcCos[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi/10]*Cos[(5*a)/b]*Fres nelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi/ 2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/4 - (S qrt[b]*Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b ]]*Sin[(3*a)/b])/8 + (Sqrt[b]*Sqrt[Pi/10]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b *ArcCos[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/8))/(b^2*c^3) - (14*d^2*((3*Sqrt[b]* Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]] )/32 - (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/32 - (Sqrt[b]*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelC[ (Sqrt[10/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi/14]* Cos[(7*a)/b]*FresnelC[(Sqrt[14/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/32 + (3*Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[ b]]*Sin[a/b])/32 - (Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b *ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/32 - (Sqrt[b]*Sqrt[Pi/10]*FresnelS[( Sqrt[10/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/32 + (Sqrt[b]* Sqrt[Pi/14]*FresnelS[(Sqrt[14/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(7 *a)/b])/32))/(b^2*c^3)
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_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(f*x)^m)*Sqrt[1 - c^2*x^2]*(d + e*x^2) ^p*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f*( n + 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}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.79 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(-\frac {d^{2} \left (5 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \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 )-5 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \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 )-3 \sqrt {-\frac {5}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )+3 \sqrt {-\frac {5}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )+\sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {7}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {7 a}{b}\right ) \operatorname {FresnelC}\left (\frac {7 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {7}{b}}\, b}\right )-\sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {7}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {7 a}{b}\right ) \operatorname {FresnelS}\left (\frac {7 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {7}{b}}\, b}\right )+\sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arccos \left (c x \right )}-\sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sin \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 {a +b \arccos \left (c x \right )}+5 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right )+\sin \left (-\frac {3 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )-3 \sin \left (-\frac {5 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {5 a}{b}\right )+\sin \left (-\frac {7 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {7 a}{b}\right )\right )}{32 c^{3} b \sqrt {a +b \arccos \left (c x \right )}}\) | \(592\) |
Input:
int(x^2*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/32*d^2/c^3/b*(5*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*c os(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)- 5*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelS( 2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)-3*(-5/b)^(1/2)*Pi ^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(5*a/b)*FresnelC(5*2^(1/2)/Pi^(1 /2)/(-5/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)+3*(-5/b)^(1/2)*Pi^(1/2)*2^(1/2 )*(a+b*arccos(c*x))^(1/2)*sin(5*a/b)*FresnelS(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1 /2)*(a+b*arccos(c*x))^(1/2)/b)+Pi^(1/2)*2^(1/2)*(-7/b)^(1/2)*(a+b*arccos(c *x))^(1/2)*cos(7*a/b)*FresnelC(7*2^(1/2)/Pi^(1/2)/(-7/b)^(1/2)*(a+b*arccos (c*x))^(1/2)/b)-Pi^(1/2)*2^(1/2)*(-7/b)^(1/2)*(a+b*arccos(c*x))^(1/2)*sin( 7*a/b)*FresnelS(7*2^(1/2)/Pi^(1/2)/(-7/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b) +(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/ b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(a+b*arccos(c*x))^(1/2)-(-3/b)^(1/2)*P i^(1/2)*2^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*a rccos(c*x))^(1/2)/b)*(a+b*arccos(c*x))^(1/2)+5*sin(-(a+b*arccos(c*x))/b+a/ b)+sin(-3*(a+b*arccos(c*x))/b+3*a/b)-3*sin(-5*(a+b*arccos(c*x))/b+5*a/b)+s in(-7*(a+b*arccos(c*x))/b+7*a/b))/(a+b*arccos(c*x))^(1/2)
Exception generated. \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x, algorithm="frica s")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{4}}{a \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{6}}{a \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\, dx\right ) \] Input:
integrate(x**2*(-c**2*d*x**2+d)**2/(a+b*acos(c*x))**(3/2),x)
Output:
d**2*(Integral(x**2/(a*sqrt(a + b*acos(c*x)) + b*sqrt(a + b*acos(c*x))*aco s(c*x)), x) + Integral(-2*c**2*x**4/(a*sqrt(a + b*acos(c*x)) + b*sqrt(a + b*acos(c*x))*acos(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*acos(c*x)) + b*sqrt(a + b*acos(c*x))*acos(c*x)), x))
\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x, algorithm="maxim a")
Output:
integrate((c^2*d*x^2 - d)^2*x^2/(b*arccos(c*x) + a)^(3/2), x)
\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x, algorithm="giac" )
Output:
integrate((c^2*d*x^2 - d)^2*x^2/(b*arccos(c*x) + a)^(3/2), x)
Timed out. \[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {x^2\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int((x^2*(d - c^2*d*x^2)^2)/(a + b*acos(c*x))^(3/2),x)
Output:
int((x^2*(d - c^2*d*x^2)^2)/(a + b*acos(c*x))^(3/2), x)
\[ \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{6}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{4}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{2}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:
int(x^2*(-c^2*d*x^2+d)^2/(a+b*acos(c*x))^(3/2),x)
Output:
d**2*(int((sqrt(acos(c*x)*b + a)*x**6)/(acos(c*x)**2*b**2 + 2*acos(c*x)*a* b + a**2),x)*c**4 - 2*int((sqrt(acos(c*x)*b + a)*x**4)/(acos(c*x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x)*c**2 + int((sqrt(acos(c*x)*b + a)*x**2)/(acos (c*x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x))