Integrand size = 27, antiderivative size = 373 \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}+\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 b^{3/2} c^2}+\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {6 a}{b}\right )}{8 b^{3/2} c^2} \] Output:
-2*d^2*x*(-c^2*x^2+1)^(5/2)/b/c/(a+b*arccos(c*x))^(1/2)+1/2*d^2*2^(1/2)*Pi ^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1 /2))/b^(3/2)/c^2+1/8*d^2*3^(1/2)*Pi^(1/2)*cos(6*a/b)*FresnelC(2*3^(1/2)/Pi ^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^2+5/8*d^2*Pi^(1/2)*cos(2 *a/b)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2))/b^(3/2)/c^2+5/8 *d^2*Pi^(1/2)*FresnelS(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a /b)/b^(3/2)/c^2+1/2*d^2*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*(a+b* arccos(c*x))^(1/2)/b^(1/2))*sin(4*a/b)/b^(3/2)/c^2+1/8*d^2*3^(1/2)*Pi^(1/2 )*FresnelS(2*3^(1/2)/Pi^(1/2)*(a+b*arccos(c*x))^(1/2)/b^(1/2))*sin(6*a/b)/ b^(3/2)/c^2
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx \] Input:
Integrate[(x*(d - c^2*d*x^2)^2)/(a + b*ArcCos[c*x])^(3/2),x]
Output:
Integrate[(x*(d - c^2*d*x^2)^2)/(a + b*ArcCos[c*x])^(3/2), x]
Time = 2.20 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.62, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5215, 5169, 3042, 3793, 2009, 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.
| \(\displaystyle \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5215 |
| \(\displaystyle -\frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b c}+\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 5169 |
| \(\displaystyle \frac {2 d^2 \int \frac {\sin ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 d^2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )^4}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {2 d^2 \int \left (\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arccos (c x))}{b}\right )}{8 \sqrt {a+b \arccos (c x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{2 \sqrt {a+b \arccos (c x)}}+\frac {3}{8 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{b^2 c^2}+\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arccos (c x)}}dx}{b}+\frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2 c^2}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 5225 |
| \(\displaystyle -\frac {12 d^2 \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{\sqrt {a+b \arccos (c x)}}d(a+b \arccos (c x))}{b^2 c^2}+\frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2 c^2}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle -\frac {12 d^2 \int \left (\frac {\cos \left (\frac {6 a}{b}-\frac {6 (a+b \arccos (c x))}{b}\right )}{32 \sqrt {a+b \arccos (c x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arccos (c x))}{b}\right )}{16 \sqrt {a+b \arccos (c x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{32 \sqrt {a+b \arccos (c x)}}+\frac {1}{16 \sqrt {a+b \arccos (c x)}}\right )d(a+b \arccos (c x))}{b^2 c^2}+\frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2 c^2}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arccos (c x)}\right )}{b^2 c^2}-\frac {12 d^2 \left (-\frac {1}{16} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{32} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arccos (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{16} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \sin \left (\frac {6 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \arccos (c x)}\right )}{b^2 c^2}+\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arccos (c x)}}\) |
Input:
Int[(x*(d - c^2*d*x^2)^2)/(a + b*ArcCos[c*x])^(3/2),x]
Output:
(2*d^2*x*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcCos[c*x]]) + (2*d^2*((3*S qrt[a + b*ArcCos[c*x]])/4 + (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*S qrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Cos[(2* a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/2 - (Sqrt[ b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[( 2*a)/b])/2 + (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[ c*x]])/Sqrt[b]]*Sin[(4*a)/b])/8))/(b^2*c^2) - (12*d^2*(Sqrt[a + b*ArcCos[c *x]]/8 - (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/16 + (Sqrt[b]*Sqrt[Pi/3]*Cos[(6*a)/b]*FresnelC[( 2*Sqrt[3/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/32 - (Sqrt[b]*Sqrt[Pi]*Cos [(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/32 - ( Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]* Sin[(2*a)/b])/32 - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*A rcCos[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/16 + (Sqrt[b]*Sqrt[Pi/3]*FresnelS[(2*S qrt[3/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(6*a)/b])/32))/(b^2*c^2)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
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_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(-(b*c)^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Subst[ Int[x^n*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, 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 = 0.64 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(-\frac {d^{2} \left (\sqrt {-\frac {6}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right )-\sqrt {-\frac {6}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {6 a}{b}\right ) \operatorname {FresnelS}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right )-8 \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \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}+8 \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \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}+10 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-10 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+5 \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )-4 \sin \left (-\frac {4 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {4 a}{b}\right )+\sin \left (-\frac {6 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {6 a}{b}\right )\right )}{16 c^{2} b \sqrt {a +b \arccos \left (c x \right )}}\) | \(449\) |
Input:
int(x*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/16*d^2/c^2/b*((-6/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos (6*a/b)*FresnelC(6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b )-(-6/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(6*a/b)*Fresnel S(6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)-8*(a+b*arccos (c*x))^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcc os(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)+8*(a+b*arccos(c*x))^(1/2)* sin(4*a/b)*FresnelS(2*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)+10*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c* x))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos( c*x))^(1/2)/b)-10*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arccos(c*x))^(1/2)*sin(2*a/b) *FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)+5*sin (-2*(a+b*arccos(c*x))/b+2*a/b)-4*sin(-4*(a+b*arccos(c*x))/b+4*a/b)+sin(-6* (a+b*arccos(c*x))/b+6*a/b))/(a+b*arccos(c*x))^(1/2)
Exception generated. \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x, algorithm="fricas" )
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {acos}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acos}{\left (c x \right )}} \operatorname {acos}{\left (c x \right )}}\, dx\right ) \] Input:
integrate(x*(-c**2*d*x**2+d)**2/(a+b*acos(c*x))**(3/2),x)
Output:
d**2*(Integral(x/(a*sqrt(a + b*acos(c*x)) + b*sqrt(a + b*acos(c*x))*acos(c *x)), x) + Integral(-2*c**2*x**3/(a*sqrt(a + b*acos(c*x)) + b*sqrt(a + b*a cos(c*x))*acos(c*x)), x) + Integral(c**4*x**5/(a*sqrt(a + b*acos(c*x)) + b *sqrt(a + b*acos(c*x))*acos(c*x)), x))
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(-c^2*d*x^2+d)^2/(a+b*arccos(c*x))^(3/2),x, algorithm="maxima" )
Output:
integrate((c^2*d*x^2 - d)^2*x/(b*arccos(c*x) + a)^(3/2), x)
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(-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/(b*arccos(c*x) + a)^(3/2), x)
Timed out. \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=\int \frac {x\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int((x*(d - c^2*d*x^2)^2)/(a + b*acos(c*x))^(3/2),x)
Output:
int((x*(d - c^2*d*x^2)^2)/(a + b*acos(c*x))^(3/2), x)
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arccos (c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{5}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x^{3}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {acos} \left (c x \right ) b +a}\, x}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:
int(x*(-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**5)/(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**3)/(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)/(acos(c* x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x))