\(\int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx\) [61]

Optimal result

Integrand size = 18, antiderivative size = 292 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}-\frac {b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {b^{3/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}+\frac {4 b \cos ^2(a+b x) \sin (a+b x)}{d^2 \sqrt {c+d x}} \] Output:

-2/3*cos(b*x+a)^3/d/(d*x+c)^(3/2)-b^(3/2)*2^(1/2)*Pi^(1/2)*cos(a-b*c/d)*Fr 
esnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/d^(5/2)-b^(3/2)*6^( 
1/2)*Pi^(1/2)*cos(3*a-3*b*c/d)*FresnelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^( 
1/2)/d^(1/2))/d^(5/2)+b^(3/2)*6^(1/2)*Pi^(1/2)*FresnelS(b^(1/2)*6^(1/2)/Pi 
^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)/d^(5/2)+b^(3/2)*2^(1/2)*Pi^ 
(1/2)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d 
)/d^(5/2)+4*b*cos(b*x+a)^2*sin(b*x+a)/d^2/(d*x+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.20 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {-4 d \cos ^3(a+b x)-3 d e^{i \left (a-\frac {b c}{d}\right )} \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )-3 d e^{-i \left (a-\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )-3 \sqrt {3} d e^{3 i \left (a-\frac {b c}{d}\right )} \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 i b (c+d x)}{d}\right )-3 \sqrt {3} d e^{-3 i \left (a-\frac {b c}{d}\right )} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 i b (c+d x)}{d}\right )+24 b (c+d x) \cos ^2(a+b x) \sin (a+b x)}{6 d^2 (c+d x)^{3/2}} \] Input:

Integrate[Cos[a + b*x]^3/(c + d*x)^(5/2),x]
 

Output:

(-4*d*Cos[a + b*x]^3 - 3*d*E^(I*(a - (b*c)/d))*(((-I)*b*(c + d*x))/d)^(3/2 
)*Gamma[1/2, ((-I)*b*(c + d*x))/d] - (3*d*((I*b*(c + d*x))/d)^(3/2)*Gamma[ 
1/2, (I*b*(c + d*x))/d])/E^(I*(a - (b*c)/d)) - 3*Sqrt[3]*d*E^((3*I)*(a - ( 
b*c)/d))*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-3*I)*b*(c + d*x))/d] - 
 (3*Sqrt[3]*d*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((3*I)*b*(c + d*x))/d]) 
/E^((3*I)*(a - (b*c)/d)) + 24*b*(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(6* 
d^2*(c + d*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.52, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3042, 3795, 3042, 3787, 3042, 3785, 3786, 3793, 2009, 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.

\(\displaystyle \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{(c+d x)^{5/2}}dx\)

\(\Big \downarrow \) 3795

\(\displaystyle -\frac {12 b^2 \int \frac {\cos ^3(a+b x)}{\sqrt {c+d x}}dx}{d^2}+\frac {8 b^2 \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {8 b^2 \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{d^2}-\frac {12 b^2 \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3787

\(\displaystyle -\frac {12 b^2 \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{d^2}+\frac {8 b^2 \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {12 b^2 \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{d^2}+\frac {8 b^2 \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle -\frac {12 b^2 \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{d^2}+\frac {8 b^2 \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3786

\(\displaystyle -\frac {12 b^2 \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{\sqrt {c+d x}}dx}{d^2}+\frac {8 b^2 \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {12 b^2 \int \left (\frac {3 \cos (a+b x)}{4 \sqrt {c+d x}}+\frac {\cos (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{d^2}+\frac {8 b^2 \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8 b^2 \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d^2}-\frac {12 b^2 \left (\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {8 b^2 \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d^2}-\frac {12 b^2 \left (\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {12 b^2 \left (\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d^2}+\frac {8 b^2 \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d^2}+\frac {4 b \sin (a+b x) \cos ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \cos ^3(a+b x)}{3 d (c+d x)^{3/2}}\)

Input:

Int[Cos[a + b*x]^3/(c + d*x)^(5/2),x]
 

Output:

(-2*Cos[a + b*x]^3)/(3*d*(c + d*x)^(3/2)) - (12*b^2*((3*Sqrt[Pi/2]*Cos[a - 
 (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b] 
*Sqrt[d]) + (Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]* 
Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/6]*FresnelS[(Sqrt[ 
b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(2*Sqrt[b]*Sqr 
t[d]) - (3*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]] 
*Sin[a - (b*c)/d])/(2*Sqrt[b]*Sqrt[d])))/d^2 + (8*b^2*((Sqrt[2*Pi]*Cos[a - 
 (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*S 
qrt[d]) - (Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]] 
*Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])))/d^2 + (4*b*Cos[a + b*x]^2*Sin[a + b 
*x])/(d^2*Sqrt[c + d*x])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

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]
 

rule 3786

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]
 

rule 3787

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]
 

rule 3793

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]))
 

rule 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

rule 3832

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]
 

rule 3833

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]
 

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(\frac {-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}}{d}\)	\(368\)
default	\(\frac {-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}}{d}\)	\(368\)

Input:

int(cos(b*x+a)^3/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/d*(-1/4/(d*x+c)^(3/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)-1/2*b/d*(-1/(d*x+c)^( 
1/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)+b/d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos((a 
*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin((a 
*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))-1/12 
/(d*x+c)^(3/2)*cos(3*b*(d*x+c)/d+3*(a*d-b*c)/d)-1/2*b/d*(-1/(d*x+c)^(1/2)* 
sin(3*b*(d*x+c)/d+3*(a*d-b*c)/d)+b/d*2^(1/2)*Pi^(1/2)*3^(1/2)/(b/d)^(1/2)* 
(cos(3*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c 
)^(1/2)/d)-sin(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2 
)*b*(d*x+c)^(1/2)/d))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {3 \, \sqrt {6} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 3 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 3 \, \sqrt {6} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, {\left (d \cos \left (b x + a\right )^{3} - 6 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \] Input:

integrate(cos(b*x+a)^3/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

-1/3*(3*sqrt(6)*(pi*b*d^2*x^2 + 2*pi*b*c*d*x + pi*b*c^2)*sqrt(b/(pi*d))*co 
s(-3*(b*c - a*d)/d)*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) + 3* 
sqrt(2)*(pi*b*d^2*x^2 + 2*pi*b*c*d*x + pi*b*c^2)*sqrt(b/(pi*d))*cos(-(b*c 
- a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 3*sqrt(2)*(p 
i*b*d^2*x^2 + 2*pi*b*c*d*x + pi*b*c^2)*sqrt(b/(pi*d))*fresnel_sin(sqrt(2)* 
sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - 3*sqrt(6)*(pi*b*d^2*x^ 
2 + 2*pi*b*c*d*x + pi*b*c^2)*sqrt(b/(pi*d))*fresnel_sin(sqrt(6)*sqrt(d*x + 
 c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 2*(d*cos(b*x + a)^3 - 6*(b*d*x 
 + b*c)*cos(b*x + a)^2*sin(b*x + a))*sqrt(d*x + c))/(d^4*x^2 + 2*c*d^3*x + 
 c^2*d^2)
 

Sympy [F]

\[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cos(b*x+a)**3/(d*x+c)**(5/2),x)
 

Output:

Integral(cos(a + b*x)**3/(c + d*x)**(5/2), x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {3 \, {\left (\sqrt {3} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} - {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}}\right )}}{16 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \] Input:

integrate(cos(b*x+a)^3/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

-3/16*(sqrt(3)*(((I - 1)*sqrt(2)*gamma(-3/2, 3*I*(d*x + c)*b/d) - (I + 1)* 
sqrt(2)*gamma(-3/2, -3*I*(d*x + c)*b/d))*cos(-3*(b*c - a*d)/d) + ((I + 1)* 
sqrt(2)*gamma(-3/2, 3*I*(d*x + c)*b/d) - (I - 1)*sqrt(2)*gamma(-3/2, -3*I* 
(d*x + c)*b/d))*sin(-3*(b*c - a*d)/d))*((d*x + c)*b/d)^(3/2) - ((-(I - 1)* 
sqrt(2)*gamma(-3/2, I*(d*x + c)*b/d) + (I + 1)*sqrt(2)*gamma(-3/2, -I*(d*x 
 + c)*b/d))*cos(-(b*c - a*d)/d) + (-(I + 1)*sqrt(2)*gamma(-3/2, I*(d*x + c 
)*b/d) + (I - 1)*sqrt(2)*gamma(-3/2, -I*(d*x + c)*b/d))*sin(-(b*c - a*d)/d 
))*((d*x + c)*b/d)^(3/2))/((d*x + c)^(3/2)*d)
 

Giac [F]

\[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int { \frac {\cos \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(b*x+a)^3/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

integrate(cos(b*x + a)^3/(d*x + c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(cos(a + b*x)^3/(c + d*x)^(5/2),x)
 

Output:

int(cos(a + b*x)^3/(c + d*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\cos \left (b x +a \right )^{3}}{\sqrt {d x +c}\, c^{2}+2 \sqrt {d x +c}\, c d x +\sqrt {d x +c}\, d^{2} x^{2}}d x \] Input:

int(cos(b*x+a)^3/(d*x+c)^(5/2),x)
 

Output:

int(cos(a + b*x)**3/(sqrt(c + d*x)*c**2 + 2*sqrt(c + d*x)*c*d*x + sqrt(c + 
 d*x)*d**2*x**2),x)