Integrand size = 26, antiderivative size = 407 \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{3}} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{18432 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2048 b^{7/2}}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2} \] Output:
45/1024*d^2*(d*x+c)^(1/2)*cos(2*b*x+2*a)/b^3-3/64*(d*x+c)^(5/2)*cos(2*b*x+ 2*a)/b-5/9216*d^2*(d*x+c)^(1/2)*cos(6*b*x+6*a)/b^3+1/192*(d*x+c)^(5/2)*cos (6*b*x+6*a)/b+5/55296*d^(5/2)*3^(1/2)*Pi^(1/2)*cos(6*a-6*b*c/d)*FresnelC(2 *b^(1/2)*3^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(7/2)-45/2048*d^(5/2)*P i^(1/2)*cos(2*a-2*b*c/d)*FresnelC(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2) )/b^(7/2)-5/55296*d^(5/2)*3^(1/2)*Pi^(1/2)*FresnelS(2*b^(1/2)*3^(1/2)/Pi^( 1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(6*a-6*b*c/d)/b^(7/2)+45/2048*d^(5/2)*Pi^(1 /2)*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d)/b^ (7/2)+15/256*d*(d*x+c)^(3/2)*sin(2*b*x+2*a)/b^2-5/2304*d*(d*x+c)^(3/2)*sin (6*b*x+6*a)/b^2
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.63 \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {e^{-\frac {6 i (b c+a d)}{d}} (c+d x)^{5/2} \left (243 e^{4 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {2 i b (c+d x)}{d}\right )+243 e^{4 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {2 i b (c+d x)}{d}\right )-\sqrt {3} \left (e^{12 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {6 i b (c+d x)}{d}\right )+e^{\frac {12 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {6 i b (c+d x)}{d}\right )\right )\right )}{41472 \sqrt {2} b \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \] Input:
Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]
Output:
((c + d*x)^(5/2)*(243*E^((4*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Ga mma[7/2, ((-2*I)*b*(c + d*x))/d] + 243*E^((4*I)*a + ((8*I)*b*c)/d)*Sqrt[(( -I)*b*(c + d*x))/d]*Gamma[7/2, ((2*I)*b*(c + d*x))/d] - Sqrt[3]*(E^((12*I) *a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[7/2, ((-6*I)*b*(c + d*x))/d] + E^(((12*I )*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[7/2, ((6*I)*b*(c + d*x))/d])))/ (41472*Sqrt[2]*b*E^(((6*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^(3/2))
Time = 0.93 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {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.
| \(\displaystyle \int (c+d x)^{5/2} \sin ^3(a+b x) \cos ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {3}{32} (c+d x)^{5/2} \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^{5/2} \sin (6 a+6 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}+\frac {45 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}\) |
Input:
Int[(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]
Output:
(45*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(1024*b^3) - (3*(c + d*x)^(5/2)*Co s[2*a + 2*b*x])/(64*b) - (5*d^2*Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(9216*b^3) + ((c + d*x)^(5/2)*Cos[6*a + 6*b*x])/(192*b) + (5*d^(5/2)*Sqrt[Pi/3]*Cos[ 6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/( 18432*b^(7/2)) - (45*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqr t[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2048*b^(7/2)) - (5*d^(5/2)*Sqrt[ Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6* b*c)/d])/(18432*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2048*b^(7/2)) + (15*d* (c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(256*b^2) - (5*d*(c + d*x)^(3/2)*Sin[6*a + 6*b*x])/(2304*b^2)
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]
Time = 22.67 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{192 b}}{d}\) | \(477\) |
| default | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{192 b}}{d}\) | \(477\) |
Input:
int((d*x+c)^(5/2)*cos(b*x+a)^3*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
2/d*(-3/128/b*d*(d*x+c)^(5/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+15/128/b*d* (1/4/b*d*(d*x+c)^(3/2)*sin(2*b/d*(d*x+c)+2*(a*d-b*c)/d)-3/4/b*d*(-1/4/b*d* (d*x+c)^(1/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+1/8/b*d*Pi^(1/2)/(b/d)^(1/2 )*(cos(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-s in(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))+1/ 384/b*d*(d*x+c)^(5/2)*cos(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-5/384/b*d*(1/12/b*d *(d*x+c)^(3/2)*sin(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/4/b*d*(-1/12/b*d*(d*x+c) ^(1/2)*cos(6*b/d*(d*x+c)+6*(a*d-b*c)/d)+1/144/b*d*2^(1/2)*Pi^(1/2)*6^(1/2) /(b/d)^(1/2)*(cos(6*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^( 1/2)*b*(d*x+c)^(1/2)/d)-sin(6*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*6^(1/ 2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))))
Time = 0.11 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.09 \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {5 \, \sqrt {3} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 5 \, \sqrt {3} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 1215 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 96 \, {\left (24 \, b^{3} d^{2} x^{2} + 2 \, {\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{6} + 48 \, b^{3} c d x + 24 \, b^{3} c^{2} + 45 \, b d^{2} \cos \left (b x + a\right )^{2} - 3 \, {\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 25 \, b d^{2} - 20 \, {\left (2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{5} - 2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{55296 \, b^{4}} \] Input:
integrate((d*x+c)^(5/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="fricas")
Output:
1/55296*(5*sqrt(3)*pi*d^3*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_cos (2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 5*sqrt(3)*pi*d^3*sqrt(b/(pi*d)) *fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) - 1215*pi*d^3*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d))) + 1215*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) + 96*(24*b^3*d^2*x^2 + 2*(48*b^ 3*d^2*x^2 + 96*b^3*c*d*x + 48*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^6 + 48*b^3*c *d*x + 24*b^3*c^2 + 45*b*d^2*cos(b*x + a)^2 - 3*(48*b^3*d^2*x^2 + 96*b^3*c *d*x + 48*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^4 - 25*b*d^2 - 20*(2*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^5 - 2*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 - 3*(b^ 2*d^2*x + b^2*c*d)*cos(b*x + a))*sin(b*x + a))*sqrt(d*x + c))/b^4
Timed out. \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/2)*cos(b*x+a)**3*sin(b*x+a)**3,x)
Output:
Timed out
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.38 \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="maxima")
Output:
-1/442368*(960*(d*x + c)^(3/2)*b^3*sin(6*((d*x + c)*b - b*c + a*d)/d) - 25 920*(d*x + c)^(3/2)*b^3*sin(2*((d*x + c)*b - b*c + a*d)/d) - 48*(48*(d*x + c)^(5/2)*b^4/d - 5*sqrt(d*x + c)*b^2*d)*cos(6*((d*x + c)*b - b*c + a*d)/d ) + 1296*(16*(d*x + c)^(5/2)*b^4/d - 15*sqrt(d*x + c)*b^2*d)*cos(2*((d*x + c)*b - b*c + a*d)/d) - 5*(-(I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d ^2)^(1/4)*cos(-6*(b*c - a*d)/d) - (I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d^2* (b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) - 1215*((I - 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) + (I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-2*( b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) - 1215*(-(I + 1)*4^(1/4)*s qrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (I - 1)*4^(1 /4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt (d*x + c)*sqrt(-2*I*b/d)) - 5*((I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^ 2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d) + (I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d ^2*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-6*I*b/d) ))*d/b^5
Result contains complex when optimal does not.
Time = 2.44 (sec) , antiderivative size = 2411, normalized size of antiderivative = 5.92 \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="giac")
Output:
-1/110592*(576*(sqrt(3)*sqrt(pi)*d*erf(-I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)* (I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sq rt(b^2*d^2) + 1)) + sqrt(3)*sqrt(pi)*d*erf(I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I* b*d/sqrt(b^2*d^2) + 1)) - 9*sqrt(pi)*d*erf(-I*sqrt(b*d)*sqrt(d*x + c)*(I*b *d/sqrt(b^2*d^2) + 1)/d)*e^(-2*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b ^2*d^2) + 1)) - 9*sqrt(pi)*d*erf(I*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^ 2*d^2) + 1)/d)*e^(-2*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)))*c^3 - d^3*((sqrt(3)*sqrt(pi)*(576*b^3*c^3 - 144*I*b^2*c^2*d - 36*b* c*d^2 + 5*I*d^3)*d*erf(-I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2* d^2) + 1)/d)*e^(-6*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1) *b^3) + 6*I*(-48*I*(d*x + c)^(5/2)*b^2*d + 144*I*(d*x + c)^(3/2)*b^2*c*d - 144*I*sqrt(d*x + c)*b^2*c^2*d + 20*(d*x + c)^(3/2)*b*d^2 - 36*sqrt(d*x + c)*b*c*d^2 + 5*I*sqrt(d*x + c)*d^3)*e^(-6*(-I*(d*x + c)*b + I*b*c - I*a*d) /d)/b^3)/d^3 + (sqrt(3)*sqrt(pi)*(576*b^3*c^3 + 144*I*b^2*c^2*d - 36*b*c*d ^2 - 5*I*d^3)*d*erf(I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2 ) + 1)/d)*e^(-6*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)* b^3) + 6*I*(-48*I*(d*x + c)^(5/2)*b^2*d + 144*I*(d*x + c)^(3/2)*b^2*c*d - 144*I*sqrt(d*x + c)*b^2*c^2*d - 20*(d*x + c)^(3/2)*b*d^2 + 36*sqrt(d*x + c )*b*c*d^2 + 5*I*sqrt(d*x + c)*d^3)*e^(-6*(I*(d*x + c)*b - I*b*c + I*a*d...
Timed out. \[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \] Input:
int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^(5/2),x)
Output:
int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^(5/2), x)
\[ \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\left (\int \sqrt {d x +c}\, \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{3} x^{2}d x \right ) d^{2}+2 \left (\int \sqrt {d x +c}\, \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{3} x d x \right ) c d +\left (\int \sqrt {d x +c}\, \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{3}d x \right ) c^{2} \] Input:
int((d*x+c)^(5/2)*cos(b*x+a)^3*sin(b*x+a)^3,x)
Output:
int(sqrt(c + d*x)*cos(a + b*x)**3*sin(a + b*x)**3*x**2,x)*d**2 + 2*int(sqr t(c + d*x)*cos(a + b*x)**3*sin(a + b*x)**3*x,x)*c*d + int(sqrt(c + d*x)*co s(a + b*x)**3*sin(a + b*x)**3,x)*c**2