\(\int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx\) [199]

Optimal result

Integrand size = 26, antiderivative size = 299 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\sqrt {d} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{3}} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{384 b^{3/2}}-\frac {3 \sqrt {d} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{128 b^{3/2}} \]

[Out]

-1/1152*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))*d^(1/2)*3^(1/2)*Pi^(1/2)/b
^(3/2)+1/1152*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)*d^(1/2)*3^(1/2)*Pi^(
1/2)/b^(3/2)+3/128*cos(2*a-2*b*c/d)*FresnelC(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*d^(1/2)*Pi^(1/2)/b^(3/2
)-3/128*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d)*d^(1/2)*Pi^(1/2)/b^(3/2)-3/64*cos(
2*b*x+2*a)*(d*x+c)^(1/2)/b+1/192*cos(6*b*x+6*a)*(d*x+c)^(1/2)/b

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b} \]

[In]

Int[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-3*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b) + (Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(192*b) - (Sqrt[d]*Sqrt[Pi/3]*Co
s[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(384*b^(3/2)) + (3*Sqrt[d]*Sqrt[Pi]
*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(3/2)) + (Sqrt[d]*Sqrt[Pi
/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(384*b^(3/2)) - (3*Sqrt[d]*Sq
rt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(3/2))

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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 3432

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 3433

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} \sqrt {c+d x} \sin (2 a+2 b x)-\frac {1}{32} \sqrt {c+d x} \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int \sqrt {c+d x} \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx \\ & = -\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {d \int \frac {\cos (6 a+6 b x)}{\sqrt {c+d x}} \, dx}{384 b}+\frac {(3 d) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{128 b} \\ & = -\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\left (d \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{384 b}+\frac {\left (3 d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{128 b}+\frac {\left (d \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{384 b}-\frac {\left (3 d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{128 b} \\ & = -\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\cos \left (6 a-\frac {6 b c}{d}\right ) \text {Subst}\left (\int \cos \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{192 b}+\frac {\left (3 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{64 b}+\frac {\sin \left (6 a-\frac {6 b c}{d}\right ) \text {Subst}\left (\int \sin \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{192 b}-\frac {\left (3 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{64 b} \\ & = -\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\sqrt {d} \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 )}{384 b^{3/2}}+\frac {3 \sqrt {d} \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 )}{128 b^{3/2}}+\frac {\sqrt {d} \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 )}{384 b^{3/2}}-\frac {3 \sqrt {d} \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 )}{128 b^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.86 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {e^{-\frac {6 i (b c+a d)}{d}} \sqrt {c+d x} \left (-27 e^{4 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )-27 e^{4 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )+\sqrt {3} \left (e^{12 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {6 i b (c+d x)}{d}\right )+e^{\frac {12 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {6 i b (c+d x)}{d}\right )\right )\right )}{1152 \sqrt {2} b \sqrt {\frac {b^2 (c+d x)^2}{d^2}}} \]

[In]

Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(Sqrt[c + d*x]*(-27*E^((4*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-2*I)*b*(c + d*x))/d] - 27*
E^((4*I)*a + ((8*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((2*I)*b*(c + d*x))/d] + Sqrt[3]*(E^((12*I)*
a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-6*I)*b*(c + d*x))/d] + E^(((12*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*
Gamma[3/2, ((6*I)*b*(c + d*x))/d])))/(1152*Sqrt[2]*b*E^(((6*I)*(b*c + a*d))/d)*Sqrt[(b^2*(c + d*x)^2)/d^2])

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {-\frac {3 d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {3 d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}+\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2304 b \sqrt {\frac {b}{d}}}}{d}\)	\(293\)
default	\(\frac {-\frac {3 d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {3 d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}+\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2304 b \sqrt {\frac {b}{d}}}}{d}\)	\(293\)

[In]

int((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2/d*(-3/128/b*d*(d*x+c)^(1/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+3/256/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)-sin(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)^(1/2)*cos(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/4608/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)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.81 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {\sqrt {3} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {3} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 27 \, \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 27 \, \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 48 \, {\left (4 \, b \cos \left (b x + a\right )^{6} - 6 \, b \cos \left (b x + a\right )^{4} + b\right )} \sqrt {d x + c}}{1152 \, b^{2}} \]

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/1152*(sqrt(3)*pi*d*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))
- sqrt(3)*pi*d*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) - 27*
pi*d*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d))) + 27*pi*d*sqrt(b/(pi*d))
*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) - 48*(4*b*cos(b*x + a)^6 - 6*b*cos(b*x + a)
^4 + b)*sqrt(d*x + c))/b^2

Sympy [F]

\[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int \sqrt {c + d x} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**(1/2)*cos(b*x+a)**3*sin(b*x+a)**3,x)

[Out]

Integral(sqrt(c + d*x)*sin(a + b*x)**3*cos(a + b*x)**3, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.47 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (\frac {48 \, \sqrt {d x + c} b^{2} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {432 \, \sqrt {d x + c} b^{2} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - {\left (-\left (i - 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {6 i \, b}{d}}\right ) - 27 \, {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) - 27 \, {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right ) - {\left (\left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {6 i \, b}{d}}\right )\right )} d}{9216 \, b^{3}} \]

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/9216*(48*sqrt(d*x + c)*b^2*cos(6*((d*x + c)*b - b*c + a*d)/d)/d - 432*sqrt(d*x + c)*b^2*cos(2*((d*x + c)*b -
b*c + a*d)/d)/d - (-(I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d) - (I + 1)*36^(1
/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) - 27*((I - 1)*4
^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) + (I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)
^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) - 27*(-(I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/
d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (I - 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*e
rf(sqrt(d*x + c)*sqrt(-2*I*b/d)) - ((I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d)
+ (I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*(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^3

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.17 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.76 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/2304*(sqrt(3)*sqrt(pi)*(12*b*c - I*d)*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) + sqrt(3)*sqrt(pi)*(12*b*c + I*d)*d*erf(I*sqr
t(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) - 12*(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)) + sqrt(3)*sqrt(pi)*d*erf(I*sqrt(3)*sqrt(b*d)*sqr
t(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 - 27*sqrt(pi)*(4*b*c - I*d)*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)*b) - 27*sqr
t(pi)*(4*b*c + I*d)*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)/(s
qrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 54*sqrt(d*x + c)*d*e^(-2*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 6*sqr
t(d*x + c)*d*e^(-6*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^(-2*(-I*(d*x + c)*b + I*b*c - I
*a*d)/d)/b + 6*sqrt(d*x + c)*d*e^(-6*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b)/d

Mupad [F(-1)]

Timed out. \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,\sqrt {c+d\,x} \,d x \]

[In]

int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^(1/2),x)

[Out]

int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^(1/2), x)