3.120 \(\int \sqrt {c+d x} \cos ^2(a+b x) \sin (a+b x) \, dx\)

Optimal. Leaf size=304 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {\sqrt {c+d x} \cos (a+b x)}{4 b}-\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b} \]

[Out]

1/72*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^(1/2)*6^(1/2)*Pi^(1/2)/b^(3/2
)-1/72*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^(1/2)*6^(1/2)*Pi^(1/2)/b^(3
/2)+1/8*cos(a-b*c/d)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^(1/2)/b^(3/2)
-1/8*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*d^(1/2)*2^(1/2)*Pi^(1/2)/b^(3/2)-1/
4*cos(b*x+a)*(d*x+c)^(1/2)/b-1/12*cos(3*b*x+3*a)*(d*x+c)^(1/2)/b

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Rubi [A]  time = 0.42, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}-\frac {\sqrt {c+d x} \cos (a+b x)}{4 b}-\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x],x]

[Out]

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

Rule 3296

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 3304

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 3305

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 3306

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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4406

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*} \int \sqrt {c+d x} \cos ^2(a+b x) \sin (a+b x) \, dx &=\int \left (\frac {1}{4} \sqrt {c+d x} \sin (a+b x)+\frac {1}{4} \sqrt {c+d x} \sin (3 a+3 b x)\right ) \, dx\\ &=\frac {1}{4} \int \sqrt {c+d x} \sin (a+b x) \, dx+\frac {1}{4} \int \sqrt {c+d x} \sin (3 a+3 b x) \, dx\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{4 b}-\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}+\frac {d \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{24 b}+\frac {d \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{8 b}\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{4 b}-\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}+\frac {\left (d \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{24 b}+\frac {\left (d \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{8 b}-\frac {\left (d \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{24 b}-\frac {\left (d \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{8 b}\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{4 b}-\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{12 b}+\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b}-\frac {\sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{12 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b}\\ &=-\frac {\sqrt {c+d x} \cos (a+b x)}{4 b}-\frac {\sqrt {c+d x} \cos (3 a+3 b x)}{12 b}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{12 b^{3/2}}-\frac {\sqrt {d} \sqrt {\frac {\pi }{6}} S\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 )}{12 b^{3/2}}-\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{4 b^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 6.62, size = 278, normalized size = 0.91 \[ -\frac {-\sqrt {2 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+\sqrt {2 \pi } \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+2 \sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (3 (a+b x))}{24 \sqrt {3} b \sqrt {\frac {b}{d}}}+\frac {\sqrt {c+d x} e^{-\frac {i (a d+b c)}{d}} \left (-\frac {e^{2 i a} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]*Cos[a + b*x]^2*Sin[a + b*x],x]

[Out]

(Sqrt[c + d*x]*(-((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d]) - (E^(((2*I)*b*c)
/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(8*b*E^((I*(b*c + a*d))/d)) - (2*Sqrt[3]*Sqrt[b/d
]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]
] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d])/(24*Sqrt[3]*b*Sqrt[b/d])

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fricas [A]  time = 0.73, size = 235, normalized size = 0.77 \[ \frac {\sqrt {6} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 9 \, \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 9 \, \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - \sqrt {6} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 24 \, \sqrt {d x + c} b \cos \left (b x + a\right )^{3}}{72 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(sqrt(6)*pi*d*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) + 9*
sqrt(2)*pi*d*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 9*sqrt(2)*
pi*d*sqrt(b/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - sqrt(6)*pi*d*sqrt(
b/(pi*d))*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) - 24*sqrt(d*x + c)*b*cos(b*x
+ a)^3)/b^2

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giac [C]  time = 2.78, size = 842, normalized size = 2.77 \[ -\frac {-\frac {i \, \sqrt {6} \sqrt {\pi } {\left (6 \, b c + i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {3 i \, b c - 3 i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - \frac {9 i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c + i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {9 i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {i \, \sqrt {6} \sqrt {\pi } {\left (6 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-3 i \, b c + 3 i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + 6 \, {\left (\frac {i \, \sqrt {6} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {3 i \, b c - 3 i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} + \frac {3 i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {3 i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {i \, \sqrt {6} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {6} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-3 i \, b c + 3 i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}\right )} c + \frac {6 \, \sqrt {d x + c} d e^{\left (\frac {3 i \, {\left (d x + c\right )} b - 3 i \, b c + 3 i \, a d}{d}\right )}}{b} + \frac {18 \, \sqrt {d x + c} d e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b} + \frac {18 \, \sqrt {d x + c} d e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b} + \frac {6 \, \sqrt {d x + c} d e^{\left (\frac {-3 i \, {\left (d x + c\right )} b + 3 i \, b c - 3 i \, a d}{d}\right )}}{b}}{144 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(-I*sqrt(6)*sqrt(pi)*(6*b*c + I*d)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)
/d)*e^((3*I*b*c - 3*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - 9*I*sqrt(2)*sqrt(pi)*(2*b*c + I*d)*d*e
rf(-1/2*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/s
qrt(b^2*d^2) + 1)*b) + 9*I*sqrt(2)*sqrt(pi)*(2*b*c - I*d)*d*erf(-1/2*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/s
qrt(b^2*d^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + I*sqrt(6)*sqrt(pi)*(6*b
*c - I*d)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*I*a*d)/d)/
(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 6*(I*sqrt(6)*sqrt(pi)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I
*b*d/sqrt(b^2*d^2) + 1)/d)*e^((3*I*b*c - 3*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + 3*I*sqrt(2)*sqrt(
pi)*d*erf(-1/2*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(
I*b*d/sqrt(b^2*d^2) + 1)) - 3*I*sqrt(2)*sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d
^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) - I*sqrt(6)*sqrt(pi)*d*erf(-1/2*sqrt
(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b
^2*d^2) + 1)))*c + 6*sqrt(d*x + c)*d*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*I*a*d)/d)/b + 18*sqrt(d*x + c)*d*e^((I*
(d*x + c)*b - I*b*c + I*a*d)/d)/b + 18*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b + 6*sqrt(d*x +
c)*d*e^((-3*I*(d*x + c)*b + 3*I*b*c - 3*I*a*d)/d)/b)/d

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maple [A]  time = 0.03, size = 296, normalized size = 0.97 \[ \frac {-\frac {d \sqrt {d x +c}\, \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{4 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{72 b \sqrt {\frac {b}{d}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-1/8/b*d*(d*x+c)^(1/2)*cos(1/d*(d*x+c)*b+(a*d-b*c)/d)+1/16/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)*(d*x+c)^(1/2)*b/d)-sin((a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)
^(1/2)*(d*x+c)^(1/2)*b/d))-1/24/b*d*(d*x+c)^(1/2)*cos(3/d*(d*x+c)*b+3*(a*d-b*c)/d)+1/144/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)*(d*x+c)^(1/2)*b/d)-sin(3
*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)))

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maxima [C]  time = 0.50, size = 422, normalized size = 1.39 \[ -\frac {{\left (\frac {24 \, \sqrt {d x + c} b^{2} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} + \frac {72 \, \sqrt {d x + c} b^{2} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}{d} + {\left (\left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) + {\left (\left (9 i - 9\right ) \, \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (9 i + 9\right ) \, \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + {\left (-\left (9 i + 9\right ) \, \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (9 i - 9\right ) \, \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) + {\left (-\left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right )\right )} d}{288 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(24*sqrt(d*x + c)*b^2*cos(3*((d*x + c)*b - b*c + a*d)/d)/d + 72*sqrt(d*x + c)*b^2*cos(((d*x + c)*b - b*
c + a*d)/d)/d + ((I - 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (I + 1)*9^(1/4)*sq
rt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + ((9*I - 9)*sqrt(2)*
sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (9*I + 9)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)
/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + (-(9*I + 9)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (9*
I - 9)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) + (-(I + 1)*9^(
1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) - (I - 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(
1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)))*d/b^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\sqrt {c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c + d x} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x)*sin(a + b*x)*cos(a + b*x)**2, x)

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