Integrand size = 12, antiderivative size = 83 \[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt {b} d} \] Output:
1/2*2^(1/2)*Pi^(1/2)*cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c))/b^( 1/2)/d+1/2*2^(1/2)*Pi^(1/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c))*sin (a)/b^(1/2)/d
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (\cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+\operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)\right )}{\sqrt {b} d} \] Input:
Integrate[Sin[a + b*(c + d*x)^2],x]
Output:
(Sqrt[Pi/2]*(Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)] + FresnelC[Sqrt [b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a]))/(Sqrt[b]*d)
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3834, 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.
\(\displaystyle \int \sin \left (a+b (c+d x)^2\right ) \, dx\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle \sin (a) \int \cos \left (b (c+d x)^2\right )dx+\cos (a) \int \sin \left (b (c+d x)^2\right )dx\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \sin (a) \int \cos \left (b (c+d x)^2\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d}\) |
Input:
Int[Sin[a + b*(c + d*x)^2],x]
Output:
(Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d) + ( Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*d)
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]
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]
Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Sin[c] In t[Cos[d*(e + f*x)^2], x], x] + Simp[Cos[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {i \sqrt {\pi }\, {\mathrm e}^{-i a} \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}+\frac {i \sqrt {\pi }\, {\mathrm e}^{i a} \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d \sqrt {-i b}}\) | \(87\) |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{2 \sqrt {d^{2} b}}\) | \(136\) |
Input:
int(sin(a+b*(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/4*I*Pi^(1/2)*exp(-I*a)/d/(I*b)^(1/2)*erf(d*(I*b)^(1/2)*x+I*b*c/(I*b)^(1/ 2))+1/4*I*Pi^(1/2)*exp(I*a)/d/(-I*b)^(1/2)*erf(-d*(-I*b)^(1/2)*x+I*b*c/(-I *b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07 \[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} \cos \left (a\right ) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + \sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) \sin \left (a\right )}{2 \, b d^{2}} \] Input:
integrate(sin(a+b*(d*x+c)^2),x, algorithm="fricas")
Output:
1/2*(sqrt(2)*pi*sqrt(b*d^2/pi)*cos(a)*fresnel_sin(sqrt(2)*sqrt(b*d^2/pi)*( d*x + c)/d) + sqrt(2)*pi*sqrt(b*d^2/pi)*fresnel_cos(sqrt(2)*sqrt(b*d^2/pi) *(d*x + c)/d)*sin(a))/(b*d^2)
\[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\int \sin {\left (a + b \left (c + d x\right )^{2} \right )}\, dx \] Input:
integrate(sin(a+b*(d*x+c)**2),x)
Output:
Integral(sin(a + b*(c + d*x)**2), x)
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {i \, b}}\right ) + {\left (-\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\frac {i \, b d x + i \, b c}{\sqrt {-i \, b}}\right )\right )}}{8 \, \sqrt {b} d} \] Input:
integrate(sin(a+b*(d*x+c)^2),x, algorithm="maxima")
Output:
-1/8*sqrt(2)*sqrt(pi)*((-(I + 1)*cos(a) + (I - 1)*sin(a))*erf((I*b*d*x + I *b*c)/sqrt(I*b)) + (-(I - 1)*cos(a) + (I + 1)*sin(a))*erf((I*b*d*x + I*b*c )/sqrt(-I*b)))/(sqrt(b)*d)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.82 \[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (i \, a\right )}}{4 \, \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (-i \, a\right )}}{4 \, \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} \] Input:
integrate(sin(a+b*(d*x+c)^2),x, algorithm="giac")
Output:
1/4*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4 ) + 1)*(x + c/d))*e^(I*a)/(sqrt(b*d^2)*(-I*b*d^2/sqrt(b^2*d^4) + 1)) - 1/4 *I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(-I*a)/(sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d^4) + 1))
Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.14 \[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\sqrt {2}\,\sqrt {\pi }\,\cos \left (a\right )\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{b\,d^2}}\,\left (b\,x\,d^2+b\,c\,d\right )}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{b\,d^2}}}{2}+\frac {\sqrt {2}\,\sqrt {\pi }\,\sin \left (a\right )\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{b\,d^2}}\,\left (b\,x\,d^2+b\,c\,d\right )}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{b\,d^2}}}{2} \] Input:
int(sin(a + b*(c + d*x)^2),x)
Output:
(2^(1/2)*pi^(1/2)*cos(a)*fresnels((2^(1/2)*(1/(b*d^2))^(1/2)*(b*c*d + b*d^ 2*x))/pi^(1/2))*(1/(b*d^2))^(1/2))/2 + (2^(1/2)*pi^(1/2)*sin(a)*fresnelc(( 2^(1/2)*(1/(b*d^2))^(1/2)*(b*c*d + b*d^2*x))/pi^(1/2))*(1/(b*d^2))^(1/2))/ 2
\[ \int \sin \left (a+b (c+d x)^2\right ) \, dx=\int \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )d x \] Input:
int(sin(a+b*(d*x+c)^2),x)
Output:
int(sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2),x)