Integrand size = 12, antiderivative size = 87 \[ \int (d x)^{3/2} \sin (f x) \, dx=-\frac {(d x)^{3/2} \cos (f x)}{f}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{2 f^{5/2}}+\frac {3 d \sqrt {d x} \sin (f x)}{2 f^2} \] Output:
-(d*x)^(3/2)*cos(f*x)/f-3/4*d^(3/2)*2^(1/2)*Pi^(1/2)*FresnelS(f^(1/2)*2^(1 /2)/Pi^(1/2)*(d*x)^(1/2)/d^(1/2))/f^(5/2)+3/2*d*(d*x)^(1/2)*sin(f*x)/f^2
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69 \[ \int (d x)^{3/2} \sin (f x) \, dx=\frac {d^2 \left (\sqrt {-i f x} \Gamma \left (\frac {5}{2},-i f x\right )+\sqrt {i f x} \Gamma \left (\frac {5}{2},i f x\right )\right )}{2 f^3 \sqrt {d x}} \] Input:
Integrate[(d*x)^(3/2)*Sin[f*x],x]
Output:
(d^2*(Sqrt[(-I)*f*x]*Gamma[5/2, (-I)*f*x] + Sqrt[I*f*x]*Gamma[5/2, I*f*x]) )/(2*f^3*Sqrt[d*x])
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3777, 3042, 3777, 25, 3042, 3786, 3832}
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.
\(\displaystyle \int (d x)^{3/2} \sin (f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d x)^{3/2} \sin (f x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {3 d \int \sqrt {d x} \cos (f x)dx}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 d \int \sqrt {d x} \sin \left (f x+\frac {\pi }{2}\right )dx}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {3 d \left (\frac {d \int -\frac {\sin (f x)}{\sqrt {d x}}dx}{2 f}+\frac {\sqrt {d x} \sin (f x)}{f}\right )}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 d \left (\frac {\sqrt {d x} \sin (f x)}{f}-\frac {d \int \frac {\sin (f x)}{\sqrt {d x}}dx}{2 f}\right )}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 d \left (\frac {\sqrt {d x} \sin (f x)}{f}-\frac {d \int \frac {\sin (f x)}{\sqrt {d x}}dx}{2 f}\right )}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {3 d \left (\frac {\sqrt {d x} \sin (f x)}{f}-\frac {\int \sin (f x)d\sqrt {d x}}{f}\right )}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {3 d \left (\frac {\sqrt {d x} \sin (f x)}{f}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \operatorname {FresnelS}\left (\frac {\sqrt {f} \sqrt {\frac {2}{\pi }} \sqrt {d x}}{\sqrt {d}}\right )}{f^{3/2}}\right )}{2 f}-\frac {(d x)^{3/2} \cos (f x)}{f}\) |
Input:
Int[(d*x)^(3/2)*Sin[f*x],x]
Output:
-(((d*x)^(3/2)*Cos[f*x])/f) + (3*d*(-((Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[f ]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/f^(3/2)) + (Sqrt[d*x]*Sin[f*x])/f))/(2*f )
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
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]
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]
Time = 0.57 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(\frac {2 \left (d x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \left (-\frac {x^{\frac {3}{2}} \sqrt {2}\, f^{\frac {3}{2}} \cos \left (f x \right )}{4 \sqrt {\pi }}+\frac {3 \sqrt {x}\, \sqrt {2}\, \sqrt {f}\, \sin \left (f x \right )}{8 \sqrt {\pi }}-\frac {3 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {x}\, \sqrt {f}}{\sqrt {\pi }}\right )}{8}\right )}{x^{\frac {3}{2}} f^{\frac {5}{2}}}\) | \(73\) |
derivativedivides | \(\frac {-\frac {d \left (d x \right )^{\frac {3}{2}} \cos \left (f x \right )}{f}+\frac {3 d \left (\frac {d \sqrt {d x}\, \sin \left (f x \right )}{2 f}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{4 f \sqrt {\frac {f}{d}}}\right )}{f}}{d}\) | \(87\) |
default | \(\frac {-\frac {d \left (d x \right )^{\frac {3}{2}} \cos \left (f x \right )}{f}+\frac {3 d \left (\frac {d \sqrt {d x}\, \sin \left (f x \right )}{2 f}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, f \sqrt {d x}}{\sqrt {\pi }\, \sqrt {\frac {f}{d}}\, d}\right )}{4 f \sqrt {\frac {f}{d}}}\right )}{f}}{d}\) | \(87\) |
Input:
int((d*x)^(3/2)*sin(f*x),x,method=_RETURNVERBOSE)
Output:
2*(d*x)^(3/2)/x^(3/2)*2^(1/2)/f^(5/2)*Pi^(1/2)*(-1/4/Pi^(1/2)*x^(3/2)*2^(1 /2)*f^(3/2)*cos(f*x)+3/8/Pi^(1/2)*x^(1/2)*2^(1/2)*f^(1/2)*sin(f*x)-3/8*Fre snelS(1/Pi^(1/2)*2^(1/2)*x^(1/2)*f^(1/2)))
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83 \[ \int (d x)^{3/2} \sin (f x) \, dx=-\frac {3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {f}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x} \sqrt {\frac {f}{\pi d}}\right ) + 2 \, {\left (2 \, d f^{2} x \cos \left (f x\right ) - 3 \, d f \sin \left (f x\right )\right )} \sqrt {d x}}{4 \, f^{3}} \] Input:
integrate((d*x)^(3/2)*sin(f*x),x, algorithm="fricas")
Output:
-1/4*(3*sqrt(2)*pi*d^2*sqrt(f/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x)*sqrt(f /(pi*d))) + 2*(2*d*f^2*x*cos(f*x) - 3*d*f*sin(f*x))*sqrt(d*x))/f^3
Time = 10.61 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int (d x)^{3/2} \sin (f x) \, dx=- \frac {7 d^{\frac {3}{2}} x^{\frac {3}{2}} \cos {\left (f x \right )} \Gamma \left (\frac {7}{4}\right )}{4 f \Gamma \left (\frac {11}{4}\right )} + \frac {21 d^{\frac {3}{2}} \sqrt {x} \sin {\left (f x \right )} \Gamma \left (\frac {7}{4}\right )}{8 f^{2} \Gamma \left (\frac {11}{4}\right )} - \frac {21 \sqrt {2} \sqrt {\pi } d^{\frac {3}{2}} S\left (\frac {\sqrt {2} \sqrt {f} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {7}{4}\right )}{16 f^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((d*x)**(3/2)*sin(f*x),x)
Output:
-7*d**(3/2)*x**(3/2)*cos(f*x)*gamma(7/4)/(4*f*gamma(11/4)) + 21*d**(3/2)*s qrt(x)*sin(f*x)*gamma(7/4)/(8*f**2*gamma(11/4)) - 21*sqrt(2)*sqrt(pi)*d**( 3/2)*fresnels(sqrt(2)*sqrt(f)*sqrt(x)/sqrt(pi))*gamma(7/4)/(16*f**(5/2)*ga mma(11/4))
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22 \[ \int (d x)^{3/2} \sin (f x) \, dx=-\frac {\sqrt {2} {\left (8 \, \sqrt {2} \left (d x\right )^{\frac {3}{2}} f^{2} \cos \left (f x\right ) - 12 \, \sqrt {2} \sqrt {d x} d f \sin \left (f x\right ) + \left (3 i + 3\right ) \, \sqrt {\pi } d^{2} \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {\frac {i \, f}{d}}\right ) - \left (3 i - 3\right ) \, \sqrt {\pi } d^{2} \left (\frac {f^{2}}{d^{2}}\right )^{\frac {1}{4}} \operatorname {erf}\left (\sqrt {d x} \sqrt {-\frac {i \, f}{d}}\right )\right )}}{16 \, f^{3}} \] Input:
integrate((d*x)^(3/2)*sin(f*x),x, algorithm="maxima")
Output:
-1/16*sqrt(2)*(8*sqrt(2)*(d*x)^(3/2)*f^2*cos(f*x) - 12*sqrt(2)*sqrt(d*x)*d *f*sin(f*x) + (3*I + 3)*sqrt(pi)*d^2*(f^2/d^2)^(1/4)*erf(sqrt(d*x)*sqrt(I* f/d)) - (3*I - 3)*sqrt(pi)*d^2*(f^2/d^2)^(1/4)*erf(sqrt(d*x)*sqrt(-I*f/d)) )/f^3
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.44 \[ \int (d x)^{3/2} \sin (f x) \, dx=-\frac {\frac {3 \, \sqrt {2} \sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {d f} \sqrt {d x} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )} f^{2}} + \frac {3 \, \sqrt {2} \sqrt {\pi } d^{3} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {d f} \sqrt {d x} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )}}{2 \, d}\right )}{\sqrt {d f} {\left (-\frac {i \, d f}{\sqrt {d^{2} f^{2}}} + 1\right )} f^{2}} - \frac {2 i \, {\left (2 i \, \sqrt {d x} d^{2} f x - 3 \, \sqrt {d x} d^{2}\right )} e^{\left (i \, f x\right )}}{f^{2}} - \frac {2 i \, {\left (2 i \, \sqrt {d x} d^{2} f x + 3 \, \sqrt {d x} d^{2}\right )} e^{\left (-i \, f x\right )}}{f^{2}}}{8 \, d} \] Input:
integrate((d*x)^(3/2)*sin(f*x),x, algorithm="giac")
Output:
-1/8*(3*sqrt(2)*sqrt(pi)*d^3*erf(-1/2*I*sqrt(2)*sqrt(d*f)*sqrt(d*x)*(I*d*f /sqrt(d^2*f^2) + 1)/d)/(sqrt(d*f)*(I*d*f/sqrt(d^2*f^2) + 1)*f^2) + 3*sqrt( 2)*sqrt(pi)*d^3*erf(1/2*I*sqrt(2)*sqrt(d*f)*sqrt(d*x)*(-I*d*f/sqrt(d^2*f^2 ) + 1)/d)/(sqrt(d*f)*(-I*d*f/sqrt(d^2*f^2) + 1)*f^2) - 2*I*(2*I*sqrt(d*x)* d^2*f*x - 3*sqrt(d*x)*d^2)*e^(I*f*x)/f^2 - 2*I*(2*I*sqrt(d*x)*d^2*f*x + 3* sqrt(d*x)*d^2)*e^(-I*f*x)/f^2)/d
Timed out. \[ \int (d x)^{3/2} \sin (f x) \, dx=\int \sin \left (f\,x\right )\,{\left (d\,x\right )}^{3/2} \,d x \] Input:
int(sin(f*x)*(d*x)^(3/2),x)
Output:
int(sin(f*x)*(d*x)^(3/2), x)
\[ \int (d x)^{3/2} \sin (f x) \, dx=\frac {\sqrt {d}\, d \left (-4 \sqrt {x}\, \cos \left (f x \right ) f x +6 \sqrt {x}\, \sin \left (f x \right )-3 \left (\int \frac {\sin \left (f x \right )}{\sqrt {x}}d x \right )\right )}{4 f^{2}} \] Input:
int((d*x)^(3/2)*sin(f*x),x)
Output:
(sqrt(d)*d*( - 4*sqrt(x)*cos(f*x)*f*x + 6*sqrt(x)*sin(f*x) - 3*int(sin(f*x )/sqrt(x),x)))/(4*f**2)