Integrand size = 16, antiderivative size = 168 \[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}-\frac {4 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {4 b^{3/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{3 d^{5/2}}+\frac {4 b \sin (a+b x)}{3 d^2 \sqrt {c+d x}} \]
-2/3*cos(b*x+a)/d/(d*x+c)^(3/2)-4/3*b^(3/2)*cos(a-b*c/d)*FresnelC(b^(1/2)* 2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(5/2)+4/3*b^(3/ 2)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*2 ^(1/2)*Pi^(1/2)/d^(5/2)+4/3*b*sin(b*x+a)/d^2/(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.13 \[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=\frac {e^{-i a} \left (-2 i e^{2 i a-\frac {i b c}{d}} \left (e^{\frac {i b (c+d x)}{d}} (-i d+2 b (c+d x))-2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )+e^{-i b x} \left (-2 d+4 i b (c+d x)-4 d e^{\frac {i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )\right )\right )}{6 d^2 (c+d x)^{3/2}} \]
((-2*I)*E^((2*I)*a - (I*b*c)/d)*(E^((I*b*(c + d*x))/d)*((-I)*d + 2*b*(c + d*x)) - (2*I)*d*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-I)*b*(c + d*x)) /d]) + (-2*d + (4*I)*b*(c + d*x) - 4*d*E^((I*b*(c + d*x))/d)*((I*b*(c + d* x))/d)^(3/2)*Gamma[1/2, (I*b*(c + d*x))/d])/E^(I*b*x))/(6*d^2*E^(I*a)*(c + d*x)^(3/2))
Time = 0.79 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3778, 25, 3042, 3778, 3042, 3787, 3042, 3785, 3786, 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.
| \(\displaystyle \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{(c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {2 b \int -\frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \int \frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \int \frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle -\frac {2 b \left (\frac {2 b \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{3 d}-\frac {2 \cos (a+b x)}{3 d (c+d x)^{3/2}}\) |
(-2*Cos[a + b*x])/(3*d*(c + d*x)^(3/2)) - (2*b*((2*b*((Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sq rt[d]) - (Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]* Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])))/d - (2*Sin[a + b*x])/(d*Sqrt[c + d*x ])))/(3*d)
3.1.46.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[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]
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[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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]
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]
Time = 0.83 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d}\) | \(180\) |
| default | \(\frac {-\frac {2 \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}}{d}\) | \(180\) |
2/d*(-1/3/(d*x+c)^(3/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)-2/3*b/d*(-1/(d*x+c)^( 1/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)+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)*b*(d*x+c)^(1/2)/d)-sin((a *d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.24 \[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 2 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \sqrt {d x + c} {\left (d \cos \left (b x + a\right ) - 2 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )\right )}\right )}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
-2/3*(2*sqrt(2)*(pi*b*d^2*x^2 + 2*pi*b*c*d*x + pi*b*c^2)*sqrt(b/(pi*d))*co s(-(b*c - a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 2*sq rt(2)*(pi*b*d^2*x^2 + 2*pi*b*c*d*x + pi*b*c^2)*sqrt(b/(pi*d))*fresnel_sin( sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + sqrt(d*x + c)* (d*cos(b*x + a) - 2*(b*d*x + b*c)*sin(b*x + a)))/(d^4*x^2 + 2*c*d^3*x + c^ 2*d^2)
\[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.77 \[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {{\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}}}{4 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \]
-1/4*(((I - 1)*sqrt(2)*gamma(-3/2, I*(d*x + c)*b/d) - (I + 1)*sqrt(2)*gamm a(-3/2, -I*(d*x + c)*b/d))*cos(-(b*c - a*d)/d) + ((I + 1)*sqrt(2)*gamma(-3 /2, I*(d*x + c)*b/d) - (I - 1)*sqrt(2)*gamma(-3/2, -I*(d*x + c)*b/d))*sin( -(b*c - a*d)/d))*((d*x + c)*b/d)^(3/2)/((d*x + c)^(3/2)*d)
\[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=\int { \frac {\cos \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos (a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \]