Integrand size = 16, antiderivative size = 114 \[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } \cos (c) \operatorname {FresnelS}\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)-\frac {b \sin \left (c+d x^2\right )}{3 x^3} \] Output:
-1/3*a/x^3-2/3*b*d*cos(d*x^2+c)/x-2/3*b*d^(3/2)*2^(1/2)*Pi^(1/2)*cos(c)*Fr esnelS(d^(1/2)*2^(1/2)/Pi^(1/2)*x)-2/3*b*d^(3/2)*2^(1/2)*Pi^(1/2)*FresnelC (d^(1/2)*2^(1/2)/Pi^(1/2)*x)*sin(c)-1/3*b*sin(d*x^2+c)/x^3
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b \cos \left (d x^2\right ) \left (2 d x^2 \cos (c)+\sin (c)\right )}{3 x^3}-\frac {2}{3} b d^{3/2} \sqrt {2 \pi } \left (\cos (c) \operatorname {FresnelS}\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )+\operatorname {FresnelC}\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right ) \sin (c)\right )+\frac {b \left (-\cos (c)+2 d x^2 \sin (c)\right ) \sin \left (d x^2\right )}{3 x^3} \] Input:
Integrate[(a + b*Sin[c + d*x^2])/x^4,x]
Output:
-1/3*a/x^3 - (b*Cos[d*x^2]*(2*d*x^2*Cos[c] + Sin[c]))/(3*x^3) - (2*b*d^(3/ 2)*Sqrt[2*Pi]*(Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x] + FresnelC[Sqrt[d]*Sq rt[2/Pi]*x]*Sin[c]))/3 + (b*(-Cos[c] + 2*d*x^2*Sin[c])*Sin[d*x^2])/(3*x^3)
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2010, 2009}
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.
\(\displaystyle \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {a}{x^4}+\frac {b \sin \left (c+d x^2\right )}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a}{3 x^3}-\frac {2}{3} \sqrt {2 \pi } b d^{3/2} \sin (c) \operatorname {FresnelC}\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2}{3} \sqrt {2 \pi } b d^{3/2} \cos (c) \operatorname {FresnelS}\left (\sqrt {d} \sqrt {\frac {2}{\pi }} x\right )-\frac {2 b d \cos \left (c+d x^2\right )}{3 x}-\frac {b \sin \left (c+d x^2\right )}{3 x^3}\) |
Input:
Int[(a + b*Sin[c + d*x^2])/x^4,x]
Output:
-1/3*a/x^3 - (2*b*d*Cos[c + d*x^2])/(3*x) - (2*b*d^(3/2)*Sqrt[2*Pi]*Cos[c] *FresnelS[Sqrt[d]*Sqrt[2/Pi]*x])/3 - (2*b*d^(3/2)*Sqrt[2*Pi]*FresnelC[Sqrt [d]*Sqrt[2/Pi]*x]*Sin[c])/3 - (b*Sin[c + d*x^2])/(3*x^3)
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {a}{3 x^{3}}+b \left (-\frac {\sin \left (d \,x^{2}+c \right )}{3 x^{3}}+\frac {2 d \left (-\frac {\cos \left (d \,x^{2}+c \right )}{x}-\sqrt {d}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (c \right ) \operatorname {FresnelS}\left (\frac {\sqrt {d}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )+\sin \left (c \right ) \operatorname {FresnelC}\left (\frac {\sqrt {d}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )\right )\right )}{3}\right )\) | \(83\) |
parts | \(-\frac {a}{3 x^{3}}+b \left (-\frac {\sin \left (d \,x^{2}+c \right )}{3 x^{3}}+\frac {2 d \left (-\frac {\cos \left (d \,x^{2}+c \right )}{x}-\sqrt {d}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (c \right ) \operatorname {FresnelS}\left (\frac {\sqrt {d}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )+\sin \left (c \right ) \operatorname {FresnelC}\left (\frac {\sqrt {d}\, \sqrt {2}\, x}{\sqrt {\pi }}\right )\right )\right )}{3}\right )\) | \(83\) |
risch | \(-\frac {a}{3 x^{3}}-\frac {i b \,d^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {i d}\, x \right ) {\mathrm e}^{-i c}}{3 \sqrt {i d}}+\frac {i b \,d^{2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i d}\, x \right ) {\mathrm e}^{i c}}{3 \sqrt {-i d}}-\frac {2 b d \cos \left (d \,x^{2}+c \right )}{3 x}-\frac {b \sin \left (d \,x^{2}+c \right )}{3 x^{3}}\) | \(97\) |
Input:
int((a+b*sin(d*x^2+c))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*a/x^3+b*(-1/3*sin(d*x^2+c)/x^3+2/3*d*(-1/x*cos(d*x^2+c)-d^(1/2)*2^(1/ 2)*Pi^(1/2)*(cos(c)*FresnelS(d^(1/2)*2^(1/2)/Pi^(1/2)*x)+sin(c)*FresnelC(d ^(1/2)*2^(1/2)/Pi^(1/2)*x))))
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=-\frac {2 \, \sqrt {2} \pi b d x^{3} \sqrt {\frac {d}{\pi }} \cos \left (c\right ) \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) + 2 \, \sqrt {2} \pi b d x^{3} \sqrt {\frac {d}{\pi }} \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {d}{\pi }}\right ) \sin \left (c\right ) + 2 \, b d x^{2} \cos \left (d x^{2} + c\right ) + b \sin \left (d x^{2} + c\right ) + a}{3 \, x^{3}} \] Input:
integrate((a+b*sin(d*x^2+c))/x^4,x, algorithm="fricas")
Output:
-1/3*(2*sqrt(2)*pi*b*d*x^3*sqrt(d/pi)*cos(c)*fresnel_sin(sqrt(2)*x*sqrt(d/ pi)) + 2*sqrt(2)*pi*b*d*x^3*sqrt(d/pi)*fresnel_cos(sqrt(2)*x*sqrt(d/pi))*s in(c) + 2*b*d*x^2*cos(d*x^2 + c) + b*sin(d*x^2 + c) + a)/x^3
\[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=\int \frac {a + b \sin {\left (c + d x^{2} \right )}}{x^{4}}\, dx \] Input:
integrate((a+b*sin(d*x**2+c))/x**4,x)
Output:
Integral((a + b*sin(c + d*x**2))/x**4, x)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=-\frac {\sqrt {d x^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, d x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, d x^{2}\right )\right )} \cos \left (c\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, i \, d x^{2}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} b d}{8 \, x} - \frac {a}{3 \, x^{3}} \] Input:
integrate((a+b*sin(d*x^2+c))/x^4,x, algorithm="maxima")
Output:
-1/8*sqrt(d*x^2)*((-(I + 1)*sqrt(2)*gamma(-3/2, I*d*x^2) + (I - 1)*sqrt(2) *gamma(-3/2, -I*d*x^2))*cos(c) + ((I - 1)*sqrt(2)*gamma(-3/2, I*d*x^2) - ( I + 1)*sqrt(2)*gamma(-3/2, -I*d*x^2))*sin(c))*b*d/x - 1/3*a/x^3
\[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=\int { \frac {b \sin \left (d x^{2} + c\right ) + a}{x^{4}} \,d x } \] Input:
integrate((a+b*sin(d*x^2+c))/x^4,x, algorithm="giac")
Output:
integrate((b*sin(d*x^2 + c) + a)/x^4, x)
Timed out. \[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=\int \frac {a+b\,\sin \left (d\,x^2+c\right )}{x^4} \,d x \] Input:
int((a + b*sin(c + d*x^2))/x^4,x)
Output:
int((a + b*sin(c + d*x^2))/x^4, x)
\[ \int \frac {a+b \sin \left (c+d x^2\right )}{x^4} \, dx=\frac {3 \left (\int \frac {\sin \left (d \,x^{2}+c \right )}{x^{4}}d x \right ) b \,x^{3}-a}{3 x^{3}} \] Input:
int((a+b*sin(d*x^2+c))/x^4,x)
Output:
(3*int(sin(c + d*x**2)/x**4,x)*b*x**3 - a)/(3*x**3)