Integrand size = 14, antiderivative size = 79 \[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\frac {\cos (2 a) \operatorname {CosIntegral}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \operatorname {CosIntegral}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8}-\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n} \]
1/2*Ci(2*b*x^n)*cos(2*a)/n+1/8*Ci(4*b*x^n)*cos(4*a)/n+3/8*ln(x)-1/2*Si(2*b *x^n)*sin(2*a)/n-1/8*Si(4*b*x^n)*sin(4*a)/n
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\frac {3 \log (x)}{8}+\frac {4 \cos (2 a) \operatorname {CosIntegral}\left (2 b x^n\right )+\cos (4 a) \operatorname {CosIntegral}\left (4 b x^n\right )-4 \sin (2 a) \text {Si}\left (2 b x^n\right )-\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n} \]
(3*Log[x])/8 + (4*Cos[2*a]*CosIntegral[2*b*x^n] + Cos[4*a]*CosIntegral[4*b *x^n] - 4*Sin[2*a]*SinIntegral[2*b*x^n] - Sin[4*a]*SinIntegral[4*b*x^n])/( 8*n)
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3907, 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.
\(\displaystyle \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 3907 |
\(\displaystyle \int \left (\frac {\cos \left (2 a+2 b x^n\right )}{2 x}+\frac {\cos \left (4 a+4 b x^n\right )}{8 x}+\frac {3}{8 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\cos (2 a) \operatorname {CosIntegral}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \operatorname {CosIntegral}\left (4 b x^n\right )}{8 n}-\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8}\) |
(Cos[2*a]*CosIntegral[2*b*x^n])/(2*n) + (Cos[4*a]*CosIntegral[4*b*x^n])/(8 *n) + (3*Log[x])/8 - (Sin[2*a]*SinIntegral[2*b*x^n])/(2*n) - (Sin[4*a]*Sin Integral[4*b*x^n])/(8*n)
3.1.72.3.1 Defintions of rubi rules used
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x _Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 2.83 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {Si}\left (4 b \,x^{n}\right ) \sin \left (4 a \right )}{8}+\frac {\operatorname {Ci}\left (4 b \,x^{n}\right ) \cos \left (4 a \right )}{8}-\frac {\operatorname {Si}\left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}+\frac {\operatorname {Ci}\left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}+\frac {3 \ln \left (b \,x^{n}\right )}{8}}{n}\) | \(66\) |
default | \(\frac {-\frac {\operatorname {Si}\left (4 b \,x^{n}\right ) \sin \left (4 a \right )}{8}+\frac {\operatorname {Ci}\left (4 b \,x^{n}\right ) \cos \left (4 a \right )}{8}-\frac {\operatorname {Si}\left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}+\frac {\operatorname {Ci}\left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}+\frac {3 \ln \left (b \,x^{n}\right )}{8}}{n}\) | \(66\) |
risch | \(\frac {3 \ln \left (x \right )}{8}+\frac {i {\mathrm e}^{-4 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right )}{16 n}-\frac {i {\mathrm e}^{-4 i a} \operatorname {Si}\left (4 b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{-4 i a} \operatorname {Ei}_{1}\left (-4 i b \,x^{n}\right )}{16 n}+\frac {i {\mathrm e}^{-2 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right )}{4 n}-\frac {i {\mathrm e}^{-2 i a} \operatorname {Si}\left (2 b \,x^{n}\right )}{2 n}-\frac {{\mathrm e}^{-2 i a} \operatorname {Ei}_{1}\left (-2 i b \,x^{n}\right )}{4 n}-\frac {{\mathrm e}^{2 i a} \operatorname {Ei}_{1}\left (-2 i b \,x^{n}\right )}{4 n}-\frac {{\mathrm e}^{4 i a} \operatorname {Ei}_{1}\left (-4 i b \,x^{n}\right )}{16 n}\) | \(154\) |
1/n*(-1/8*Si(4*b*x^n)*sin(4*a)+1/8*Ci(4*b*x^n)*cos(4*a)-1/2*Si(2*b*x^n)*si n(2*a)+1/2*Ci(2*b*x^n)*cos(2*a)+3/8*ln(b*x^n))
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\frac {\cos \left (4 \, a\right ) \operatorname {Ci}\left (4 \, b x^{n}\right ) + 4 \, \cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) + 3 \, n \log \left (x\right ) - \sin \left (4 \, a\right ) \operatorname {Si}\left (4 \, b x^{n}\right ) - 4 \, \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )}{8 \, n} \]
1/8*(cos(4*a)*cos_integral(4*b*x^n) + 4*cos(2*a)*cos_integral(2*b*x^n) + 3 *n*log(x) - sin(4*a)*sin_integral(4*b*x^n) - 4*sin(2*a)*sin_integral(2*b*x ^n))/n
\[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\int \frac {\cos ^{4}{\left (a + b x^{n} \right )}}{x}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.51 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.39 \[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\frac {{\left ({\rm Ei}\left (4 i \, b x^{n}\right ) + {\rm Ei}\left (-4 i \, b x^{n}\right ) + {\rm Ei}\left (4 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (4 \, a\right ) + 4 \, {\left ({\rm Ei}\left (2 i \, b x^{n}\right ) + {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (2 \, a\right ) + 12 \, n \log \left (x\right ) + {\left (i \, {\rm Ei}\left (4 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-4 i \, b x^{n}\right ) + i \, {\rm Ei}\left (4 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (4 \, a\right ) - 4 \, {\left (-i \, {\rm Ei}\left (2 i \, b x^{n}\right ) + i \, {\rm Ei}\left (-2 i \, b x^{n}\right ) - i \, {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + i \, {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (2 \, a\right )}{32 \, n} \]
1/32*((Ei(4*I*b*x^n) + Ei(-4*I*b*x^n) + Ei(4*I*b*e^(n*conjugate(log(x)))) + Ei(-4*I*b*e^(n*conjugate(log(x)))))*cos(4*a) + 4*(Ei(2*I*b*x^n) + Ei(-2* I*b*x^n) + Ei(2*I*b*e^(n*conjugate(log(x)))) + Ei(-2*I*b*e^(n*conjugate(lo g(x)))))*cos(2*a) + 12*n*log(x) + (I*Ei(4*I*b*x^n) - I*Ei(-4*I*b*x^n) + I* Ei(4*I*b*e^(n*conjugate(log(x)))) - I*Ei(-4*I*b*e^(n*conjugate(log(x)))))* sin(4*a) - 4*(-I*Ei(2*I*b*x^n) + I*Ei(-2*I*b*x^n) - I*Ei(2*I*b*e^(n*conjug ate(log(x)))) + I*Ei(-2*I*b*e^(n*conjugate(log(x)))))*sin(2*a))/n
\[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\int { \frac {\cos \left (b x^{n} + a\right )^{4}}{x} \,d x } \]
Timed out. \[ \int \frac {\cos ^4\left (a+b x^n\right )}{x} \, dx=\int \frac {{\cos \left (a+b\,x^n\right )}^4}{x} \,d x \]