Integrand size = 19, antiderivative size = 130 \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x^{1+m} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2 i+2 i m+3 b n}{4 b n},-\frac {2 i+2 i m-b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \] Output:
2*x^(1+m)*cos(a+b*ln(c*x^n))^(3/2)*hypergeom([-3/2, -1/4*(2*I+2*I*m+3*b*n) /b/n],[-1/4*(2*I+2*I*m-b*n)/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(2+2*m-3*I*b *n)/(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)
Time = 1.64 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.57 \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{1+m} \left (6 b^2 n^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i+2 i m-3 b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(2+2 m+i b n) \left (4 (1+m) \cos ^2\left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{(2+2 m+i b n) (2+2 m-3 i b n) (2+2 m+3 i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \] Input:
Integrate[x^m*Cos[a + b*Log[c*x^n]]^(3/2),x]
Output:
(x^(1 + m)*(6*b^2*n^2*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F 1[1, -1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n ), -E^((2*I)*(a + b*Log[c*x^n]))] + (2 + 2*m + I*b*n)*(4*(1 + m)*Cos[a + b *Log[c*x^n]]^2 + 3*b*n*Sin[2*(a + b*Log[c*x^n])])))/((2 + 2*m + I*b*n)*(2 + 2*m - (3*I)*b*n)*(2 + 2*m + (3*I)*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4997, 4995, 888}
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.
\(\displaystyle \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 4997 |
\(\displaystyle \frac {x^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4995 |
\(\displaystyle \frac {x^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}+\frac {3 i b}{2}} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \int \left (c x^n\right )^{-\frac {3 i b}{2}+\frac {m+1}{n}-1} \left (e^{2 i a} \left (c x^n\right )^{2 i b}+1\right )^{3/2}d\left (c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-3\right ),-\frac {2 i m-b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(-3 i b n+2 m+2) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\) |
Input:
Int[x^m*Cos[a + b*Log[c*x^n]]^(3/2),x]
Output:
(2*x^(1 + m)*Cos[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[-3/2, (-3 - ((2 *I)*(1 + m))/(b*n))/4, -1/4*(2*I + (2*I)*m - b*n)/(b*n), -(E^((2*I)*a)*(c* x^n)^((2*I)*b))])/((2 + 2*m - (3*I)*b*n)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b ))^(3/2))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] : > Simp[Cos[d*(a + b*Log[x])]^p*(x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p ) Int[(e*x)^m*((1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)), x], x] /; Fr eeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[x ^((m + 1)/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
\[\int x^{m} {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}d x\]
Input:
int(x^m*cos(a+b*ln(c*x^n))^(3/2),x)
Output:
int(x^m*cos(a+b*ln(c*x^n))^(3/2),x)
Exception generated. \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*cos(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x**m*cos(a+b*ln(c*x**n))**(3/2),x)
Output:
Timed out
\[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^m*cos(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")
Output:
integrate(x^m*cos(b*log(c*x^n) + a)^(3/2), x)
\[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^m*cos(a+b*log(c*x^n))^(3/2),x, algorithm="giac")
Output:
integrate(x^m*cos(b*log(c*x^n) + a)^(3/2), x)
Timed out. \[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^m\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2} \,d x \] Input:
int(x^m*cos(a + b*log(c*x^n))^(3/2),x)
Output:
int(x^m*cos(a + b*log(c*x^n))^(3/2), x)
\[ \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^{m} \sqrt {\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}\, \cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )d x \] Input:
int(x^m*cos(a+b*log(c*x^n))^(3/2),x)
Output:
int(x**m*sqrt(cos(log(x**n*c)*b + a))*cos(log(x**n*c)*b + a),x)