Integrand size = 19, antiderivative size = 130 \[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x^{1+m} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m-b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \] Output:
2*x^(1+m)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(1/2)*hypergeom([1/2, -1/4*(2*I+2 *I*m-b*n)/b/n],[-1/4*(2*I+2*I*m-5*b*n)/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/( 2+2*m+I*b*n)/cos(a+b*ln(c*x^n))^(1/2)
Time = 0.66 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i+2 i m-3 b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{(2+2 m+i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \] Input:
Integrate[x^m/Sqrt[Cos[a + b*Log[c*x^n]]],x]
Output:
(2*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x^(1 + m)*Hypergeometric2F1[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)*Sqrt[Cos[a + b*Log[c*x^n]]])
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx\) |
\(\Big \downarrow \) 4997 |
\(\displaystyle \frac {x^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \frac {\left (c x^n\right )^{\frac {m+1}{n}-1}}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4995 |
\(\displaystyle \frac {x^{m+1} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \left (c x^n\right )^{-\frac {m+1}{n}-\frac {i b}{2}} \int \frac {\left (c x^n\right )^{\frac {i b}{2}+\frac {m+1}{n}-1}}{\sqrt {e^{2 i a} \left (c x^n\right )^{2 i b}+1}}d\left (c x^n\right )}{n \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 x^{m+1} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i m-b n+2 i}{4 b n},-\frac {2 i m-5 b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(i b n+2 m+2) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}\) |
Input:
Int[x^m/Sqrt[Cos[a + b*Log[c*x^n]]],x]
Output:
(2*x^(1 + m)*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2 , -1/4*(2*I + (2*I)*m - b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -( E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + 2*m + I*b*n)*Sqrt[Cos[a + b*Log[c*x ^n]]])
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])
\[\int \frac {x^{m}}{\sqrt {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}}d x\]
Input:
int(x^m/cos(a+b*ln(c*x^n))^(1/2),x)
Output:
int(x^m/cos(a+b*ln(c*x^n))^(1/2),x)
Exception generated. \[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m/cos(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {x^{m}}{\sqrt {\cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \] Input:
integrate(x**m/cos(a+b*ln(c*x**n))**(1/2),x)
Output:
Integral(x**m/sqrt(cos(a + b*log(c*x**n))), x)
\[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \] Input:
integrate(x^m/cos(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")
Output:
integrate(x^m/sqrt(cos(b*log(c*x^n) + a)), x)
\[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \] Input:
integrate(x^m/cos(a+b*log(c*x^n))^(1/2),x, algorithm="giac")
Output:
integrate(x^m/sqrt(cos(b*log(c*x^n) + a)), x)
Timed out. \[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {x^m}{\sqrt {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \] Input:
int(x^m/cos(a + b*log(c*x^n))^(1/2),x)
Output:
int(x^m/cos(a + b*log(c*x^n))^(1/2), x)
\[ \int \frac {x^m}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {x^{m} \sqrt {\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}d x \] Input:
int(x^m/cos(a+b*log(c*x^n))^(1/2),x)
Output:
int((x**m*sqrt(cos(log(x**n*c)*b + a)))/cos(log(x**n*c)*b + a),x)